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 1–4 cover 1.6.A.1; 5–8 cover 1.6.A.2; 9–19 cover 1.6.A.3; question 20 covers them all.
The table gives values of a polynomial function p for large inputs. Which statement best describes the behavior of p as x increases without bound?
| x | 10 | 100 | 1,000 | 10,000 |
| p(x) | −20 | −9,200 | −992,000 | −99,920,000 |
Select A, B, C, or D to check your answer.
Which of the following is NOT possible for a nonconstant polynomial function p?
Select A, B, C, or D to check your answer.
A student computes f(5) = −1,875 for f(x) = x⁴ − 100x² and concludes that lim x→∞ f(x) = −∞. Which statement is correct?
Select A, B, C, or D to check your answer.
Let h(x) = x³ − 12x². (a) Find h(10), h(100), and h(1,000). (b) Describe what happens to the outputs as x increases without bound, and write your answer in limit notation. (c) h(10) is negative. Explain why that does not change your answer to (b).
The table gives values of a polynomial function p. Which statement is supported by the table?
| x | −10 | −100 | −1,000 |
| p(x) | −1,060 | −1,000,600 | −1,000,006,000 |
Select A, B, C, or D to check your answer.
For the polynomial function g(x) = x⁴ − 5x², the table shows outputs for large negative inputs. What is lim x→−∞ g(x)?
| x | −10 | −100 | −1,000 |
| g(x) | 9,500 | 99,950,000 | 999,995,000,000 |
Select A, B, C, or D to check your answer.
The graph of a polynomial function p is shown. Which statement describes its end behavior?
Select A, B, C, or D to check your answer.
Let q(x) = −2x² + 5x. (a) Find q(−10), q(−100), and q(−1,000). (b) Write lim x→−∞ q(x). (c) Find q(10), q(100), and q(1,000), and write lim x→∞ q(x). (d) Describe how the two ends of the graph compare.
What is the leading term of p(x) = 5x − 2x³ + x² − 9?
Select A, B, C, or D to check your answer.
The polynomial function p is given by p(x) = −3x⁵ + 2x² − x. Which statement about the end behavior of p is true?
Select A, B, C, or D to check your answer.
Which of the following polynomial functions satisfies lim x→∞ f(x) = ∞ and lim x→−∞ f(x) = ∞?
Select A, B, C, or D to check your answer.
Let p(x) = −(x − 1)²(x + 2)(x − 4). Which statement describes the end behavior of p?
Select A, B, C, or D to check your answer.
Let p(x) = 2x⁴ − x⁵ + 3x² − 7. (a) Identify the leading term, the degree, and the sign of the leading coefficient. (b) Write the end behavior using limit notation for both x → ∞ and x → −∞. (c) Explain why the term 3x² does not affect the end behavior.
The graph of a polynomial function g is shown. It has three real zeros. Which statement must be true?
Select A, B, C, or D to check your answer.
Which of the following graphs could be the graph of p(x) = −x⁴ + 3x² + 2?
Select A, B, C, or D to check your answer.
For large input values, p(x) = x³ − 50x² + 7 behaves like x³. Which statement best explains why?
Select A, B, C, or D to check your answer.
Let p(x) = −2(x + 1)²(x − 3)³. (a) Find the degree and the leading term of p without multiplying it out. (b) Write the end behavior of p in limit notation. (c) Use your answer to (b) to explain why the graph of p must cross the x-axis at least once. Then identify where it crosses and where it is tangent.
Let p(x) = 0.1x⁵ − 1000x². Which statement is true?
Select A, B, C, or D to check your answer.
A polynomial function p satisfies lim x→∞ p(x) = ∞ and lim x→−∞ p(x) = ∞. Which statement must be true?
Select A, B, C, or D to check your answer.
Let p(x) = −x⁴ + 5x² − 4. (a) Write the end behavior of p in limit notation. (b) Factor p and find all of its zeros. (c) Is p even, odd, or neither? Justify. (d) Solve p(x) > 0.