Functions Unit Test Review

**Honors Algebra 2: Functions Unit Test Review** **Name:** **1. Describe the interval shown using an inequality, set notation, and interval notation.** [Image of a number line with an interval from -4 to -3, excluding -3] Inequality: -4 ≤ x < -3 Set Notation: {x | -4 ≤ x < -3} Interval Notation: [-4, -3) **2. Describe the interval shown using an inequality, set notation, and interval notation.** [Image of a number line with an interval from -2 to 6, excluding -2 and 6] Inequality: -2 < x < 6 Set Notation: {x | -2 < x < 6} Interval Notation: (-2, 6) **3. Describe the domain and range of the graph using an inequality, set notation, and interval notation. Then describe its end behavior.**   [Image of a graph of a parabola opening upwards, with vertex at (-2, -4)] Domain: Inequality: -4 ≤ x ≤ 2 Set Notation: {x | -4 ≤ x ≤ 2} Interval Notation: [-4, 2] Range: Inequality: -4 ≤ y ≤ 4 Set Notation: {y | -4 ≤ y ≤ 4} Interval Notation: [-4, 4] End Behavior: As x → -∞, f(x) → ∞ As x → ∞, f(x) → ∞ **4. Let y = -3x + 2 on the domain (-1, 2].** Part A: Graph the function. [Image of a line graph with points (-1, 5) and (2, -4)] Part B: Write the range using interval notation. (-4, 5] **5. Steven's average running time for the 100-meter dash is 12 seconds. Determine a function that models this situation. Determine a domain from the situation and identify the range. Use interval notation for the domain and range.**   Rate = 100 meters / 12 seconds = 25/3 meters/second t = time in seconds d(t) = distance in meters d(t) = (25/3)t D: [0, 12] R: [0, 100] Shelley is studying the relationship between her car's mileage (miles per gallon) and speed (miles per hour). The table shows the data Shelley found.   Use it to answer questions 6-9.   | Speed (miles per hour) | 30 | 40 | 50 | 60 | 70 | |-----------------------|---|---|---|---|---| | Mileage (miles per gallon) | 34.0 | 33.5 | 31.5 | 29.0 | 27.5 | 6. What is the best fit linear regression equation for this data, rounded to nearest hundredths? M(s) = -0.17s + 39.85 7. What is the slope of the regression line telling us? For every additional mph, the mileage decreases by 0.175 mpg. 8. What is the y-intercept telling us? Is this an interpolation or an extrapolation? How much confidence do you have in this answer?   At 0 mph, the car is expected to have 39.85 mpg. Extrapolation - not much confidence, as we cannot get mileage when not moving. 9. Predict y when x = 55 using the linear regression. Interpret your answer with a complete sentence. Is this an interpolation or an extrapolation? How much confidence do you have in this answer?   M(55) = 30.205 A car that is going 55 mph has an estimated 30.205 mpg. Interpolation - yes, confident in this answer, as our data set includes similar values. **Use the graph below for questions 10-13.** [Image of a graph with a curve that increases, decreases, and then increases again] 10. Describe the end behavior. As x → ∞, f(x) → ∞ As x → -∞, f(x) → ∞ 11. What is the average rate of change on the interval [-3, 1]? (-3, -2) and (1, 2) (2 - (-2)) / (1 - (-3)) = 1 12. On what interval(s) is the graph increasing? (-∞, -3) U (1, ∞) 13. On what interval(s) is the graph decreasing? (-3, 1) **Use the graph below for questions 14-16.** [Image of a graph with points (-1, -1), (0, 0), (1, 0), (2, 1), (3, 0), (4, 0), and (5, 2)] 14. Find the average rate of change on the interval [-1, 1]. (-1, -1) and (1, 0) (0 - (-1)) / (1 - (-1)) = 1/2 15. Identify the domain and range. D: [-1, 5] R: [-1, 2] 16. Identify the zero(s), maximum(s) and minimum(s). Write as coordinates. Zero(s): (1, 0), (4, 0) Local maximum(s): (1, 0) Local minimum(s): (3, -1) Absolute maximum: (5, 2) Absolute minimum: (-1, -1) **A fireworks projectile is launched at 20 meters per second from a 60-meter cliff. The table shows the height, y, in meters the projectile is above a field after x seconds. Use the table for questions 17-18.** | Time (seconds) | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | Height (meters) | 60 | 75 | 80 | 75 | 60 | 35 | 17. Over what interval(s) is the projectile's height increasing? (0, 2) 18. Over what interval(s) is the projectile's height decreasing? (2, 5) 19. Given f(x) = 1/2x - 5, choose True or False for each statement. A) f^(-1)(y) = 2(y + 5) **True** False B) f^(-1)(-4) = 2 **True** False C) f^(-1)(y) = -1/2(y + 5) **False** True D) f^(-1)(y) = 2y + 10 **True** False **20. Find the inverse of f(x) = -2x - 3.** y = -2x - 3 y + 3 = -2x (y + 3)/-2 = x f^(-1)(y) = -(y + 3)/2 or y + 3/-2 **Part B: Interpret the meaning of the inverse that you found in Part A.** This function inputs °F and outputs °C. **21. Find the inverse of f(x) = (3x + 1)/8.** y = (3x + 1)/8 8y = 3x + 1 6y - 1 = 3x (6y - 1)/3 = x f^(-1)(y) = (6y - 1)/3 **Part C: Assume that we only want to consider positive degrees Celsius in this context. What is the domain and range (use inequality notation) of the inverse function?** D: C ≥ 0 R: F ≥ 32 **22. The formula to convert temperatures from degrees Fahrenheit to degrees Celsius is C = (5/9)(F - 32), where C represents the degrees Celsius and F represents the degrees in Fahrenheit.** **Part A: Find an equation for the inverse function.** C = (5/9)(F - 32) 9C/5 = F - 32 9C/5 + 32 = F