Algebra 2 Forms of Quadratic Functions Test Solutions and Scoring (PDF)

Algebra 2 math test solutions: Forms of Quadratic Functions – Maths PDF

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This quadratic forms test answers for Algebra 2 covers forms of quadratic functions (Common Core A-SSE.B.3a, A-SSE.B.3b, F-IF.C.8a), with a free PDF to print.

Use this answer key and scoring guide to grade the test on the forms of quadratic functions (A-SSE.B.3a). Each solution shows the full reasoning, and the scoring note explains how many points to award for correct setup, correct work, and a correct final answer. Partial credit rewards a sound method even when a small arithmetic slip changes the result.

Algebra 2 · Common Core State Standards: A-SSE.B.3a, A-SSE.B.3b, F-IF.C.8a

Quadratic forms test answers: worked solutions

Question 1 – Zeros from factored form [2 pts]

Zeros: $x = -6$ and $x = 2$. Axis: $x = \frac{-6 + 2}{2} = -2$.

Zeros -6, 2; axis x = -2

Scoring: 1 point for the zeros, 1 point for the axis.

Question 2 – Complete the square [3 pts]

Take half of the $x$-coefficient $6$: $3$. Square it: $9$.
Add and subtract $9$: $x^{2} + 6x + 1$ becomes $(x^{2} + 6x + 9) + 1 – 9$.
Write the perfect square and combine constants: $y = (x + 3)^{2} – 8$.

$(x + 3)^{2} – 8$; vertex $(-3, -8)$

Scoring: 1 point for the correct number to add and subtract, 1 point for vertex form, 1 point for the vertex.

Question 3 – Vertex from standard form [2 pts]

$x = -\frac{-12}{2(3)} = 2$. $g(2) = 12 – 24 + 7 = -5$. Since $a = 3 > 0$, it is a minimum.

Vertex (2, -5), a minimum

Scoring: 1 point for the x-coordinate, 1 point for the y-coordinate and classification.

Question 4 – Equation from a graph [2 pts]

$f(x) = a(x – 2)^{2} + 4$. Substitute $(0, 0)$: $0 = 4a + 4$, so $a = -1$.

y = -(x – 2)² + 4.

$f(x) = -(x – 2)^{2} + 4$

Scoring: 1 point for the setup, 1 point for a = -1 and the equation.

Question 5 – Projectile [2 pts]

$t = -\frac{96}{2(-16)} = 3$. $h(3) = -144 + 288 = 144$.

144 feet at t = 3 seconds

Scoring: 1 point for the time, 1 point for the height.

Question 6 – Revenue [2 pts]

Factored form: $R(p) = -2p(p – 30)$, zeros 0 and 30.

The vertex is halfway: $p = 15$, and $R(15) = -450 + 900 = 450$.

Vertex (15, 450).

R(p) = -2p(p – 30); price $15, revenue $450

Scoring: 1 point for the factored form, 1 point for the price.

Question 7 – Choose the form [2 pts]

$(x – 2)(x – 4) = 0$: zeros 2 and 4.
Axis $x = 3$; $f(3) = 9 – 18 + 8 = -1$.

(a) 2 and 4; (b) (3, -1)

Scoring: 1 point for each part.

Question 8 – Complete the square, a = 2 [3 pts]

Factor $2$ from the $x^{2}$ and $x$ terms: $2(x^{2} – 4x) + 3$.
Take half of the $x$-coefficient $-4$: $-2$. Square it: $4$.
Add and subtract $4$ inside the parentheses: the subtracted amount becomes $2 \cdot 4 = 8$ outside.
Write the perfect square and combine constants: $y = 2(x – 2)^{2} – 5$.

$2(x – 2)^{2} – 5$

Scoring: 1 point for factoring out 2, 1 point for completing the square correctly, 1 point for the final form.

Question 9 – Find the error [2 pts]

Vertex form is $(x – h)^{2} + k$. Here $x + 3 = x – (-3)$, so $h = -3$.

The vertex is (-3, -1)

Scoring: 1 point for the sign explanation, 1 point for the vertex.

Total score

Total

20 points

Key facts behind this quadratic forms test answers

Key facts

Standard form ax² + bx + c: y-intercept c.
Factored form a(x – r)(x – s): zeros r and s.
Vertex form a(x – h)² + k: vertex (h, k) and extreme value k.
Complete the square: add and subtract (b/2a)².
a > 0 opens up (minimum); a < 0 opens down (maximum).

How to use this quadratic forms test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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