SAT Quadratic Equations and Parabolas: Practice Questions & Study Guide
Solving quadratics, identifying the vertex and intercepts of parabolas, and interpreting these features in applied contexts.
Understanding Quadratic Equations and Parabolas
Quadratic equations and their parabolas connect several Advanced Math skills. A quadratic function f(x) = ax^2 + bx + c produces a parabola that opens upward when a > 0 and downward when a < 0. Key features include the vertex, axis of symmetry, y-intercept c, and x-intercepts found by solving ax^2 + bx + c = 0.
Vertex form f(x) = a(x - h)^2 + k reveals the vertex (h, k) and axis of symmetry x = h. From standard form, use h = -b/(2a), then evaluate k = f(h). In a context such as projectile height or area optimization, distinguish the input at the vertex from the maximum or minimum output.
The discriminant b^2 - 4ac determines the number of real zeros: a positive value gives two, zero gives one repeated zero, and a negative value gives none. This relationship can be represented in an equation or in how a parabola meets the x-axis.
The roots also satisfy useful coefficient relationships: for ax^2 + bx + c = 0, their sum is -b/a and their product is c/a. These can connect a factored form or a pair of roots back to the coefficients.
Key Rules & Formulas
Memorize these rules — they come up directly in practice questions.
Standard form: f(x) = ax^2 + bx + c. The vertex x-coordinate is h = -b/(2a).
For f(x) = 2x^2 - 8x + 3: h = -(-8)/(2*2) = 8/4 = 2. Then f(2) = 8 - 16 + 3 = -5, so vertex is (2, -5).
Vertex form: f(x) = a(x - h)^2 + k. Vertex is (h, k); axis of symmetry is x = h.
f(x) = 3(x - 1)^2 + 4 has vertex (1, 4) and axis of symmetry x = 1.
Discriminant b^2 - 4ac: > 0 means 2 real zeros; = 0 means 1 repeated zero; < 0 means no real zeros.
For x^2 - 4x + 4 = 0: discriminant = 16 - 16 = 0 → one repeated zero at x = 2.
Sum of roots = -b/a; product of roots = c/a (for ax^2 + bx + c = 0).
For 2x^2 - 6x + 4 = 0: sum of roots = 6/2 = 3; product = 4/2 = 2. Roots are 1 and 2 ✓.
If zeros are r and s, the quadratic (with leading coefficient a) can be written as a(x - r)(x - s).
Zeros at x = -1 and x = 4 with a = 1: f(x) = (x + 1)(x - 4) = x^2 - 3x - 4.
Quadratic Equations and Parabolas Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
What is the vertex of the parabola y = (x - 3)^2 + 5?
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Correct answer: A. (3, 5)
Explanation
The equation is in vertex form y = (x - h)^2 + k, where (h, k) is the vertex. Here h = 3 and k = 5, so the vertex is (3, 5).
The parabola y = x^2 - 4x + 3 has two x-intercepts. What are they?
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Correct answer: B. x = 1 and x = 3
Explanation
Set y = 0: x^2 - 4x + 3 = 0 → (x-1)(x-3) = 0 → x = 1 or x = 3.
A parabola opens downward and has vertex (2, 8). Which of the following correctly describes the function?
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Correct answer: C. f(x) = -(x - 2)^2 + 8 and has a maximum value of 8
Explanation
A downward-opening parabola has a negative leading coefficient, so the form is -(x-2)^2 + 8. The vertex (2, 8) is a maximum since the parabola opens downward.
What is the x-coordinate of the vertex of y = 2x^2 - 12x + 7?
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Correct answer: D. x = 3
Explanation
Use h = -b/(2a) = -(-12)/(2*2) = 12/4 = 3. The vertex's x-coordinate is 3.
A ball is thrown upward and its height in feet after t seconds is given by h(t) = -16t^2 + 64t + 4. What is the maximum height reached by the ball?
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Correct answer: A. 68 feet
Explanation
The maximum occurs at t = -b/(2a) = -64/(2*(-16)) = -64/(-32) = 2 seconds. The maximum height is h(2) = -16(4) + 64(2) + 4 = -64 + 128 + 4 = 68 feet.
If the sum of the roots of 3x^2 + bx - 12 = 0 is 4, what is the value of b?
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Correct answer: B. b = -12
Explanation
By Vieta's formulas, sum of roots = -b/a = -b/3 = 4 → -b = 12 → b = -12.
For the equation x^2 + (k-2)x + (k+1) = 0 to have no real solutions, which condition must be true?
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Correct answer: C. (k-2)^2 - 4(k+1) < 0
Explanation
No real solutions means the discriminant is negative: b^2 - 4ac < 0, where b = k-2, a = 1, c = k+1. So (k-2)^2 - 4(1)(k+1) < 0.
The parabola y = ax^2 + bx + c passes through (0, 3), (1, 0), and (-1, 8). What is the value of a + b + c?
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Correct answer: D. 0
Explanation
From (0,3): c = 3. From (1,0): a + b + c = 0. From (-1,8): a - b + c = 8. Since a + b + c = 0, the answer is 0. (We can verify: a - b + 3 = 8 → a - b = 5; a + b = -3; adding: 2a = 2, a=1, b=-4. Check (1,0): 1-4+3 = 0 ✓. Check (-1,8): 1+4+3 = 8 ✓.)
If one root of 2x^2 - 9x + k = 0 is x = 4, what is the other root?
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Correct answer: A. x = 1/2
Explanation
Substitute x = 4: 2(16) - 9(4) + k = 0 → 32 - 36 + k = 0 → k = 4. Now the equation is 2x^2 - 9x + 4 = 0. By Vieta's, sum of roots = 9/2, so the other root = 9/2 - 4 = 1/2.
The equation x^2 - 8x + (m + 1) = 0 has two positive real roots, counting a repeated root twice. Which of the following gives all possible values of m?
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Correct answer: C. -1 < m ≤ 15
Explanation
For real roots counting multiplicity, the discriminant must satisfy 64 - 4(m+1) ≥ 0, so m ≤ 15. For both roots to be positive, their product m+1 must be positive, so m > -1; their sum is 8. Combined, -1 < m ≤ 15. At m = 15, the discriminant is 0 and x = 4 is a repeated positive root.
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Common Mistakes to Avoid
Use this checklist to catch common traps while working through Quadratic Equations and Parabolas questions, then add your own patterns as you review missed answers.
- !Using h = b/(2a) instead of h = -b/(2a)—forgetting the negative sign is the most common vertex error.
- !Confusing vertex form (x - h)^2: the formula is (x - h), so vertex at x = 3 appears as (x - 3)^2, NOT (x + 3)^2.
- !Solving a quadratic by taking the square root and forgetting the ± (e.g., from x^2 = 9, writing x = 3 but not x = -3).
- !Reporting the x-coordinate of a contextual vertex when the question asks for the maximum or minimum value, which is the y-coordinate.
- !Forgetting that the y-intercept is c (when x = 0, f(0) = c) and confusing it with other features.
Strategy Tips: Quadratic Equations and Parabolas
For any parabola question, immediately identify what feature you need (vertex? zeros? intercept?) and use the most direct method for that feature.
Use the graphing calculator to graph the parabola and read off the vertex and intercepts—this is often faster than algebra, especially for finding the vertex.
For an unrestricted quadratic, compare the vertex output with what the question asks. If the context restricts the domain, also check the relevant endpoints.
For 'exactly one solution' questions, set the discriminant equal to zero: b^2 - 4ac = 0 and solve for the parameter.
Other Advanced Math Subtopics
Equivalent Expressions
Rewriting algebraic expressions into equivalent forms through factoring, expanding, and applying algebraic identities.
Nonlinear Equations in One Variable
Solving quadratic, radical, and rational equations, and understanding the conditions under which extraneous solutions arise.
Nonlinear Functions
Understanding polynomial, radical, and rational functions—evaluating them, identifying their key features, and interpreting them in context.
Exponential Functions
Modeling growth and decay with exponential functions, interpreting the base and exponent, and comparing exponential to linear growth.
Master Quadratic Equations and Parabolas
Continue with this skill in the question bank, or test how it holds up under section timing before returning to the surrounding Adv. Math skills.