As $x \to -\infty$, $f(x) \to +\infty$

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

Question 1

Which of the following statements is TRUE about the graph of any polynomial function $f(x)$?

Accepted Answer

It is continuous and smooth for all real $x$.

Question 2

Given the polynomial $f(x) = x^4 - 5x^2 + 4$, what is the end behavior of its graph as $x 	o -\infty$?

Accepted Answer

As $x 	o -\infty$, $f(x) 	o +\infty$

Question 3

Suppose $f(x) = (x-2)^3(x+1)^2$. Which of the following is TRUE about the zeros of $f(x)$?

Accepted Answer

$x = 2$ is a zero of multiplicity 3 and $x = -1$ is a zero of multiplicity 2

Question 4

A power function $f(x) = x^n$ is symmetric with respect to the y-axis if:

Accepted Answer

$n$ is even

Question 5

Which of the following best describes the number of turning points for a polynomial of degree $n$?

Accepted Answer

At most $n-1$ turning points

Question 6

Given the rational function $R(x) = \frac{x^2-4}{x-2}$, which of the following statements is correct?

Accepted Answer

There is a hole at $x = 2$

Question 7

A rational function $R(x) = \frac{p(x)}{q(x)}$ has a vertical asymptote at $x = a$ if:

Accepted Answer

$q(a) = 0$ and $p(a) 
eq 0$

Question 8

Consider the function $f(x) = -2x^6 + 2x^2$. Which of the following describes its end behavior as $x 	o \infty$?

Accepted Answer

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

Question 9

If $f(x) = x^3 - 4x^2 + x - 6$, what is the smallest possible degree of $f(x)$?

Accepted Answer

3

Question 10

Which of the following is the correct definition of a zero of a polynomial $f(x)$?

Accepted Answer

A value $c$ such that $(x-c)$ is a factor of $f(x)$

Question 11

Given the rational function $R(x) = \frac{x^2+1}{x^2-4}$, what are the vertical asymptotes?

Accepted Answer

$x = 2$ and $x = -2$

Question 12

Suppose you fit a quadratic polynomial to model the data for coal consumption over time. What is the general form of the quadratic polynomial?

Accepted Answer

$f(x) = ax^2 + bx + c$

Question 13

If a rational function $R(x)$ has a hole at $x = a$, which of the following must be true?

Accepted Answer

$p(a) = 0$ and $q(a) = 0$

Question 14

Which of the following is TRUE about the horizontal asymptote of $R(x) = \frac{2x^2 + 1}{x^2 - 3}$?

Accepted Answer

The horizontal asymptote is $y = 2$

Question 15

A polynomial of degree $n$ has how many complex zeros, counting multiplicity?

Accepted Answer

Exactly $n$ complex zeros

Question 16

Given $f(x) = x^5$, which of the following describes its symmetry?

Accepted Answer

Symmetric with respect to the origin

Question 17

Suppose $f(x) = x^2 + 4x + 3$. What are the real zeros of $f(x)$?

Accepted Answer

$x = -3$ and $x = 1$

Question 18

Which of the following is NOT a feature of a polynomial function?

Accepted Answer

The graph has sharp corners

Question 19

If $f(x) = x^3 - 3x^2 + 2x$, what is the sum of the multiplicities of all its real zeros?

Accepted Answer

3

Polynomial and Rational Functions: Concepts, Properties, and Applications

Study Guide - Practice Questions