Q: Which of the following best describes the end behavior of the polynomial function $$f(x) = 2x^4$ - 5x^3$ + x - 7$$?

As $$x \to \infty$$, $$f(x) \to \infty$$; as $$x \to -\infty$$, $$f(x) \to \infty$$

Q: If a polynomial function $$f(x)$$ has degree 4 and a positive leading coefficient, what is the end behavior of its graph?

Both ends go up as $$x \to \pm\infty$$

Question 1

Which of the following best describes the end behavior of the polynomial function $$f(x) = 2x^4$ - 5x^3$ + x - 7$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o \infty$$; as $$x 	o -\infty$$, $$f(x) 	o \infty$$

Question 2

Given the polynomial $$f(x) = (x+2)^2$(x-4)$$, what is the behavior of the graph at $$x = -2$$?

Accepted Answer

The graph touches the x-axis at $$x = -2$$ and turns around

Question 3

A polynomial function of degree 5 can have at most how many turning points?

Accepted Answer

$$4$$

Question 4

If a polynomial function $$f(x)$$ has a zero at $$x = 3$$ with multiplicity 3, what does the graph do at $$x = 3$$?

Accepted Answer

It crosses the x-axis and flattens out at $$x = 3$$

Question 5

Which of the following is the correct process for graphing a polynomial function?

Accepted Answer

Determine end behavior, find intercepts, analyze zeros and multiplicities, find turning points, and sketch the graph

Question 6

Given $$f(x) = x^4$ - 2x^2$ + 1$$, what is the y-intercept of the function?

Accepted Answer

$$1$$

Question 7

A cubic model is used to predict the cost $$C (in $$thousands of dollars) of printing $$x$$ thousand books: $$C(x) = 0.0035x^3$ - 0.0593x^2$ + 1.90x + 119.23. $$What is the predicted cost of printing 20 thousand books?

Accepted Answer

$$C(20) \approx 161.9$$

Question 8

Which of the following statements about the domain of a polynomial function is true?

Accepted Answer

The domain is all real numbers

Question 9

Suppose a polynomial function $$f(x)$$ has zeros at $$x = -1 ($$multiplicity 3) and $$x = 3 ($$multiplicity 1). Which statement is true about the graph at these points?

Accepted Answer

The graph crosses the x-axis at both $$x = -1$$ and $$x = 3$$

Question 10

A graphing utility shows the x-intercepts of a polynomial as $$x = -5.31, -1.10, -0.27, 1.48. $$What is the minimum possible degree of the polynomial?

Accepted Answer

$$4$$

Question 11

If a polynomial function $$f(x)$$ has degree 4 and a positive leading coefficient, what is the end behavior of its graph?

Accepted Answer

Both ends go up as $$x 	o \pm\infty$$

Question 12

A real-world scenario: The cost to produce $$x$$ thousand units of a product is modeled by a cubic function. If the cost increases rapidly after a certain point, what does this suggest about the leading coefficient of the cubic model?

Accepted Answer

The leading coefficient is positive

Question 13

Which of the following is NOT a step in graphing a polynomial function by hand?

Accepted Answer

Find the vertical asymptotes

Question 14

Given the polynomial $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is NOT a zero of the function?

Accepted Answer

$$x = 0$$

Question 15

A polynomial function $$f(x) is $$increasing on the interval $$(-\infty, -2)$$ and decreasing on $$(-2, 1). $$What can you conclude about $$x = -2$$?

Accepted Answer

$$x = -2 is a $$local maximum

Question 16

If a cubic function of best fit is used to model data, which of the following is a valid reason for choosing a cubic model over a linear or quadratic model?

Accepted Answer

The data has two turning points

Question 17

Suppose you use a graphing utility to find the minimum value of a polynomial function at $$x = 0.85$$ and $$f(0.85) = -8.5. $$What does this point represent?

Accepted Answer

A local minimum

Question 18

Which of the following statements about the range of a degree 4 polynomial function is true?

Accepted Answer

The range is bounded below or above, depending on the leading coefficient

Question 19

A company uses a cubic regression model to predict costs. If the model is $$C(x) = 0.0035x^3$ - 0.0593x^2$ + 1.90x + 119.23$$, what is the real-world meaning of the constant term $$119.23$$?

Accepted Answer

It is the fixed cost when no units are produced

Question 20

If a polynomial function $$f(x)$$ has a degree of 3 and a negative leading coefficient, what is the end behavior of the graph?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o \infty$$

Graphing Polynomial Functions and Building Cubic Models

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