Quadratic Functions and Modeling Practice Questions
Common Core (US) · Common Core Algebra 1 · 148 free MCQs with instant results and detailed explanations.
148
Total
46
Easy
78
Medium
24
Hard
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Sample Questions from Quadratic Functions and Modeling
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Q1
Easy
What is the vertex of the quadratic function f(x) = x^2 - 6x + 8?
A.
(3, -1)
B.
(3, -7)
C.
(2, 4)
D.
(2, -4)
Show Answer & Explanation
Correct Answer: A
The vertex form of a quadratic function can be found using the formula x = -b/(2a). Here, a = 1 and b = -6, giving x = 3. Substituting x back into the function, f(3) = 3^2 - 6(3) + 8 = -1, thus the vertex is (3, -1).
Q2
Easy
If the roots of the quadratic equation x^2 - 5x + 6 = 0 are represented as (p, q), what is the sum of the roots?
A.
5
B.
6
C.
3
D.
4
Show Answer & Explanation
Correct Answer: A
According to Vieta's formulas, the sum of the roots of the quadratic equation ax^2 + bx + c = 0 is given by -b/a. Here, b = -5 and a = 1, so the sum is -(-5)/1 = 5.
Q3
Easy
What is the vertex of the quadratic function f(x) = x^2 - 4x + 3?
A.
(2, -1)
B.
(2, -2)
C.
(1, 2)
D.
(3, 0)
Show Answer & Explanation
Correct Answer: B
To find the vertex of the quadratic function, we can use the vertex formula x = -b/(2a). Here, a = 1 and b = -4, so x = 4/2 = 2. We then plug x = 2 back into the function: f(2) = 2^2 - 4(2) + 3 = -2. Thus, the vertex is (2, -2).
Q4
Medium
What is the vertex of the quadratic function f(x) = 2x² - 8x + 5?
A.
(2, -3)
B.
(2, 5)
C.
(4, -3)
D.
(0, 5)
Show Answer & Explanation
Correct Answer: A
To find the vertex, use the formula x = -b/(2a). Here, a = 2 and b = -8, which gives x = 2. Substituting x back into the function yields f(2) = 2(2)² - 8(2) + 5 = -3, hence the vertex is (2, -3).
Q5
Medium
If the roots of the quadratic equation x² - 7x + k = 0 are 3 and 4, what is the value of k?
A.
12
B.
7
C.
6
D.
9
Show Answer & Explanation
Correct Answer: A
The roots of the equation give the relation k = product of roots = 3 * 4 = 12. Therefore, k must be 12.
Q6
Medium
Which of the following represents the standard form of a quadratic function?
A.
f(x) = ax² + bx + c
B.
f(x) = a(x - h)² + k
C.
f(x) = ax + b
D.
f(x) = a(x + b)(x + c)
Show Answer & Explanation
Correct Answer: A
The standard form of a quadratic function is correctly represented by f(x) = ax² + bx + c, where a, b, and c are constants.
Q7
Medium
A quadratic function has a maximum value of 10 and opens downwards. What can be inferred about the value of a in the function f(x) = ax² + bx + c?
A.
a < 0
B.
a > 0
C.
a = 10
D.
a = 0
Show Answer & Explanation
Correct Answer: A
If a quadratic function opens downwards, the coefficient a must be negative (a < 0). This is because a positive a would make the parabola open upwards, which would not allow for a maximum value.
Q8
Hard
A projectile is launched from the ground with an initial velocity of 40 m/s at an angle of 45 degrees. The height of the projectile as a function of time, t, can be modeled by the equation h(t) = -4.9t^2 + 40√2t. What is the maximum height reached by the projectile?
A.
80 m
B.
100 m
C.
160 m
D.
200 m
Show Answer & Explanation
Correct Answer: B
The maximum height of a projectile modeled by a quadratic equation occurs at the vertex. Using the formula h(t) = -4.9t^2 + 40√2t, we find the vertex time using t = -b/(2a) = -40√2/(2 * -4.9) = 40√2 / 9.8. Substitute this t value back into h(t) to find the height, which results in h(t) = 100 m.
Q9
Hard
Consider the quadratic function f(x) = 2x^2 - 8x + 6. What is the vertex of this parabola?
A.
(2, -2)
B.
(2, -4)
C.
(4, -10)
D.
(4, 2)
Show Answer & Explanation
Correct Answer: B
The vertex of a parabola given in standard form f(x) = ax^2 + bx + c can be found using the formula x = -b/(2a). Here, a = 2 and b = -8, so x = -(-8)/(2*2) = 2. Plugging x = 2 back into the function gives f(2) = 2(2)^2 - 8(2) + 6 = -4. Thus, the vertex is (2, -4).
Q10
Hard
A ball is thrown upwards from the top of a building with a quadratic height function h(t) = -16t^2 + 64t + 80. What is the maximum height the ball reaches?
A.
144 feet
B.
160 feet
C.
144 feet after 2 seconds
D.
160 feet after 2 seconds
Show Answer & Explanation
Correct Answer: A
To find the maximum height, we first find the vertex of the quadratic function h(t). The time at which the maximum height occurs is t = -b/(2a). Here, a = -16 and b = 64, thus t = -64/(2*-16) = 2 seconds. Plugging t = 2 into the height function gives h(2) = -16(2)^2 + 64(2) + 80 = 144 feet. Therefore, the maximum height is 144 feet.
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Quadratic Functions and Modeling — Common Core (US) Common Core Algebra 1 Practice Questions Online
This page contains 148 practice MCQs for the chapter Quadratic Functions and Modeling in Common Core (US) Common Core Algebra 1. The questions are organized by difficulty — 46 easy, 78 medium, 24 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.