Exponential growth and decay #3 - Questions and Answers

Question 1 Explain how zeros and end behavior of polynomial functions and their graphs are related to the degree and the factors of the polynomial. Answer A polynomial of degree n will have identically n complex zeros according to the Fundamental Theorem of Algebra. Any non-real complex zeros of the polynomial must appear in conjugate pairs if the polynomial's coefficients are all real (which is the case for the polynomials under discussion here). A polynomial of degree 7, for instance, will either contain 0, 2, 4, or 6 non-real zeros, meaning it will either have 7, 5, 3, or 1 real zeros. Take the polynomial as an illustration. f(x) = x^5 - 6x^4 - 3x^3 + 66x^2 - 58x - 120 Graph has three real roots: x = -3, x = -1, x = 4. The two remaining roots are complex. This means that we can write it as a product of factors where, for each real root a, x - a will be a factor. Factor f as f(x) = (x - 4)(x + 1)(x + 3)(x² - 6x + 10) and indeed, if solve x² - 6x + 10 = 0, you will find that it has no real solution. A polynomial with zero real roots cannot be factored into linear factors x - a where a is a real number. The polynomial f(x) = x^4 + 6x^3 + 11x^2 + 6x + 10, shown below, has no real roots and therefore cannot be factored using only real numbers. We need to focus on the word with the highest degree in order to more clearly comprehend the final behavior. Take a fourth-degree polynomial like for example f(x) = x^4 + 10x^3 - x^2 + 1 As x becomes a very large positive number or very large negative number, the x^4 term will become larger much faster than the other terms, and the other terms will become irrelevant. As x^4 can never be a negative number if x is real, On both sides of x, f(x) will get closer to positive infinity. In a similar vein, for a function like f(x) = -x^4 - 10x^3 + x^2 - 1 then -x^4 will become a very large negative number, regardless of the other terms. We already mentioned that odd-degree exponents can be either positive or negative in odd-degree functions. Consider, for instance, the function f(x) = x^3 + 2x^2 + 3x - 1 the x^3 term will become a very large positive number as x approaches positive infinity, and a very large negative number as x → -∞, eclipsing the other terms of lower degrees. Similarly, the function f(x) = -x^3 - 2x^2 - 3x + 1 will show the opposite behavior -x^3 will become a very large negative number as x becomes very large to the positive side, and vice versa. Thus the end behavior of an odd- degree polynomial will always have opposite ends - either it will "start" at -∞ and "end" at ∞, or with these directions switched if the leading coefficient is negative. Question 2 For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. y = 11,701(0.97)^x Answer The formula for an exponential function is represented by its base raised to a power, f(x) = ab^x where a is the initial value and b is the growth rate. If the base, b > 1, then it shows exponential growth. If it is 0 Note that the base is between 0 and 1 in the function. Therefore, this equation represents exponential decay. Question 3 Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change. y = 3(1.88)^t, y = 3(1.88)^t Answer y = 3(1.88)^t We are given the function: y = 3(1 + 0.88)^t The function represents exponential growth because 1.88 > 1. r = 0.88 We determine the rate of change: Question 4 Determine whether each function is an example of exponential growth or decay. Then find the y-intercept. y = 3e^x Answer y = 3e^x y = 3e^0 y = 3(1) y=3 The e function has both a positive constant and a positive exponent so it represents exponential growth. We find the y-intecept by plugging in 0 for x and then solving for y. Question 5 How do graphs and equations reveal key features of exponential growth and decay functions? Answer If function is increasing-exponential growth, D = R, R# = {y ∈ R | y > 0}, y-intecept (0, c), asymptote x-axis If function is decreasing-exponential decay, D = R, R = {y ∈ R | y > 0}, y- intecept (0, a), asymptote x-axis