Q: A rocket's acceleration (in Gs) over 200 seconds is modeled by the table: $$f(0) = 1.34$$, $$f(200) = 2.2. $$What is the average rate of change of acceleration?

$$\\frac{2.2 - 1.34}{200 - 0} = 0.0043$$

Q: If a polynomial function $$f(x) is $$even, which of the following must be true?

$$f(x) = f(-x)$$ \\text{ for all } x$

Q: If a polynomial function $$f(x)$$ has a leading coefficient that is negative and an odd degree, what is the end behavior as $$x \\to \\infty$$?

$$f(x) \\to -1 as$$ $$x \\to \\infty$$

Q: Given the function $$f(x) = x^3$ - 3x + 1$$, what is the domain and range?

Domain: $$x \\in \\mathbb{R}$$, Range: $$f(x) \\in \\mathbb{R}$$

Question 1

According to the Location Principle, if $$f(a)  0$$ for a polynomial function $$f(x)$$, what can be concluded about the interval $$[a, b]$$?

Accepted Answer

There is at least one real zero between $$x = a$$ and $$x = b.$$

Question 2

Given the polynomial $$f(x) = x^4$ - 2x^3$ - x^2$ + 1$$, use the Location Principle and the table below to estimate the intervals where the real zeros are located:

$$x$$: $$-2$$, $$-1$$, $$0$$, $$1$$, $$2$$, $$3$$, $$4$$

$$f(x)$$: $$29$$, $$3$$, $$1$$, $$-1$$, $$-3$$, $$19$$, $$113$$

Between which consecutive integer values of $$x$$ does a real zero occur?

Accepted Answer

Between $$x = 0$$ and $$x = 1$$, and between $$x = 2$$ and $$x = 3$$

Question 3

A polynomial function $$f(x)$$ has a degree $$n. $$What is the maximum number of relative extrema (maxima or minima) it can have?

Accepted Answer

$$n - 1$$

Question 4

Given $$f(x) = x^3$ + x^2$ - 5x - 2$$, use a table to estimate the locations of the relative maximum and minimum. If $$f(-2) = 4$$, $$f(-1) = 3$$, $$f(0) = -2$$, $$f(1) = -5$$, $$f(2) = 0$$, which $$x-$$values are closest to the relative maximum and minimum?

Accepted Answer

Maximum near $$x = -2$$, minimum near $$x = 1$$

Question 5

A quartic regression model for U.S. backpack sales is $$y = 2x^4$ - 3.9x^3$ + 26.4x^2$ - 43.6x + 1139.9. $$What is the estimated sales in 2015, if $$x = 15$$ represents the year 2015?

Accepted Answer

$$y \approx 3523$$

Question 6

Which of the following statements about the end behavior of polynomial functions is correct?

Accepted Answer

The end behavior is determined only by the leading coefficient and the degree.

Question 7

Given the function $$f(x) = 0.0000903x^4$ - 0.0166x^3$ + 0.762x^2$ + 6.317x + 7.708$$ modeling the number of pilots in thousands, what is the relevant domain for this function?

Accepted Answer

$$x \geq 0$$

Question 8

A rocket's acceleration (in Gs) over 200 seconds is modeled by the table: $$f(0) = 1.34$$, $$f(200) = 2.2. $$What is the average rate of change of acceleration?

Accepted Answer

$$\frac{2.2 - 1.34}{200 - 0} = 0.0043$$

Question 9

If a polynomial function $$f(x) is $$even, which of the following must be true?

Accepted Answer

$$f(x) = f(-x)$$ \text{ for all } x$

Question 10

A canister's volume is modeled by $$V(x) = \pi(x^3$ + 3x^2$) + \frac{4}{3}\pi x^3$, where x$ is the radius in millimeters. What is a restriction on the domain of this model?

Accepted Answer

$$x \u003e 0$$

Question 11

Given the cubic function $$g(x)$$ with values $$g(2) = -2$$ and $$g(3) = -2$$, could there be a zero between $$x = 2$$ and $$x = 3$$?

Accepted Answer

No, because $$g(x)$$ does not change sign between $$x = 2$$ and $$x = 3$$

Question 12

If a polynomial function $$f(x)$$ changes from increasing to decreasing at $$x = a$$, what is $$f(a)$$ called?

Accepted Answer

A relative maximum

Question 13

Which polynomial function best models a data set with 6 turning points and odd degree?

Accepted Answer

A seventh-degree polynomial

Question 14

If a polynomial function $$f(x)$$ has a leading coefficient that is negative and an odd degree, what is the end behavior as $$x \to \infty$$?

Accepted Answer

$$f(x) \to -1 as$$ $$x \to \infty$$

Question 15

A scatter plot of U.S. backpack sales over time is best fit by a quartic polynomial. What does this suggest about the trend in sales?

Accepted Answer

Sales increase rapidly after a certain point, showing a nonlinear trend.

Question 16

If a polynomial function $$f(x)$$ has no symmetry, which of the following is true?

Accepted Answer

It is neither even nor odd.

Question 17

Given the function $$f(x) = x^3$ - 3x + 1$$, what is the domain and range?

Accepted Answer

Domain: $$x \in \mathbb{R}$$, Range: $$f(x) \in \mathbb{R}$$

Question 18

If a polynomial function $$f(x)$$ has a y-intercept at $$(0, 7.708)$$ and no x-intercept in the relevant domain, what does this indicate about the function?

Accepted Answer

The function does not cross the x-axis in the relevant domain.

Analyzing Graphs of Polynomial Functions: Zeros, Extrema, and Modeling

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