Study on the flyACT

Study on the fly›ACT›Mathematics › Quadratic and polynomial functions

MathematicsFree

Quadratic and polynomial functions — ACT practice questions

On the ACT Mathematics test, Quadratic and polynomial functions covers graphs and equations of quadratics and higher-degree polynomials, including intercepts, vertices, and behavior at a root. A student must convert among factored, standard, and vertex forms, read x-intercepts from linear factors, tell whether the graph touches or crosses the axis at a given root, and write a parabola from a vertex and another point. Items are multiple choice: intercept pairs, statements about touching or crossing, or candidate equations. Traps include mixing intercepts with the vertex, dropping a sign, and treating a repeated factor as a simple crossing.

  • Section: Mathematics 45 questions on the paper
  • 42 practice questions in the app

Sample questions

Question 1

The polynomial function pp is defined by p(x)=x2(x−5)p(x)=x^{2}(x-5). At x=0x=0, the graph of y=p(x)y=p(x):

  1. Acrosses the xx-axis
  2. Btouches the xx-axis without crossing
  3. Cdoes not intersect the xx-axis
  4. Dhas a local minimum
Show answer

Answer: B — touches the xx-axis without crossing

The factor xx appears twice, so x=0x=0 is a zero of even multiplicity. The graph meets the xx-axis there and turns back instead of crossing.

Question 2

The graph of y=q(x)y=q(x) is a parabola with vertex (1,0)(1,0) that passes through (0,−3)(0,-3). Which of the following equations defines qq?

  1. Aq(x)=−3(x−1)2q(x)=-3(x-1)^2
  2. Bq(x)=3(x−1)2q(x)=3(x-1)^2
  3. Cq(x)=−3(x+1)2q(x)=-3(x+1)^2
  4. Dq(x)=−(3x−1)2q(x)=-(3x-1)^2
Show answer

Answer: A — q(x)=−3(x−1)2q(x)=-3(x-1)^2

Vertex (1,0)(1,0) means q(x)=a(x−1)2q(x)=a(x-1)^2. The point (0,−3)(0,-3) gives a(0−1)2=−3a(0-1)^2=-3, so a=−3a=-3. Thus q(x)=−3(x−1)2q(x)=-3(x-1)^2.

Question 3

The function ff is defined by f(x)=(x−4)(x+2)f(x)=(x-4)(x+2). What are the xx-intercepts of the graph of y=f(x)y=f(x)?

  1. A-4 and 2
  2. B-2 and 4
  3. C2 and 4
  4. D-4 and -2
Show answer

Answer: B — -2 and 4

Set f(x)=0f(x)=0: x−4=0x-4=0 or x+2=0x+2=0, so x=4x=4 and x=−2x=-2. Sign errors swap the intercepts.

Practice 42 Quadratic and polynomial functions questions in the app

More ACT topics

Mathematics

Everything on the ACT

Study on the fly is an independent study tool and is not affiliated with, sponsored by, or endorsed by any testing organization, school, or government body.

ACT® is a registered trademark of ACT Education Corp., which is not affiliated with, and does not endorse, this product.

All practice questions and passages are original works created for Study on the fly and are not reproduced from any official test.