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

Question 1

Given the polynomial $f(x) = (x + 2)^3 - 3$, what is the parent function, and what transformation has been applied to it?

Accepted Answer

Parent function: $f(x) = x^3$; transformation: shift left 2 units and down 3 units

Question 2

What is the degree of the polynomial $f(x) = x^4 - 2x^2 + 1$?

Accepted Answer

Degree 4

Question 3

A polynomial function has zeros at $x = -2$, $x = 3$, and $x = 6$. Which of the following is a possible function?

Accepted Answer

$f(x) = (x + 2)(x - 3)(x - 6)$

Question 4

Given $f(x) = -2(x - 1)^2 + 4$, which transformation is applied to the parent function $f(x) = x^2$?

Accepted Answer

Reflect over the x-axis, vertically stretch by 2, shift right 1 unit, and up 4 units

Question 5

Which of the following could be the graph of a polynomial with zeros at $x = -1$, $x = 2$, and $x = 4$?

Accepted Answer

A graph crossing the x-axis at $x = -1$, $x = 2$, and $x = 4$

Question 6

If $f(x) = x^3 - 2x^2 + x - 5$, what is the y-intercept of the function?

Accepted Answer

$-5$

Question 7

Which of the following is the domain of the rational function $f(x) = \dfrac{2x + 1}{x^2 - 4}$?

Accepted Answer

All real numbers except $x = 2$ and $x = -2$

Question 8

Given $f(x) = \dfrac{3x}{x - 2}$, what is the vertical asymptote?

Accepted Answer

$x = 2$

Question 9

For the rational function $f(x) = \dfrac{x^2 - 9}{x + 3}$, what is the x-intercept?

Accepted Answer

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

Question 10

Which of the following describes the end behavior of $f(x) = -x^4 + 2x^2 - 1$?

Accepted Answer

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

Question 11

A chemical reaction rate is modeled by $R(t) = 5t^2 - 3t + 2$. What is the instantaneous rate at $t = 1$?

Accepted Answer

$R(1) = 5(1)^2 - 3(1) + 2 = 4$

Question 12

Which of the following is the correct factorization of $x^3 - 4x^2 + 4x$?

Accepted Answer

$x(x^2 - 4x + 4)$

Question 13

A company’s profit is modeled by $P(x) = -2x^2 + 8x - 6$. What is the maximum profit, and at what value of $x$ does it occur?

Accepted Answer

Maximum at $x = 2$, $P(2) = 2$

Question 14

Which of the following is the correct graph for $f(x) = -x^2 + 4$?

Accepted Answer

A downward opening parabola with vertex at $(0, 4)$

Question 15

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

Accepted Answer

$x = -1$

Question 16

For the rational function $f(x) = \dfrac{x^2 - 1}{x - 1}$, what is the simplified form and its domain?

Accepted Answer

Simplified: $f(x) = x + 1$, domain: all real numbers except $x = 1$

Question 17

Which of the following is the correct equation for a polynomial with zeros at $x = 1$ (multiplicity 2) and $x = -3$?

Accepted Answer

$f(x) = (x - 1)^2(x + 3)$

Question 18

A rational function $f(x) = \dfrac{2x^2 + 3x - 2}{x - 1}$ has a hole at $x = 1$. What is the y-value of the hole?

Accepted Answer

$f(1) = 2(1)^2 + 3(1) - 2 = 3$

Question 19

Which of the following is the correct parent function for $f(x) = (x - 2)^5 + 4$?

Accepted Answer

$f(x) = x^5$

Polynomial and Rational Functions: Study Notes for College Algebra

Study Guide - Practice Questions