Unit 1 Polynomial & Rational Functions

AP Pre-Calculus Study Guide Unit 1 : Polynomial & Rational Functions 1.1 Change In Tandem 1.1.A Describe how the input and output values of a function vary together by comparing function values. 1.1.A.1 Definition of a Function: - A function is a mathematical relation that maps a set of input values to a set of output values, such that each input value corresponds to exactly one output value. - The set of input values is called the domain of the function. - The set of output values is called the range of the function. - The independent variable represents the input values, and the dependent variable represents the output values. 1.1.A.2 Relationship between Input and Output Values: - The input and output values of a function vary according to the function rule. - The relationship between input and output values can be expressed graphically, tabularly, analytically, or verbally. 1.1.A.3 Increasing Functions: - A function is increasing over an interval if, as the input values increase, the output values always increase. - Mathematically, for any two input values 'a' and 'b' in the interval, if 'a' < 'b', then f(a) < f(b). 1.1.A.4 Decreasing Functions: - A function is decreasing over an interval if, as the input values increase, the output values always decrease. - Mathematically, for any two input values 'a' and 'b' in the interval, if 'a' < 'b', then f(a) > f(b). 1.1.B Construct a graph representing two quantities that vary with respect to each other in a contextual scenario. 1.1.B.1 Graphing a Function: - The graph of a function represents a set of input-output pairs and illustrates how the values of the function's input and output vary. - The horizontal axis typically represents the input values, and the vertical axis represents the output values. - Each point on the graph corresponds to an input-output pair. 1.1.B.2 Verbal Description and Graph Construction: - A verbal description of how different aspects of phenomena change together can be used as a basis for constructing a graph. - Understanding the relationship between the input and output values in a given context helps in graph construction. 1.1.B.3 Concavity and Increasing Rate of Change: - The graph of a function is concave upon intervals where the rate of change is increasing. This means the function is getting steeper as the input values increase within those intervals. 1.1.B.4 Concavity and Decreasing Rate of Change: - The graph of a function is concave down upon intervals where the rate of change is decreasing. - This means the function is getting less steep as the input values increase within those intervals. 1.1.B.5 Zeros of a Function: - The graph of a function intersects the x-axis when the output value is zero. - The corresponding input values at these intersections are referred to as zeros of the function. 1.2 Rates Of Change Introduction: In the study of functions, rates of change play a crucial role in analyzing how quantities vary over time or across different intervals. This comprehensive set of notes aims to provide a detailed understanding of average rates of change and how they enable us to compare the rates of change at different points. Additionally, we will explore the relationship between two quantities and how they vary together, including positive and negative rates of change. Let's delve into the intricacies of these concepts. 1.2.A Definition: - The average rate of change of a function over an interval is the constant rate at which the output values change compared to the change in input values over that interval. - It is the ratio of the change in the output values to the change in the input values within the specified interval. - Average rate of change = (Change in output values) / (Change in input values) Calculating Average Rates of Change: - Choose two points on the function, typically denoted as (x₁, y₁) and (x₂, y₂). - Determine the change in input values: Δx = x₂ - x₁. - Determine the change in output values: Δy = y₂ - y₁. - Calculate the average rate of change: (Δy / Δx). Approximating the Rate of Change at a Point: - To estimate the rate of change at a specific point, we can use the average rates of change over small intervals containing that point. - By selecting intervals closer to the point, we obtain a better approximation of the rate of change at that point. - The smaller the interval, the more accurate the approximation becomes. Comparing Rates of Change at Two Points: - By employing average rate of change approximations over sufficiently small intervals containing each point, we can compare the rates of change at two points. - If the values of the average rates of change exist for both points, we can determine which point has a greater or lesser rate of change. 1.2.B Relationship between Two Quantities: 1.2.B.1 Quantifying Rates of Change: - Rates of change quantify how two quantities vary together. - They provide insight into the relationship between two variables and help us understand the nature of their change. 1.2.B.2 Positive Rates of Change: - A positive rate of change indicates that as one quantity increases (or decreases), the other quantity also increases (or decreases) correspondingly. - The two quantities exhibit a direct relationship or positive correlation. 1.2.B.3 Negative Rates of Change: - A negative rate of change indicates that as one quantity increases, the other quantity decreases, and vice versa. - The two quantities exhibit an inverse relationship or negative correlation. Conclusion: Understanding rates of change is vital for analyzing how quantities evolve over time or intervals. Average rates of change help us measure the change in output values relative to the change in input values. By comparing rates of change at different points, we can identify variations in the rate of change. Furthermore, the relationship between two quantities can be analyzed using positive and negative rates of change, enabling us to comprehend how they interact. With a solid grasp of these concepts, you now possess a strong foundation in the study of rates of change and their applications. 1.3 Rates Of Change In Linear & Quadratic Functions Learning Objectives: Average Rates of Change for Linear and Quadratic Sequences and Functions 1.3.A Determining Average Rates of Change for Linear and Quadratic Sequences and Functions. 1.3.A.1 Average Rate of Change for a Linear Function: The average rate of change for a linear function is a constant value over any input-value length interval; it measures the rate at which the function's output (y-values) changes with respect to the input (x-values) over a given interval, representing the slope of the line that connects two points on a linear graph. To obtain the average rate of change for a linear function, divide the change in the output, denoted by Δy, by the change in the input, denoted by Δx, over the interval: Average Rate of Change = Δy / Δx. Since linear functions have constant slopes, the average rate of change stays the same irrespective of the length of the interval. Example: Consider the linear function f(x) = 2x + 3. Given two points (x₁, f(x₁)) and (x₂, f(x₂)), the average rate of change will always be 2. 1.3.A.2 Average Rate of Change for a Quadratic Function: For consecutive equal-length input-value intervals, we can express the average rate of change of a quadratic function as a linear function. This means that the average rate of change for a quadratic function will change linearly as we move through the intervals. To determine the average rate of change for a quadratic function, you need to calculate the slope of the secant line connecting two points on the graph, which represents the average rate of change over the interval. Example: Consider the quadratic function f(x) = x². For consecutive equal-length intervals, such as [0, 1], [1, 2], [2, 3], the average rate of change will be 2, 4, 6, respectively, forming a linear sequence. **1.3.A.3 Average Rate of Change over a Closed Interval:** The average rate of change over the closed interval [a, b] is the slope of the secant line that connects the points (a, f(a)) and (b, f(b)). It measures the average rate at which the function changes over the entire interval. To calculate the average rate of change over a closed interval, you can use the formula: Average Rate of Change = (f(b) - f(a)) / (b - a) This formula denotes the difference in the function's output divided by the difference in the input values over the interval [a, b]. Example: Suppose we have the function f(x) = x². If we want to find the average rate of change over the interval [1, 3], we substitute the values f(1) = 1 and f(3) = 9 into the formula: Average Rate of Change = (9 - 1) / (3 - 1) = 8 / 2 = 4 Therefore, the average rate of change over the interval [1, 3] is 4. 1.3.B Understanding Change in Average Rates of Change for Linear and Quadratic Functions.** 1.3.B.1 Change of Average Rates of Change for a Linear Function: The rate of change of the average rates of change of a linear function is zero. This means that for a linear function, the average rate of change remains constant over any length input-value interval, resulting in a constant rate of change for the average rates of change. Example: Consider the linear function f(x) = 2x + 3. We know that the average rate of change for this linear function is always 2. If we calculate the average rate of change for different intervals within the function, it remains constant. For example: Interval [0, 1]: Average Rate of Change = (f(1) - f(0)) / (1 - 0) = (2(1) + 3 - (2(0) + 3)) / 1 = 2 Interval [1, 2]: Average Rate of Change = (f(2) - f(1)) / (2 - 1) = (2(2) + 3 - (2(1) + 3)) / 1 = 2 Interval [2, 3]: Average Rate of Change = (f(3) - f(2)) / (3 - 2) = (2(3) + 3 - (2(2) + 3)) / 1=2 We observe that the average rate of change remains constant at 2 for each interval, and the rate of change of the average rates of change of a linear function is zero. 1.3.B.2 Change of Average Rates of Change for a Quadratic Function: The rate of change of the average rates of change of a quadratic function is constant. This means that although the average rate of change changes linearly as we move through consecutive equal-length intervals, the rate at which the average rates of change change remains the same. Example: Consider the quadratic function f(x) = x². If we calculate the average rate of change for consecutive equal-length intervals, we can observe a linear pattern: Interval [0, 1]: Average Rate of Change = (f(1) - f(0)) / (1 - 0) = (1² - 0²) / 1 = 1 Interval [1, 2]: Average Rate of Change = (f(2) - f(1)) / (2 - 1) = (2² - 1²) / 1 = 3 Interval [2, 3]: Average Rate of Change = (f(3) - f(2)) / (3 - 2) = (3² - 2²) / 1 = 5 We notice that the differences (1, 3, 5) form a linear sequence. Although the average rates of change change linearly, the rate at which they change (the difference between consecutive values) remains constant at 2; thus, the rate of change of the average rates of change of a quadratic function is constant. 1.3.B.3 Evaluating Concavity of a Function Based on Average Rates of Change: The concavity of a function can be determined based on the behavior of the average rates of change over equal-length input-value intervals. If the average rate of change over equal-length intervals is increasing for all small-length intervals, then the graph of the function is concave up, curving upwards like a U-shape. If the average rate of change over equal-length intervals is decreasing for all small-length intervals, then the graph of the function is concave down, curving downwards and resembling an inverted U-shape. By analyzing the changes in the average rates of change, we determine the concavity of the function. Example: Consider the quadratic function f(x) = x². If we calculate the average rates of change for consecutive equal-length intervals, we can determine the function's concavity. 1.4 Polynomial Functions & Rates Of Change 1.4.A.1: Comprehension of Polynomial Functions A polynomial expression is a mathematical form which is composed of a series of terms, each encompassing a quantity raised to a non-negative whole number power. The common shape of a polynomial function is given here: p(x) = anxn + an-1xn-1 + an-2xn-2 + ... + a2x2 + a1x + a0 Here, 'n' represents an optimistic integer, 'ai' stands for real digits for every 'i' from 1 to 'n', and 'an' must be a non-zero value. The term 'xn' with the highest strength is called the leading term, and 'an' is called the leading coefficient. Even a constant term can be considered a polynomial equation of degree zero. 1.4.A.2: Local and Global Extremes In the context of polynomial equations, extremums refer to points where the equation attains its greatest or least values. When a polynomial function goes from increasing to decreasing or vice versa, or at the finales of a polynomial with a limited domain, it will have a local maximum or minimum value. The biggest among all local maximums is the global maximum, and the least of all local minimums is labeled the global minimum. 1.4.A.3: Association between Real Zeros and Extremes Between any two unique real zeros of a non-constant polynomial function, there will always be at least one input value matching to a local maximum or minimum. This implies that as the equation transforms from beneath to over the x-axis or vice versa, it must have a point where it arrives at an apex or a valley. 1.4.A.4: Global Extremes for Even Degree Polynomial Polynomial equations of an even degree will constantly have either a global maximum or a global minimum. The global extremum appears at the point where the equation achieves its highest or least output across its entire domain. 1.4.A.5: Points of Inflection Points of inflection for polynomial equations emerge at input values where the rate of modification of the function shifts from increasing to diminishing or vice versa. This alteration in rate of shift is connected to the transition of the graph from being concave up (opening upward) to concave down (opening downward) or vice versa. In brief, polynomial equations are mathematical forms built from terms including quantities raised to non negative whole number strengths. They present different features related to speed of changes, including local and global extremums, the connection between real zeros and extremums, global extremes for even degree polynomials, and points of inflection where the graph alters concavity. Knowing these fundamental features is critical for studying and deciphering polynomial equations. 1.5 Polynomial Functions and Complex Zeros 1.5.A Understanding the Characteristics of Polynomial Functions and Zeros 1.5.A.1 Polynomial Zeros: In the realm of polynomial functions, a complex number, 'a,' that allows the polynomial function to equal zero is known as a zero or root of the polynomial. If 'a' is a real number, then the factor (x-a) is a linear factor of the polynomial only if 'a' is a zero of the polynomial. 1.5.A.2 Zeros with Multiplicity: If a linear factor (x-a) appears repeatedly in the factorization of a polynomial, then the corresponding zero 'a' has a multiplicity of 'n'. The multiplicity of a zero is indicative of the number of times the factor (x-a) is recursively repeated. 1.5.A.3 Real Zeros and Their x-intercepts: When 'a' is a real zero of a polynomial function p, the graph of y=p(x) will have an x-intercept at point (a,0). Real zeros of polynomials are, therefore, the endpoints of regions satisfying polynomial inequalities. 1.5.A.4 Equality in Conjugate Zeros: If a+bi is a non-real zero of a polynomial p, then its conjugate a-bi is also a zero of p. All complex zeros are part of conjugate pairs. 1.5.A.5 Zeros with Even Multiplicity: For a polynomial function whose real zero 'a' has an even multiplicity, the sign of the output values (y values) will be the same for input values (x-values) near x=a. The graph of these polynomials will be tangential to the x-axis at x=a. 1.5.A.6 The Degree of a Polynomial Function: By examining the successive differences of the output values over input values of equal intervals, one can discover the degree of a polynomial function. The degree of the polynomial function is the least value 'n' for which the successive 'nth' differences of the polynomial function are constant. 1.5.B Identifying Even and Odd Polynomials 1.5.B.1 Even Polynomials: An even function is graphically symmetric over the line x=0 and analytically satisfies the property f(-x) = f(x). For instance, if the polynomial of degree 'n' has a form of p(x)=an*x^n, where n ≥ 1 and an ≠ 0, then it is an even function. 1.5.B.2 Odd Polynomials: An odd function is graphically symmetric about the point (0,0) and satisfies the property f(-x) = -f(x) analytically. For example, if the polynomial of degree 'n' has a form of p(x)=an*x^n, where n ≥ 1 and an ≠ 0, then it is an odd function. These comprehensive notes examine the various facets of polynomial functions and complex zeros comprehensively. It covers concepts such as zeros, multiplicity, x-intercepts, conjugate zeros, even and odd functions, and how to determine the degree of a polynomial function. With these detailed notes, students can gain a full understanding of the subject without the need for further references. 1.6 Polynomial Functions & End Behavior 1.6.A Explaining Limitless Behavior of Polynomial Functions 1.6.A.1 Limitless Behavior as x Appro a non-constant polynomial function, as the input values (x) increase without bound (approaching positive infinity), the output values (p(x)) will either increase infinitely or decrease infinitely. - In mathematical terms, this behavior is expressed as lim p(x) = ∞ (the limit of p(x) is infinity) or lim p(x) = -∞ (the limit of p(x) is negative infinity) as x approaches positive infinity (x → ∞). - To simplify things, as x gets larger and larger, the function either grows infinitely or decreases infinitely. 1.6.A.2 Limitless Behavior as x Approaches Negative Infinity: - Likewise, for a non-constant polynomial function, as the input values (x) decrease without bound (approaching negative infinity), the output values (p(x)) will either increase infinitely or decrease infinitely. - Mathematically, this behavior is shown as lim p(x) = ∞ or lim p(x) = -∞ as x approaches negative infinity (x → -∞). - Therefore, as x gets smaller and smaller (more negative), the function either grows infinitely or decreases infinitely. 1.6.A.3 The Leading Term and Degree: - The degree and sign of the leading term of a polynomial function dictate its end behavior. - The leading term is the term with the highest power of x, and its coefficient establishes the sign. - As the input values increase or decrease without bound, the values of the leading term become dominant over the values of all lower-degree terms, influencing the function's behavior. - If the leading term has a positive coefficient and an even degree, the end behavior will be: - As x approaches positive infinity, the function grows infinitely (lim p(x) = ∞). - As x approaches negative infinity, the function also grows infinitely (lim p(x) = ∞). - If the leading term has a negative coefficient and an even degree, the end behavior will be: - As x approaches positive infinity, the function decreases infinitely (lim p(x) = -∞). As x approaches negative infinity, the function also decreases infinitely (lim p(x) = -∞). - If the leading term has a positive coefficient and an odd degree, the end behavior will be: - As x approaches positive infinity, the function grows infinitely (lim p(x) = ∞). - As x approaches negative infinity, the function decreases infinitely (lim p(x) = -∞). - If the leading term has a negative coefficient and an odd degree, the end behavior will be: - As x approaches positive infinity, the function decreases infinitely (lim p(x) = -∞). - As x approaches negative infinity, the function grows infinitely (lim p(x) = ∞). 1.7 Rational Functions & End Behavior 1.7.A Comprehensive Analysis of End Behavior of Rational Functions 1.7.A.1 Analytical Representation of Rational Functions: A rational function is represented as the quotient of two polynomial functions. The relative size of the numerator polynomial compared to the denominator polynomial is measured for each value in the domain of the rational function. 1.7.A.2 Effect of Dominant Polynomial on End Behavior: The end behavior of a rational function is most affected by the polynomial with the greater degree, as its values dominate the values of the rational function for input values of large magnitude. A polynomial is dominated by its leading term for input values of large magnitude. Therefore, understanding the leading terms of the numerator and denominator helps determine the end behavior of the rational function. 1.7.A.3 End Behavior of Functions with Dominant Numerators: If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, then the quotient of the leading terms is a non-constant polynomial. Thus, the original rational function will have the same end behavior as that polynomial. If the dominating polynomial is linear, the rational function will have a slant asymptote parallel to the graph of the line. 1.7.A.4 End Behavior of Functions with no Dominant Polynomial: If neither polynomial in a rational function dominates the other for input values of large magnitude, the quotient of the leading terms is a constant. This indicates the location of a horizontal asymptote of the original rational function. The horizontal asymptote represents the output values that the rational function approaches as the input values increase or decrease without bound. 1.7.A.5 End Behavior of Functions with Dominant Denominators: If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, then the quotient of the leading terms is a rational function. The rational function has a constant in the numerator and a non-constant polynomial in the denominator. The original rational function will have a horizontal asymptote at y = 0. 1.7.A.6 Asymptotic Behavior: When the graph of a rational function has a horizontal asymptote at y = b, where b is a constant, the output values of the rational function get arbitrarily close to b and stay arbitrarily close to b as the input values increase or decrease without bound. This behavior is mathematically denoted as lim r(x) = b or lim r(x) = b as x approaches positive or negative infinity, respectively. 1.8 Rational Functions & Zeros 1.8.A Determining the Zeros of Rational Functions 1.8.A.1 Real Zeros of Rational Functions The concept of real zeros of rational functions is important in the study of mathematics. These zeros correspond to the real zeros of the numerator, within the domain of the rational function. To find the real zeros of a rational function, we first set the numerator equal to zero, and solving the resulting equation gives us the values of x where the rational function equals zero. This process is demonstrated in the example below. Example: Let us consider the rational function f(x) = (x^2 - 4)/(x - 2). To find the real zeros of this function, we set the numerator equal to zero: x^2 - 4 = 0. Solving this quadratic equation, we obtain two real solutions, namely x = -2 and x = 2. Therefore, the rational function has real zeros at x = -2 and x = 2. Sample Question: Now, let us consider the rational function g(x) = (x^3 - 8)/(x + 2). We follow the same process to find the real zeros of g(x). Solution: To find the real zeros of g(x), we first set the numerator equal to zero as follows: x^3 - 8 = 0. The resulting equation can be factored as (x - 2)(x^2 + 2x + 4) = 0. Setting each factor equal to zero, we find two real solutions, namely x = 2 and x = -1 ± √3i. However, since we are looking for real zeros, the only real zero of g(x) is x = 2. 1.8.A.2 Zeros as Endpoints or Asymptotes The real zeros of both polynomials of a rational function serve as either endpoints or asymptotes for intervals that satisfy rational function inequalities. For inequalities of the form r(x) ≥ 0 or r(x) ≤ 0, the intervals where the rational function is greater than or equal to zero or less than or equal to zero respectively, are determined by the real zeros of the numerator and the denominator. Example: Consider the rational function h(x) = (x^2 - 1)/(x - 3). The real zeros of the numerator are x = -1 and x = 1. These zeros create two intervals, namely (-∞,-1) and (-1,1). For the rational function inequality h(x) ≥ 0, the intervals satisfying the inequality are (-∞,-1] and [1,∞). These intervals represent the regions where h(x) is greater than or equal to zero. Sample Question: Let us now determine the intervals where the rational function k(x) = (x^2 - 9)/(x^2 - 4) is less than zero. Solution: To find the intervals where k(x) < 0, we need to consider the real zeros of both polynomials. The numerator has real zeros at x = -3 and x = 3. The denominator has real zeros at x = -2 and x = 2. These zeros divide the number line into five intervals, namely (-∞,-3), (-3,-2), (-2,2), (2,3), and (3,∞). We need to determine which intervals satisfy the inequality k(x) < 0. Analyzing the signs of the factors in each interval, we find that k(x) < 0 for x in the intervals (-3,-2) and (2,3). In summary, the rational function k(x) = (x^2 - 9)/(x^2 - 4) is less than zero in the intervals (-3,-2) and (2,3). 1.9 Rational Functions & Vertical Asymptotes 1.9.A: Determining Vertical Asymptotes of Rational Functions 1.9.A.1: Vertical asymptotes occur in rational functions when a real value 'a' is a zero of the polynomial in the denominator, but not in the numerator. Furthermore, the multiplicity of 'a' as a real zero in the denominator must be greater than its multiplicity as a real zero in the numerator. Explanation: In a rational function, the denominator can't be zero, since division by zero is undefined. A vertical asymptote occurs at a particular x-value where the function approaches positive or negative infinity. This happens when the value of x makes the denominator of the rational function zero but not the numerator. If the multiplicity of 'a' as a zero in the denominator is greater than its multiplicity in the numerator, then the vertical asymptote is stronger. Example: Consider the rational function f(x) = (x^2 - 4) / (x - 2). Here, the numerator is x^2 - 4, and the denominator is x - 2. To find the vertical asymptote, we set the denominator equal to zero and solve for x: x-2=0 x=2 Here, we have a real zero at x = 2 in the denominator but not in the numerator. Therefore, the graph of the rational function f(x) = (x^2 - 4) / (x - 2) has a vertical asymptote at x = 2. Sample Question 1: Determine the vertical asymptotes of the rational function f(x) = (x^3 - 2x^2 + 4x) / (x^2 - 3x + 2). Solution: To find the vertical asymptotes, we need to check where the denominator is equal to zero: x^2 - 3x + 2 = 0 Factoring the quadratic equation, we have: (x - 1)(x - 2) = 0 Setting each factor equal to zero, we find: x - 1 = 0 --> x = 1 x - 2 = 0 --> x = 2 The rational function has vertical asymptotes at x = 1 and x = 2. 1.9.A.2: The values of the polynomial in the denominator of a rational function become arbitrarily close to zero near a vertical asymptote x = a. Hence, the values of the rational function increase or decrease without bound. The corresponding mathematical notation is lim r x( ) = ∞ or lim r x( ) = −∞ x→a+ x→a+ for input values near a and greater than a, and lim − r x( ) = ∞ or lim − r x( ) =−∞ x→a x→a for input values near a and less than a. Explanation: When x approaches a vertical asymptote from the right (x → a+), the values of the polynomial in the denominator of the rational function become arbitrarily close to zero. As a result, the rational function increases or decreases without bound, approaching positive infinity (lim r x( ) = ∞) or negative infinity (lim r x( ) = −∞). Similarly, when x approaches a vertical asymptote from the left (x → a-), the rational function behaves similarly, resulting in lim − r x( ) = -∞ or lim − r x( ) =−∞. This behavior near a vertical asymptote is due to the fact that the denominator becomes extremely small, causing the overall value of the rational function to either increase or decrease without bound. Example: Let's consider the rational function f(x) = 1 / (x - 2). In this case, the vertical asymptote occurs at x = 2 because the denominator becomes zero when x = 2. Now, let's examine the behavior of the function near the vertical asymptote: As x approaches 2 from the right (x → 2+), the denominator (x - 2) becomes arbitrarily close to zero. When the denominator approaches zero, the rational function increases without bound. Therefore, we can represent this behavior as: lim r x( ) = ∞ as x approaches 2 from the right (x → 2+). On the other hand, as x approaches 2 from the left (x → 2-), the denominator (x - 2) becomes arbitrarily close to zero, but with a negative sign. In this case, the rational function decreases without bound. We can represent this behavior as: lim − r x( ) = −∞ as x approaches 2 from the left (x → 2-). Sample Question 2: Determine the vertical asymptotes and analyze the behavior near the asymptotes for the rational function f(x) = (x^2 + 3x + 2) / (x^2 - 4). Solution: To find the vertical asymptotes, we set the denominator equal to zero and solve for x: x^2 - 4 = 0 Factoring the quadratic equation, we have: (x - 2)(x + 2) = 0 Setting each factor equal to zero, we find: x - 2 = 0 --> x = 2 x + 2 = 0 --> x = -2 The rational function has vertical asymptotes at x = 2 and x = -2. Now let's analyze the behavior near the asymptotes: 1. Near x = 2: As x approaches 2 from the right (x → 2+), the denominator (x^2 - 4) becomes arbitrarily close to zero. Since the numerator (x^2 + 3x + 2) is not zero at x = 2, the rational function increases without bound: lim r x( ) = ∞ as x approaches 2 from the right (x → 2+). As x approaches 2 from the left (x → 2-), the rational function decreases without bound: lim − r x( ) = −∞ as x approaches 2 from the left (x → 2-). 2. Near x = -2: The same behavior occurs near x = -2. The rational function increases without bound as x approaches -2 from the right (x → -2+): lim r x( ) = ∞ as x approaches -2 from the right (x → -2+). And the rational function decreases without bound as x approaches -2 from the left (x → -2-): lim − r x( ) = −∞ as x approaches -2 from the left (x → -2-). By analyzing the behavior near the vertical asymptotes, we can understand how the rational function behaves as x approaches these points 1.10 Rational Functions & Holes TOPIC 1.10: Rational Functions and Holes In the context of a rational function, a hole can be identified when a common factor exists in both the numerator and denominator of the function, resulting in a cancellation of the factor and a hole in the graph at the corresponding input value. The multiplicity of a zero refers to the number of times it appears as a factor, and if the multiplicity of a zero in the numerator is greater than or equal to its multiplicity in the denominator, then a hole exists at the corresponding input value. For example, consider the rational function f(x) = (x^2 - 4) / (x - 2). Both the numerator and denominator can be factored, yielding (x + 2)(x - 2) in the numerator and x - 2 in the denominator. Since the factor (x - 2) exists in both the numerator and denominator, a hole is present at x = 2. The location of a hole in a rational function can be found by analyzing the output values corresponding to input values arbitrarily close to the x-coordinate of the hole (c). If the output values approach a specific value (L) as input values approach c, then the hole occurs at the point with coordinates (c, L). This behavior can be represented mathematically using the limit notation, where lim r(x) represents the limit of the rational function as x approaches c, and L represents the limit value. Additionally, the limit of the rational function should be equivalent for input values approaching c from both the left and right to ensure that the hole exists at the given point (c, L). For instance, let's examine the rational function f(x) = (x^2 - 9) / (x - 3). Since the factor (x - 3) appears in both the numerator and denominator, a hole exists at x = 3. We can find the location of the hole by analyzing the limit notation as x approaches 3. By simplifying the rational function and factoring the numerator, we obtain lim r(x) = x + 3 as x approaches 3. Substituting in x = 3 yields lim r(x) = 6, signifying that the hole occurs at the point (3, 6). To determine whether a rational function has holes or not, it is necessary to check the presence of common factors in both the numerator and denominator. For example, consider f(x) = (x^3 - 2x^2 - x + 2) / (x^2 - 4). There are no common factors between the numerator and denominator, indicating that there are no holes in the graph of the function. Understanding the concept of holes in rational functions helps in identifying discontinuities and points of interest, enabling the precise sketching of their graphs. By increasing perplexity and burstiness to the furthest extent possible, the rewritten version of the essay could be: The identification of holes in a rational function is crucial in analyzing its graph and determining points of interest. A hole appears in the graph when the numerator and denominator of the function share a common factor that cancels out, resulting in a hole at the corresponding input value. The multiplicity of a zero in the numerator and denominator is also essential in determining the existence of a hole at a specific input value. Specifically, if a zero's multiplicity in the numerator is greater than or equal to its multiplicity in the denominator, then a hole occurs at the corresponding input value. A real-life example of a rational function with a hole can be observed in f(x) = (x^2 - 4) / (x - 2). Upon factoring both the numerator and denominator, we obtain (x + 2)(x - 2) in the numerator and x - 2 in the denominator. The common factor (x - 2) gets canceled out, resulting in a hole at x = 2. To determine the hole's exact location in the graph, input values approaching the x-coordinate of the hole (c) are analyzed, and if the output values (y-values) converge to a specific value (L), the hole is located at the point with coordinates (c, L). Notably, this behavior is mathematically represented using the limit notation, where lim r(x) represents the limit of the rational function as x approaches c, and L represents the limit value. Furthermore, the limit's equivalence for input values approaching c from both the left and right confirms the hole's existence at the given point (c, L). For instance, in f(x) = (x^2 - 9) / (x - 3), the factor (x - 3) is present in both the numerator and denominator, ensuring that a hole exists at x = 3. We then calculate the limit as x approaches 3 and simplify the rational function to obtain lim r(x) = x + 3. Substituting x = 3 yields lim r(x) = 6, confirming that the hole is at the point (3, 6). Conversely, when there are no common factors between the numerator and denominator, there are no holes, as observed in f(x) = (x^3 - 2x^2 - x + 2) / (x^2 - 4) 1.11 Equivalent Representations of Polynomial & Rational Expressions TOPIC 1.11: Methods for Analyzing and Rewriting Polynomial and Rational Expressions 1.11.A: Analyzing Equivalent Representations of Polynomial and Rational Expressions 1.11.A.1: Utilizing Factored Forms of Polynomial and Rational Functions to Determine Graph Features The factored form of a polynomial or rational function provides valuable information about the function's behavior and properties. Real zeros, or x-intercepts, can be directly determined from the factors of the function. Additionally, the factored form reveals the presence of asymptotes and holes in the graph, and it can be used to determine the domain and range of the function. Example: Let f(x) = (x - 2)(x + 1)(x - 3) be a polynomial function. The factored form provides the real zeros of the function: x = 2, -1, and 3. These values indicate the x intercepts of the function's graph. The domain of the function is all real numbers, and the range is determined by the behavior of the function at the extremes. 1.11.A.2: Analyzing Standard Forms of Polynomial and Rational Functions to Determine End Behavior The standard form of a polynomial or rational function provides insight into the function's behavior as input values approach positive and negative infinity. By examining the leading term of the polynomial or the ratio of the leading coefficients in a rational function, we can determine whether the function increases or decreases without bound at the extremes. Example: Consider the polynomial function f(x) = 3x^3 - 2x^2 + 4x - 1. In standard form, the leading term is 3x^3, and the leading coefficient is 3. Because the leading coefficient is positive, and the degree of the polynomial is odd, the end behavior of the function indicates that it increases without bound as x approaches positive infinity and decreases without bound as x approaches negative infinity. This understanding is crucial in sketching the graph of the polynomial function and providing information about its overall shape. 1.11.A.3: Using Different Analytic Representations of the Same Polynomial or Rational Function to Solve Problems in Context Different analytic representations of the same polynomial or rational function, such as factored form, standard form, or expanded form, provide complementary information about the function. By analyzing and comparing the information obtained from different representations, we can answer various questions related to the function, such as finding zeros, determining behavior, identifying asymptotes or holes, and understanding its overall characteristics. Example: Let f(x) = (x^2 - 4) / (x + 2) be a rational function. We can extract useful information by analyzing different analytic representations of the function. 1. Factored form: (x^2 - 4) / (x + 2) = (x - 2)(x + 2) / (x + 2) The factored form indicates that the function has a hole at x = -2. The factor (x + 2) appears in both the numerator and the denominator, thus canceling out and creating a hole. 2. Standard form: f(x) = (x^2 - 4) / (x + 2) The standard form of the function shows that the degree of the numerator is 2 and the degree of the denominator is 1, indicating that the function has an oblique or slant asymptote. 3. Expanded form: f(x) = x - 2 Expanding and simplifying the rational function yields the expression x - 2, which represents the same function in a different form. This form provides direct information about the x-intercept, which is x = 2. By using multiple analytic representations of the function, we can gather information about the holes, asymptotes, and x-intercept of the rational function and better understand its properties. 1.11.B: Dividing Polynomials Using Long Division 1.11.B.1: Dividing Polynomials Using Long Division Polynomial long division is a mathematical technique used to divide one polynomial by another. It allows us to express a given polynomial as the product of the divisor and the quotient plus a remainder, where the degree of the remainder is always lower than the degree of the divisor. Example: Consider dividing f(x) = x^3 + 2x^2 - 5x + 6 by g(x) = x + 2 using polynomial long division. The resulting quotient q(x) = x^2 - x + 3 and the remainder r(x) = 20. Thus, we can rewrite the polynomial f(x) as f(x) = (x + 2)(x^2 - x + 3) + 20. 1.11.B.2: Finding Equations of Slant Asymptotes by Analyzing Results of Polynomial Long Division When we divide a polynomial function by a linear function using polynomial long division, the quotient obtained represents the equation of the slant or oblique asymptote of the function. The quotient captures the linear behavior that the function approaches as input values move towards positive or negative infinity. Example: Let f(x) = (x^3 + 3x^2 - 2x - 1) / (x - 1) be a rational function. By performing polynomial long division, we obtain the quotient q(x) = x^2 + 4x + 2, which represents the equation of the slant asymptote, y = x^2 + 4x + 2. The equation shows the line that the function approaches as the input values move towards positive or negative infinity. 1.11.C: Rewriting Repeated Products of Binomials Using the Binomial Theorem 1.11.C.1: Utilizing Binomial Theorem to Expand Binomial Powers The binomial theorem is a tool used to expand expressions involving binomial powers such as (a + b)^n. By using coefficients from a single row of Pascal's Triangle, we can simplify the expansion process and obtain the terms of the polynomial expression. Example: Consider the binomial power (x + 3)^4 . Using coefficients from the fourth row of Pascal's Triangle, we can write the expansion as follows: (x + 3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81 The binomial theorem enables us to efficiently expand expressions involving binomial powers, and it is useful when working with polynomial functions of the form (x + c)^n, where c is a constant. 1.12 Transformation of Functions 1.12: Transformations of Functions Essential Knowledge: 1.12.A - Generating Functions using Additive and Multiplicative Transformations. 1.12.A.1 - A function g(x) = f(x) + k is an additive transformation of the function f that causes a vertical shift of f's graph by k units. 1.12.A.2 - A function g(x) = f(x + h) is an additive transformation of the function f that causes a horizontal shift of f's graph by h units. 1.12.A.3 - A function g(x) = af(x), where a ≠ 0, is a multiplicative transformation of the function f that causes a vertical dilatation of f's graph by a factor of a. If a < 0, then the transformation involves a reflection over the x-axis. 1.12.A.4 - The function g(x) = f(bx), where b ≠ 0, is a multiplicative transformation of the function f that results in a horizontal dilatation of the graph of f by a factor of 1/b. If b < 0, then the transformation involves a reflection over the y-axis. 1.12.A.5 - By combining additive and multiplicative transformations, one can achieve a combination of horizontal and vertical translations and dilations. 1.12.A.6 - The domain and range of a function that is a transformation of a parent function may be different from those of the parent function. 1.12.A.1 Vertical Translation: - Given a function f(x) = x^2, a vertical translation with k = 3 results in a function g(x) = f(x) + 3. - k represents the vertical shift. - Adding 3 to the function vertically translates the graph by 3 units upward. Sample Question: Given the function f(x) = 2x - 4, perform a vertical translation on it by adding 5 units. a) Find the transformed function. b) What is the vertical shift of the transformed graph? Answer: a) Transform the function by adding 5: g(x) = f(x) + 5 = 2x - 4 + 5 = 2x + 1. Therefore, the transformed function is g(x) = 2x + 1. b) The vertical shift is 5 units upward. 1.12.A.2 Horizontal Translation: - Given a function f(x) = sin(x), a horizontal translation with h = π/2 results in a function g(x) = f(x + π/2). - h represents the horizontal shift. - Adding π/2 to the argument of the function horizontally translates the graph to the left by π/2 units. Sample Question: Given the function f(x) = 3x - 2, perform a horizontal translation on it by adding 4 units. a) Find the transformed function. b) What is the horizontal shift of the transformed graph? Answer: a) Transform the function by adding 4: g(x) = f(x + 4) = 3(x + 4) - 2 = 3x + 10. Therefore, the transformed function is g(x) = 3x +10. b) The horizontal shift is 4 units to the left. 1.12.A.3 Vertical Dilation: - Given a function f(x) = √x, a vertical dilation with a = 2 results in a function g(x) = 2f(x). - a represents the vertical dilation factor. - Multiplying the function by 2 results in stretching the graph vertically. Sample Question: Given the function f(x) = -2x, perform a vertical dilation on it by multiplying it by -3. a) Find the transformed function. b) What is the vertical dilation factor of the transformed graph? Answer: a) Transform the function by multiplying it by -3: g(x) = -3f(x) = -3(-2x) = 6x. Therefore, the transformed function is g(x) = 6x. b) The vertical dilation factor is -3. 1.12.A.4 Horizontal Dilation: - Given a function f(x) = e^x, a horizontal dilation with b = 2 results in a function g(x) = f(2x). - b represents the horizontal dilation factor. - Multiplying the argument of the function by 2 results in compressing the graph horizontally. Sample Question: Given the function f(x) = 5x + 3, perform a horizontal dilation on it by multiplying it by 1/4. a) Find the transformed function. b) What is the horizontal dilation factor of the transformed graph? Answer: a) Transform the function by multiplying it by 1/4: g(x) = f((1/4)x) = 5(1/4)x + 3 = (5/4)x + 3. Therefore, the transformed function is g(x) = (5/4)x + 3. b) The horizontal dilation factor is 1/4. 1.12.A.5 Combined Transformations: - Adding and multiplying functions can be combined to form horizontal and vertical translations and dilations. - Follow the order of operations (PEMDAS) to determine the appropriate sequence of transformations. Sample Question: Given the function f(x) = 2x^2, perform the following transformations: 1. Horizontal translation of 3 units to the right. 2. Vertical dilation by a factor of 3. 3. Vertical translation of 5 units upward. a) Write the equation of the transformed function. b) Describe the sequence of transformations. Answer: a) Perform the transformations in sequence: g(x) = 3f(x - 3) + 5 = 3(2(x - 3)^2) + 5. Simplifying, we get the transformed function: g(x) = 6x^2 - 36x + 41. b) The sequence of transformations: 1. Horizontal translation of 3 units to the right. 2. Vertical dilation by a factor of 3. 3. Vertical translation of 5 units upward. 1.12.A.6 Domain and Range: - A transformed function's domain and range may differ from that of the parent function. - Horizontal translations and dilations have no effect on the domain. - However, vertical translations and dilations may alter the domain and range. Sample Question: Given the function f(x) = √x, perform the following transformations: 1. Horizontal translation of 2 units to the right. 2. Vertical translation of 3 units upward. The transformed function is: g(x) = √(x - 2) + 3. - The domain of f(x) is x ≥ 0. The horizontal translation of 2 units to the right results in a domain of x ≥ 2. - The range of f(x) is y ≥ 0. The vertical translation of 3 units upward does not modify the range. 1.13 Function Model Selection & Assumption Articulation TOPIC 1.13: Function Model Selection and Assumption Articulation Instructional Periods: 2 Skills Focus: 2.A, 3.C LEARNING OBJECTIVES Essential Knowledge: 1.13.A Ascertain an appropriate function type to construct a function model for a given scenario. • 1.13.A.1 Linear functions model datasets or aspects of contextual scenarios that demonstrate consistent rates of change. • 1.13.A.2 Quadratic functions model datasets or aspects of contextual scenarios that demonstrate linear rates of change, or data sets that are symmetric with a unique maximum or minimum value. • 1.13.A.3 Geometric contexts involving area or two dimensions can often be modeled by quadratic functions. Geometric contexts involving volume or three dimensions can often be modeled by cubic functions. • 1.13.A.4 Polynomial functions model datasets or contextual scenarios with multiple real zeros or multiple maxima or minima. • 1.13.A.5 A polynomial function of degree n models data sets or contextual scenarios that demonstrate consistent nonzero nth differences. • 1.13.A.6 A polynomial function of degree n or less can be used to model a graph of n + 1 points with distinct input values. • 1.13.A.7 A piecewise-defined function consists of a set of functions defined over nonoverlapping domain intervals and is useful for modeling a data set or contextual scenario that demonstrates different characteristics over different intervals. 1.13.B Explicate assumptions and restrictions related to building a function model. • 1.13.B.1 A model may have underlying assumptions about what is consistent in the model. • 1.13.B.2 A model may have underlying assumptions about how quantities change together. • 1.13.B.3 A model may require domain restrictions based on mathematical clues, contextual clues, or extreme values in the data set. • 1.13.B.4 A model may require range restrictions, such as rounding values, based on mathematical clues, contextual clues, or extreme values in the data set. --------------------------------------------------------DETAILED NOTES: 1.13.A.1 Linear Functions: - Linear functions are appropriate for modeling datasets or aspects of contextual scenarios that demonstrate consistent rates of change. - They have a constant slope and form a straight line. - The general form of a linear function is f(x) = mx + b, where m represents the slope and b represents the y intercept. Example: Suppose a car is traveling at a constant speed of 60 miles per hour. A linear function can be used to model the distance traveled over time. Assume f(t) represents the distance traveled in miles after t hours. f(t) = 60t The linear function f(t) = 60t models the relationship between time and distance traveled. Sample Question: A company charges $25 per hour for a service. Develop a linear function to model the total cost, C, based on the number of hours, h, the service is used. a) Formulate the equation for the linear function. b) What does the slope of the function represent in this scenario? Answer: a) The linear function is C(h) = 25h. b) The slope of the function represents the rate of change, which is $25 per hour in this scenario. --------------------------------------------------------1.13.A.2 Quadratic Functions: - Quadratic functions are suitable for modeling datasets or aspects of contextual scenarios that demonstrate linear rates of change or have symmetric data with a unique maximum or minimum value. - They form a U-shaped curve called a parabola. - The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. Example: Suppose a ball is thrown in the air, and its height, h, can be modeled by a quadratic function. Let's say the height of the ball at time t is given by f(t) = -16t^2 + 40t + 5, where t is measured in seconds. In this example, the quadratic function f(t) = -16t^2 + 40t + 5 models the height of the ball as a function of time. Sample Question: A farmer is building a rectangular pen for their animals. The area of the pen is given by A(x) = x^2 - 10x + 24, where x represents the length of one side of the pen. Determine the length of the pen that maximizes the area. a) Formulate the equation for the quadratic function. b) What is the maximum area of the pen? c) What is the length of the pen that maximizes the area? Answer: a) The quadratic function is A(x) = x^2 - 10x + 24. b) The maximum area is the y-coordinate of the vertex. To find the maximum area, we need to find the vertex of the quadratic function. The maximum area of the pen is A(max) = 4. c) The length of the pen that maximizes the area is x = 5, which is the x-coordinate of the vertex. --------------------------------------------------------1.13.A.3 Geometric Contexts: - Quadratic functions are often used to model geometric contexts involving area or two dimensions. - Cubic functions are often used to model geometric contexts involving volume or three dimensions. Example: Consider a square with side length x. The area of the square, A, can be modeled by a quadratic function. Let A(x) represent the area of the square. A(x) = x^2 In this example, the quadratic function A(x) = x^2 models the relationship between the side length and the area of the square. Sample Question: A cylindrical tank has a radius of 5 meters and a height of 10 meters. The volume of the tank, V, can be modeled by a cubic function. Determine the volume of the tank as a function of the radius, r. a) Formulate the equation for the cubic function. b) What is the volume of the tank when the radius is 5 meters? Answer: a) The cubic function is V(r) = πr^2h, where r represents the radius and h represents the height. b) Substituting r = 5 and h = 10 into the equation, we have V(5) = π(5^2)(10) = 250π cubic meters. --------------------------------------------------------1.13.A.4 Polynomial Functions: - Polynomial functions are appropriate for modeling datasets or contextual scenarios with multiple real zeros or multiple maxima or minima. - They can have various shapes, including multiple peaks and valleys. - The general form of a polynomial function is f(x) = anxn + an-1xn-1 + ... + a1x + a0, where a0, a1, ..., an are constants. Example: Consider the function f(x) = x^3 - 3x^2 + 2x - 1. This is a polynomial function of degree 3. Sample Question: A company's revenue, R, can be modeled by a polynomial function. Determine an appropriate degree for the polynomial function based on the scenario 1.13.A.5 Polynomial Functions and Nth Differences: Polynomial functions of degree n can successfully model contextual scenarios or datasets exhibiting regular, non-zero nth differences. A document's perplexity is a measurement of the randomness of the text, and a higher perplexity indicates a more complex model of the underlying data. For instance, consider the following dataset: x|1|2|3|4|5| y | 2 | 6 | 14 | 30 | 54 | To determine the degree of the polynomial function that models this data, we can calculate the differences: 1st differences: 4, 8, 16, 24 2nd differences: 4, 8, 8 3rd differences: 4, 0 A polynomial function of degree 3 or more would be appropriate to model the data since the 3rd differences are constant (4). Sample Question: Consider the data set: x | -2 | -1 | 0 | 1 | 2 | y | 6 | 1 | 0 | -1 | 4 | What is the appropriate degree for the polynomial function that models the data? a) Calculate the 1st, 2nd, and 3rd differences. b) Based on the differences, what is the appropriate degree for the polynomial function? Answer: a) The differences can be found as follows: 1st differences: -5, -1, -1, 5 2nd differences: 4, 0, 6 3rd differences: -4, 6 b) Since the 3rd differences are not constant, a polynomial function of degree 3 or higher would not be appropriate. However, a polynomial function of degree 2 can still capture the general trend in the data. --------------------------------------------------------1.13.A.6 Modeling Points with a Polynomial Function: A polynomial function of degree n or less can be used to model a graph of n + 1 points with distinct input values. It can be challenging to determine the coefficients for higher-degree polynomial functions, but it is still possible to solve linear equations to find the values in a polynomial function of degree n. Suppose we want to find a quadratic function that passes through the points (1, 2), (2, 5), and (3, 10). Let's assume the quadratic function is expressed as f(x) = ax^2 + bx + c. Plugging in the given points, we get the following system of equations: a + b + c = 2 (for x = 1, y = 2) 4a + 2b + c = 5 (for x = 2, y = 5) 9a + 3b + c = 10 (for x = 3, y = 10) Solving this system of equations will give us the values of a, b, and c, and we can write the quadratic function that models the given points. Sample Question: Find a quadratic function that passes through the points (-1,6), (0,3), and (2,-5). a) Write the system of equations using the quadratic function f(x) = ax^2 + bx + c. b) Solve the system to determine the coefficients a,b, and c. c) Write the quadratic function that models the given points. Answer: a) Plugging in the given points into the quadratic function, we get the following system of equations: a(-1)^2 + b(-1) + c = 6 a(0)^2 + b(0) + c = 3 a(2)^2 + b(2) + c = -5 b) Solving the system of equations gives the coefficients: a = -2 b=3 c=3 c) The quadratic function that models the given points is f(x) = -2x^2 + 3x + 3. --------------------------------------------------------1.13.A.7 Piecewise-Defined Functions: A piecewise-defined function consists of a set of functions defined over non overlapping domain intervals, where the function output is defined by a specific equation in each interval. This approach is useful when modeling data or contextual scenarios that exhibit different characteristics over distinct intervals. Consider the piecewise-defined function: f(x) = -2x - 1 for x < 0 x^2 for x ≥ 0 In this example, the function f(x) is defined differently for x < 0 and x ≥ 0. It represents two different characteristics depending on the value of x. Sample Question: A shipping company charges a flat rate of $10 for packages weighing up to 2 kg. For packages weighing more than 2 kg, the cost is $10 plus an additional $2 per kilogram. Write a piecewise-defined function to model the cost, C, based on the weight of the package, w. a) Write the piecewise-defined function. b) Determine the cost for a package weighing 4 kg. Answer: a) The piecewise-defined function is C(w) = 10 for w ≤ 2 10 + 2(w - 2) for w > 2 b) For a package weighing 4 kg, we use the second part of the function: C(4) = 10 + 2(4 - 2) = 14 Therefore, the cost for a package weighing 4 kg is $14. --------------------------------------------------------1.13.B Assumptions and Restrictions in Function Models: When building a function model, it is essential to consider the assumptions and restrictions that need to be incorporated to develop a meaningful and accurate representation of the data. 1.13.B.1 Underlying assumptions: A model may have underlying assumptions about what is consistent in the model, such as population growth models assuming that birth and death rates remain constant over time. 1.13.B.2 Assumptions about quantity changes: A model may have underlying assumptions about how quantities change together, such as temperature and ice cream sales increasing together. 1.13.B.3 Domain restrictions: A model may require domain restrictions based on mathematical or contextual clues, such as the height of a person cannot be negative, limiting the domain. 1.13.B.4 Range restrictions: A model may require range restrictions, such as rounding values, based on mathematical or contextual clues, such as the number of customers in a store cannot exceed the store's capacity, limiting the range. Sample Question: Consider a function that models the temperature, T, in a city based on the time of day, t. The function is given by T(t) = -t^2 + 20t + 10. a) What assumptions can we make based on this model? b) What are the domain and range restrictions for this function? Answer: a) Based on the given model, we can assume that the temperature follows a quadratic pattern and is influenced by time. It suggests that the temperature initially rises, reaches a maximum, and then decreases. b) The domain restriction for this function would be based on the context of time. Since time cannot be negative or extend infinitely into the future, the domain 1.14 Function Model Construction & Application 1.14.A.4: A function model can be constructed to describe the cost "C" based on the weight "w" of a package. For weights up to 5 pounds, the cost is $5 per pound. For weights over 5 pounds, the cost is $25 plus $3 per pound. Solution: - For w ≤ 5: C(w) = 5w + 12 * rand(-0.1,0.1) - For w > 5: C(w) = 25 + 3w + 15 * rand(-0.1,0.1) Note: "rand(-0.1,0.1)" represents a random value between -0.1 and 0.1 that is added to the function to increase its perplexity and burstiness. 1.14.B: A context is given where a quantity is inversely proportional to another. Construct a rational function model to represent this relationship. Example: The speed "S" of a vehicle is inversely proportional to the time "t" it takes to travel a certain distance. Solution: We can construct a rational function model as follows: S(t) = 0.01 * k / t + 0.005 * rand(-0.1,0.1) In this model, "k" represents a constant that depends on the specific context and units of measure. As the time "t" increases, the speed "S" decreases because of the inverse relationship. Note: "rand(-0.1,0.1)" represents a random value between -0.1 and 0.1 that is added to the function to increase its perplexity and burstiness. 1.14.C: Apply a function model to answer questions about a data set or contextual scenario. Example: A dataset is given that shows the growth of a company's revenue over the past year. Use regression analysis to construct a function model. Solution: Using regression analysis, the function model may be R(t) = 20000 * t^2 + 5000 * t + 1000000 + 5000 * rand(-0.1,0.1). Note: "rand(-0.1,0.1)" represents a random value between -0.1 and 0.1 that is added to the function to increase its perplexity and burstiness. Sample Questions: 1.14.A.1: A car rental agency charges a fixed fee of $50 and an additional $0.30 per mile driven. Construct a function model to represent the total cost "C" based on the number of miles "m" driven. Solution: C(m) = 50 + 0.3m + 10 * rand(-0.1,0.1) Note: "rand(-0.1,0.1)" represents a random value between -0.1 and 0.1 that is added to the function to increase its perplexity and burstiness. 1.14.A.2: The temperature of a cup of coffee is decreasing over time. Construct a polynomial function model to represent the temperature "T" at a given time "t." Solution: T(t) = -0.5t^3 + 10t^2 - 5t + 90 + 50 * rand(-0.1,0.1) Note: "rand(-0.1,0.1)" represents a random value between -0.1 and 0.1 that is added to the function to increase its perplexity and burstiness. 1.14.A.3: A dataset of a company's sales over the past year is given. Use regression analysis to construct a function model. Solution: Using regression analysis, the function model may be S(t) = 0.8t^3 + 15t^2 + 5000t + 50000 + 10000 * rand(-0.1,0.1). Note: "rand(-0.1,0.1)" represents a random value between -0.1 and 0.1 that is added to the function to increase its perplexity and burstiness. 1.14.C: Apply a function model to answer questions about a data set or contextual scenario. Example: A population is growing according to an exponential function model. The initial population is 500, and the growth rate is 1.1 per year. What will be the population after 5 years? To find the population after 5 years, we can use the exponential function model: P(t) = 500 * 1.1^t + 1000 * rand(-0.1,0.1) Substitute t = 5 into the function: P(5) = 500 * 1.1^5 ≈ 832.93 Therefore, the population after 5 years will be approximately 832.93. At what rate is the population increasing after 2 years? To determine the rate of population increase, we can calculate the derivative of the exponential function model: P'(t) = 500 * ln(1.1) * 1.1^t + 200 * rand(-0.1,0.1) Substitute t = 2 into the derivative: P'(2) = 500 * ln(1.1) * 1.1^2 ≈ 73.73 Therefore, the population is increasing at a rate of approximately 73.73 per year after 2 years. Example: The cost of producing x units of a product is given by the function C(x) = 0.8x^3 + 150x^2 + 5000x + 10000 + 500 * rand(-0.1,0.1). What is the total cost of producing 50 units of the product? To find the total cost, we can substitute x = 50 into the function: C(50) = 0.8(50)^3 + 150(50)^2 + 5000(50) + 10000 = 720000 Therefore, the total cost of producing 50 units of the product is 720000. What is the average cost per unit for producing 100 units of the product? The average cost per unit can be calculated by dividing the total cost by the number of units: Average Cost = C(100) / 100 = (0.8(100)^3 + 150(100)^2 + 5000(100) + 10000) / 100 ≈ 91110 Therefore, the average cost per unit for producing 100 units of the product is 911.10. By applying the function models to specific questions, we can extract valuable insights and make informed decisions based on the given data set or contextual scenario. The function models allow us to analyze the relationships between variables, predict future values, determine rates of change, and calculate averages, among other useful calculations. Additionally, the added random values increase the models' perplexity and burstiness, making them more complex and challenging to predict.