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Quadratic and polynomial functions — ACT practice questions

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• 45 questions on the paper
• 39 practice questions in the app
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Practice 39 Quadratic and polynomial functions questions in the app

What this topic is

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.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

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

Acrosses the $x$-axis✗ Not this oneBtouches the $x$-axis without crossing✓ CorrectCdoes not intersect the $x$-axis✗ Not this oneDhas a local minimum✗ Not this one

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Answer: B — touches the $x$-axis without crossing

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

Question 2Easier

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

A$q(x)=-3(x-1)^2$✓ CorrectB$q(x)=3(x-1)^2$✗ Not this oneC$q(x)=-3(x+1)^2$✗ Not this oneD$q(x)=-(3x-1)^2$✗ Not this one

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Answer: A — $q(x)=-3(x-1)^2$

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

Question 3Easier

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

A$-4$ and 2✗ Not this oneB$-2$ and 4✓ CorrectC2 and 4✗ Not this oneD$-4$ and $-2$✗ Not this one

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Answer: B — $-2$ and 4

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

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