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 associated parabolas are arguably the most heavily tested topic in the Advanced Math section. A quadratic function f(x) = ax^2 + bx + c produces a parabola that opens upward when a > 0 and downward when a < 0. The key features are the vertex (the maximum or minimum point), the axis of symmetry, the y-intercept (c), and the x-intercepts (zeros, found by solving ax^2 + bx + c = 0).
The vertex form f(x) = a(x - h)^2 + k directly reveals the vertex (h, k) and the axis of symmetry x = h. Converting from standard to vertex form requires completing the square. The vertex x-coordinate can also be found quickly using h = -b/(2a) without completing the square; then substitute to find k = f(h). The test frequently gives a quadratic in a real-world context (projectile height, area optimization) and asks for the maximum or minimum value—that's always the y-coordinate of the vertex.
For finding zeros, the most important concept is the relationship between the discriminant (b^2 - 4ac) and the number of x-intercepts: positive discriminant means two real zeros (parabola crosses the x-axis twice), zero discriminant means the vertex touches the x-axis exactly once (one repeated zero, the parabola is tangent to the axis), and negative discriminant means no real zeros (the parabola doesn't touch the x-axis). The test tests this graphically (asking which graph has a given discriminant property) and algebraically (asking for the value of a parameter that gives exactly one solution).
Vieta's formulas provide a shortcut for sum and product of roots: for ax^2 + bx + c = 0, the sum of roots is -b/a and the product of roots is c/a. If the test gives you two roots and asks for a coefficient, these relations let you write the answer immediately without using the quadratic formula.
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?
Show explanation
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: A. 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: A. 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: B. 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: B. 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: A. 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: A. (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: A. 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?
Show explanation
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 real roots, both positive. Which of the following gives all possible values of m?
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Correct answer: C. -1 < m ≤ 15
Explanation
For two real roots: discriminant ≥ 0: 64 - 4(m+1) ≥ 0 → 60 ≥ 4m → m ≤ 15. For both roots positive: product of roots = m+1 > 0 → m > -1, and sum of roots = 8 > 0 ✓ (automatically). Combined: -1 < m ≤ 15.
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Common Mistakes to Avoid
These are the most frequent errors students make on Quadratic Equations and Parabolas questions. Knowing them in advance prevents costly point losses.
- !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).
- !Misinterpreting a contextual maximum/minimum: students often report the x-coordinate when the question asks for the maximum VALUE (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.
When a contextual question asks for maximum or minimum, the answer is always the y-coordinate of the vertex (k in vertex form, or f(-b/2a) in standard form).
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
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