11.4.A.1Multiple Choice
Which of the following is a polynomial function?
Select A, B, C, or D to check your answer.
Directions: Work each problem on your own paper first. For multiple choice, select A, B, C, or D. You will see right away whether you are correct. If you miss, you will see the most likely reason, and you can try again. For short response, type your answer, then compare it with the model solution. Questions 1–3 cover 1.4.A.1; 4–7 cover 1.4.A.2; 8–10 cover 1.4.A.3; 11–14 cover 1.4.A.4; 15–19 cover 1.4.A.5; question 20 covers them all.
Which of the following is a polynomial function?
Select A, B, C, or D to check your answer.
What are the degree and the leading coefficient of p(x) = 5x² − 3x⁶ + 2x − 8?
Select A, B, C, or D to check your answer.
Let p(x) = (3x − 2)(x² + 5)(1 − x). (a) Write p in the form aₙxⁿ + … + a₀ to show that p is a polynomial function. (b) State the degree, the leading term, and the leading coefficient. (c) Explain how you could find the leading term without multiplying everything out.
The table gives values of a polynomial function f on the domain −2 ≤ x ≤ 4, with both endpoints included. Assume f changes direction only where the table indicates. Which statement is correct?
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
| f(x) | 3 | 10 | 5 | −6 | −17 | −22 | −15 |
Select A, B, C, or D to check your answer.
The graph shows f(x) = x⁴ − 4x². Which statement is correct?
Select A, B, C, or D to check your answer.
A polynomial function P is defined on 0 ≤ x ≤ 6, with endpoints included. It is increasing on (0, 2), decreasing on (2, 5), and increasing on (5, 6). Which statement must be true?
Select A, B, C, or D to check your answer.
A company's profit, in thousands of dollars, is P(x) = −x³ + 9x² − 15x, where x is the number of units produced in hundreds and 0 ≤ x ≤ 8. The table gives values; assume P changes direction only where the table indicates. (a) Identify every local extremum, including endpoints. (b) Find the global maximum and the global minimum. (c) Interpret the global maximum and the global minimum in context.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P(x) | 0 | −7 | −2 | 9 | 20 | 25 | 18 | −7 | −56 |
A polynomial function has real zeros at x = −5, x = −1, x = 2, and x = 7. What is the least number of local extrema the function must have?
Select A, B, C, or D to check your answer.
A nonconstant polynomial function f has f(−2) = 0 and f(3) = 0. Which statement must be true?
Select A, B, C, or D to check your answer.
The table gives values of p(x) = x(x − 4)(x + 2), which has zeros at x = −2, 0, and 4. Assume p changes direction only where the table indicates. Which statement about the stretch between the zeros x = 0 and x = 4 is correct?
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
| p(x) | 0 | 5 | 0 | −9 | −16 | −15 | 0 |
Select A, B, C, or D to check your answer.
Which of the following functions must have a global minimum but no global maximum?
Select A, B, C, or D to check your answer.
Let f(x) = −2x² + 8x − 3. (a) Find the vertex. (b) Does f have a global maximum or a global minimum on all real numbers? Give its value. (c) Now restrict the domain to 3 ≤ x ≤ 6. Find the global maximum and the global minimum on that domain.
Let g(x) = 3x² − 18x + 20. What is the vertex of g, and what kind of extremum is it?
Select A, B, C, or D to check your answer.
Let g(x) = (x − 2)² − 9, with the domain restricted to 3 ≤ x ≤ 8 (endpoints included). What is the global minimum of g on this domain?
Select A, B, C, or D to check your answer.
The table gives values of a polynomial function f. Using average rates of change on intervals of length 1, where is the point of inflection of f?
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| f(x) | 0 | 16 | 20 | 18 | 16 | 20 | 36 |
Select A, B, C, or D to check your answer.
For a polynomial function f, the average rates of change over [0, 1], [1, 2], [2, 3], [3, 4], [4, 5], and [5, 6] are 3, 7, 9, 9, 5, and −1. Which statement is correct?
Select A, B, C, or D to check your answer.
Let g(x) = x³ − 6x². (a) Find the average rate of change of g on each of [0, 1], [1, 2], [2, 3], [3, 4], [4, 5], and [5, 6]. (b) Describe how the rates change and where g is concave up and concave down. (c) Find the point of inflection. Is it also a local maximum or minimum? Explain.
Which of the following graphs shows a polynomial function with exactly two points of inflection?
Select A, B, C, or D to check your answer.
A polynomial function f is increasing on both sides of x = 4, and the rates of change of f change from increasing to decreasing at x = 4. What does the graph have at the point (4, f(4))?
Select A, B, C, or D to check your answer.
The graph shows f(x) = x⁴ − 8x² + 7. (a) State the degree and the leading coefficient. (b) f has four distinct real zeros. What is the least number of local extrema it must have, and what are all of its local extrema? (c) Does f have a global maximum or a global minimum? Find any that exist. (d) Estimate the x-values of the points of inflection and describe the concavity.