Algebra 1 Practice Test: Quadratic Equations and Functions
- Quadratic Equations and Functions
- 15 questions
- 20 minutes
These 15 questions practice solving quadratic equations by factoring, square roots, completing the square and the quadratic formula, the discriminant, vertex and axis of symmetry, graphs, transformations and word problems. The vertex of y = ax2 + bx + c is at x = −b2a.
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Question 1
What are the solutions of x2 − 7x + 12 = 0?
Correct answer: D
Two numbers with product 12 and sum −7 are −3 and −4, so (x − 3)(x − 4) = 0 and x = 3 or x = 4. The numbers −3 and −4 appear in the factors, but the solutions have the opposite signs.
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Question 2
What are the solutions of 2x2 = 50?
Correct answer: C
Divide by 2: x2 = 25, so x = 5 or x = −5. Taking the square root of 50 without dividing by 2 gives ±52, and forgetting the negative root gives "5 only".
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Question 3
What are the solutions of x2 + 4x − 3 = 0?
Correct answer: D
x = −4 ± 16 + 122 = −4 ± 272 = −2 ± 7. Dividing only the −4 by 2 gives −2 ± 27.
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Question 4
How many real solutions does 3x2 − 6x + 5 = 0 have?
Correct answer: A
The discriminant is b2 − 4ac = 36 − 60 = −24. A negative discriminant means there are no real solutions; the parabola does not touch the x-axis.
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Question 5
What is the vertex of the graph of y = x2 − 6x + 5?
Correct answer: C
x = −−62(1) = 3, and y = 9 − 18 + 5 = −4. Using x = −3 (a sign error) gives y = 32.
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Question 6
Which statement about the function f(x) = −2(x + 1)2 + 8 is true?
Correct answer: A
In vertex form a(x − h)2 + k the vertex is (h, k) = (−1, 8). Because a = −2 is negative, the parabola opens downward, so 8 is a maximum. (x + 1) means h = −1, not 1.
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Question 7
What are the x-intercepts of the graph of y = (x − 2)(x + 5)?
Correct answer: A
Set each factor equal to 0: x − 2 = 0 gives x = 2, and x + 5 = 0 gives x = −5. The point (0, −10) is the y-intercept.
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Question 8
What is the axis of symmetry of the graph of y = 2x2 + 8x − 1?
Correct answer: C
x = −b2a = −84 = −2. Forgetting the 2 in the denominator gives x = −4.
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Question 9
The graph of a quadratic function is shown.
Which equation represents the graph?
Correct answer: D
The graph crosses the x-axis at −1 and 3, so the factors are (x + 1) and (x − 3), and it opens upward, so a > 0. Check the vertex: at x = 1, (2)(−2) = −4. With a negative sign the parabola would open downward.
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Question 10
A ball is thrown upward. Its height in feet after t seconds is h(t) = −16t2 + 64t + 5. What is the maximum height of the ball?
Correct answer: B
The maximum occurs at t = −642(−16) = 2. Then h(2) = −64 + 128 + 5 = 69 feet. The value 53 is the height after 1 second.
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Question 11
Which equation results from completing the square on x2 + 10x + 7 = 0?
Correct answer: B
Move the constant: x2 + 10x = −7. Add (102)2 = 25 to both sides: (x + 5)2 = 18. Adding 7 instead of subtracting it gives 32.
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Question 12
How does the graph of y = (x − 4)2 + 2 compare with the graph of y = x2?
Correct answer: C
The vertex moves from (0, 0) to (4, 2). Inside the square, x − 4 shifts the graph right, not left; the +2 outside shifts it up.
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Question 13
The length of a rectangle is 3 meters more than its width, and its area is 70 square meters. What is the width?
Correct answer: B
w(w + 3) = 70 gives w2 + 3w − 70 = 0, so (w + 10)(w − 7) = 0. A width cannot be negative, so w = 7. 10 meters is the length.
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Question 14
What is the average rate of change of f(x) = x2 − 3x from x = 1 to x = 4?
Correct answer: B
f(4) = 16 − 12 = 4 and f(1) = 1 − 3 = −2. The average rate of change is 4 − (−2)4 − 1 = 63 = 2. The value 6 is the change in f before dividing by 3.
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Question 15
Which equation has the solutions x = 3 and x = −12?
Correct answer: A
The factors are (x − 3) and (2x + 1), and (x − 3)(2x + 1) = 2x2 − 5x − 3. The equation 2x2 + 5x − 3 = 0 has solutions −3 and 12, with both signs reversed.
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