Polynomial Functions Flashcards

What is the end behavior of f(x) = −2x⁵ + 3x² − 1?

Answer: f(x) → −∞ as x → ∞ and f(x) → ∞ as x → −∞

The leading term −2x⁵ has odd degree and a negative coefficient, so the graph falls to the right and rises to the left.

What is the average rate of change of f(x) = x² − 4x + 1 over the interval [1, 5]?

Answer: 2

f(5) = 6 and f(1) = −2, so the average rate of change is (6 − (−2))/(5 − 1) = 2.

A quadratic function has average rates of change of 3 on [0, 1] and 7 on [1, 2]. If this pattern continues, what is its average rate of change on [2, 3]?

Answer: 11

For a quadratic, average rates of change over consecutive equal-length intervals change by a constant amount; here it increases by 4, so the next value is 11.

Which is a polynomial of least degree with real coefficients that has 2 and 1 + i as zeros?

Answer: x³ − 4x² + 6x − 4

Complex zeros of real-coefficient polynomials come in conjugate pairs, so 1 − i is also a zero, and (x − 2)(x² − 2x + 2) = x³ − 4x² + 6x − 4.

The point (4, 5) lies on the graph of y = f(x). Which point must lie on the graph of g(x) = −f(x − 3) + 2?

Answer: (7, −3)

Replacing x with x − 3 shifts the graph right 3, giving x = 7; the negative reflects the y-value to −5 and +2 raises it to −3.

What is the greatest possible number of relative extrema (turning points) of a polynomial function of degree 6?

Answer: 5

A polynomial of degree n has at most n − 1 turning points, so a degree 6 polynomial has at most 5.

Let f(x) = (x + 1)²(x − 3)³. Which describes the behavior of the graph of f at x = 3?

Answer: It crosses the x-axis and flattens near the zero

The zero x = 3 has odd multiplicity 3, so the graph crosses the axis, and because the multiplicity is greater than 1 it flattens near the intercept.

On an interval, a function's average rates of change over consecutive equal-length subintervals are all positive and increasing. What does this indicate about the graph of the function on that interval?

Answer: Increasing and concave up

Positive rates of change mean the function is increasing, and rates that grow larger mean the graph is concave up.

What is the remainder when f(x) = x³ − 2x² + 5x − 1 is divided by x − 2?

Answer: 9

By the Remainder Theorem, the remainder equals f(2) = 8 − 8 + 10 − 1 = 9.

What is the range of the quadratic function g(x) = −(x + 2)² + 5?

Answer: (−∞, 5]

The graph of y = x² is shifted left 2, reflected over the x-axis, and shifted up 5, so it opens downward with vertex (−2, 5) and maximum value 5.