Q: Given the polynomial function $f(x) = 2x^4 - 6x^3 + 6x - 2$, what is the end behavior of the graph as $x \to \infty$ and $x \to -\infty$?

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

Question 1

Given the polynomial function $f(x) = 2x^4 - 6x^3 + 6x - 2$, what is the end behavior of the graph as $x 	o \infty$ and $x 	o -\infty$?

Accepted Answer

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

Question 2

For the polynomial $f(x) = 3x^2 + 12x + 12$, what is the $y$-intercept?

Accepted Answer

$y = 12$

Question 3

Which of the following best describes the domain of any polynomial function?

Accepted Answer

$(-\infty, \infty)$

Question 4

Given $f(x) = (x-3)^2(x+1)$, what is the multiplicity of the zero at $x = 3$ and how does the graph behave at this point?

Accepted Answer

Multiplicity 2; the graph touches and turns at the $x$-axis

Question 5

A chemical reaction rate is modeled by $f(x) = x^3 - 2x^2 + x$. What are the $x$-intercepts of the function?

Accepted Answer

$x = 0, 1, 0$

Question 6

For $f(x) = x^3 - 4x$, which intervals should be used to analyze the graph between the $x$-intercepts?

Accepted Answer

$(-\infty, 0), (0, \infty)$

Question 7

A medical dosage model is given by $f(x) = x^2 - 6x + 8$. What is the range of this function?

Accepted Answer

$[4, \infty)$

Question 8

Which of the following is NOT a step in graphing a polynomial function according to the study materials?

Accepted Answer

Calculate the derivative at each $x$-intercept

Question 9

If $f(x) = x^4 - 2x^2 + 1$, what is the $y$-intercept and how do you find it?

Accepted Answer

$y = 1$, by evaluating $f(0)$

Question 10

A polynomial $f(x)$ has zeros at $x = -2$ (multiplicity 1), $x = 1$ (multiplicity 2), and $x = 3$ (multiplicity 1). Which zero does the graph touch but not cross?

Accepted Answer

$x = 1$

Question 11

For $f(x) = x^3 - 3x^2 + 2x$, what is the $y$-intercept?

Accepted Answer

$y = 0$

Question 12

Which of the following best describes the range of $f(x) = x^3$?

Accepted Answer

$(-\infty, \infty)$

Question 13

A polynomial function $f(x)$ has degree 4 and a positive leading coefficient. What is the end behavior?

Accepted Answer

Both ends rise

Question 14

If $f(x) = (x-2)^3$, what is the behavior of the graph at $x = 2$?

Accepted Answer

Crosses the $x$-axis at $x = 2$

Question 15

Which of the following is a correct step for determining intervals of unknown behavior in a polynomial graph?

Accepted Answer

Break the graph at each $x$-intercept

Question 16

For $f(x) = x^2(x-1)$, what is the multiplicity of the zero at $x = 0$?

Accepted Answer

2

Question 17

A real-world population model is given by $f(x) = -x^2 + 4x - 3$. What is the maximum value of the function?

Accepted Answer

$f(2) = 3$

Question 18

Which of the following is true about the graph of $f(x) = x^2$?

Accepted Answer

It is symmetric about the $y$-axis

Question 19

For $f(x) = x^3 - 2x^2 + x$, what is the degree of the polynomial and what does it tell you about the number of turning points?

Accepted Answer

Degree 3; up to 2 turning points

Question 20

A polynomial $f(x)$ has zeros at $x = -1$ (multiplicity 2) and $x = 2$ (multiplicity 1). Which statement is true about the graph at $x = -1$?

Accepted Answer

The graph touches and turns at $x = -1$

Graphing Polynomial Functions: Intervals, End Behavior, and Domain/Range

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