Quadratic Functions
Grade 10 · Math · Study Guide
Understanding parabolas, vertex form, and the key features of quadratic functions.
A quadratic function has the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a U-shaped curve called a parabola. The direction it opens — up or down — depends entirely on the sign of a. If a > 0, the parabola opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point.
The most powerful form for understanding a parabola's shape is vertex form: f(x) = a(x - h)² + k. Here, (h, k) is the vertex — the highest or lowest point on the curve. The axis of symmetry is the vertical line x = h, which divides the parabola into two mirror-image halves. From standard form ax² + bx + c, you can find the vertex's x-coordinate using h = -b/(2a).
The x-intercepts (also called roots or zeros) are where the parabola crosses the x-axis. You find them by solving ax² + bx + c = 0. The discriminant, b² - 4ac, tells you how many real solutions exist: positive means two x-intercepts, zero means one (the vertex touches the axis), and negative means none — the parabola floats entirely above or below the x-axis.
Real-world applications are everywhere. Projectile motion — a ball thrown in the air — follows a quadratic path. Business analysts use quadratic models to find profit-maximizing price points. Engineers use parabolic shapes in satellite dishes and bridges because of their reflective and structural properties.
| Discriminant Value | Number of Real Roots | Graph Behavior |
|---|---|---|
| b² - 4ac > 0 | 2 real roots | Parabola crosses x-axis twice |
| b² - 4ac = 0 | 1 real root | Parabola touches x-axis at vertex |
| b² - 4ac < 0 | 0 real roots | Parabola does not cross x-axis |
Key terms
- Parabola
- The U-shaped graph of a quadratic function.
- Vertex
- The maximum or minimum point of a parabola, located at (h, k) in vertex form.
- Axis of Symmetry
- The vertical line x = h that divides the parabola into two symmetric halves.
- Discriminant
- The expression b² - 4ac that determines the number of real roots of a quadratic.
- Vertex Form
- f(x) = a(x - h)² + k; reveals the vertex and direction of opening directly.
Flashcards
- Standard form of a quadratic
- f(x) = ax² + bx + c where a ≠ 0
- Vertex form
- f(x) = a(x - h)² + k; vertex is at (h, k)
- Axis of symmetry formula
- x = -b/(2a) from standard form
- Discriminant
- b² - 4ac; determines number and type of roots
- What does a > 0 mean for a parabola?
- Opens upward, has a minimum vertex
- What does a < 0 mean for a parabola?
- Opens downward, has a maximum vertex
Practice questions
What is the vertex of f(x) = 2(x - 3)² + 5?
- (3, 5)
- (-3, 5)
- (3, -5)
- (-3, -5)
Show answer
Answer: (3, 5)
In vertex form a(x-h)²+k, h=3 and k=5, so the vertex is (3,5).
The parabola f(x) = -4x² + 2x - 1 opens downward.
Show answer
Answer: True
The leading coefficient is -4, which is negative, so the parabola opens downward.
For f(x) = ax² + bx + c, the axis of symmetry is x = ___.
Show answer
Answer: -b/(2a)
The axis of symmetry passes through the vertex whose x-coordinate is -b/(2a).
If the discriminant of a quadratic is negative, the parabola has how many x-intercepts?
- 0
- 1
- 2
- 3
Show answer
Answer: 0
A negative discriminant means b²-4ac < 0, so there are no real solutions and no x-intercepts.
The vertex form f(x) = a(x - h)² + k directly shows the y-intercept of the parabola.
Show answer
Answer: False
Vertex form directly reveals the vertex (h,k) and direction, not the y-intercept. To find y-intercept, set x=0.
Fun facts
- The word 'parabola' comes from the Greek word meaning 'to throw beside' — the ancient Greeks discovered it while studying conic sections around 200 BCE.
- Every parabolic satellite dish uses the reflective property of parabolas: signals coming in parallel to the axis all bounce to the exact focus point where your receiver sits.
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