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Quadratic and polynomial functions — ACT practice questions

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Practice 42 Quadratic and polynomial functions questions in the app

What this topic is

Graph features are the focus of Quadratic and polynomial functions.

In factored form, each factor shows an x-intercept, so y = (x - 1)(x + 3) crosses the x-axis at 1 and -3. Vertex form, y = a(x - h)² + k, shows the turning point (h, k) directly. A repeated factor changes the graph's behavior at a zero: a squared factor makes the graph touch the axis and turn back instead of crossing it.

ACT Mathematics asks for equations matching given intercepts or a vertex. Sign errors, such as reading the factor x - 1 as an intercept at -1, appear among wrong choices.

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 $y = p (x)$ :

crosses the

x

-axis✗ Not this onetouches the

x

-axis without crossing✓ Correctdoes not intersect the

x

-axis✗ Not this onehas a local minimum✗ Not this one

Show answer

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 $q$ ?

q (x) = - 3 (x - 1)^{2}

✓ Correct

q (x) = 3 (x - 1)^{2}

✗ Not this one

q (x) = - 3 (x + 1)^{2}

✗ Not this one

q (x) = - (3 x - 1)^{2}

✗ Not this one

Show answer

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 $y = f (x)$ ?

- 4

and 2✗ Not this one

- 2

and 4✓ Correct2 and 4✗ Not this one

- 4

and

- 2

✗ Not this one

Show answer

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.

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 42 Quadratic and polynomial functions questions in the app

Quadratic and polynomial functions — ACT practice questions

What this topic is

Sample questions

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Mathematics