Lesson 2 • 40 min read
Quadratic Functions
In This Lesson
1What is a Quadratic Function?
A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a ≠ 0. The graph of a quadratic function is always a parabola.
Examples of Quadratic Functions:
f(x) = x²
Simplest form
f(x) = 2x² - 3x + 1
Standard form
f(x) = -(x - 2)² + 5
Vertex form
Key Characteristics:
- • Highest power of x is 2 (degree 2)
- • Graph is always a U-shaped curve (parabola)
- • Has exactly one vertex (turning point)
- • Has an axis of symmetry through the vertex
2Standard & Vertex Forms
Standard Form
f(x) = ax² + bx + c
Best for:
- • Finding y-intercept (c)
- • Using quadratic formula
- • Identifying a, b, c coefficients
Vertex Form
f(x) = a(x - h)² + k
Best for:
- • Finding vertex: (h, k)
- • Graphing quickly
- • Finding axis of symmetry: x = h
Converting Between Forms
Example: Convert f(x) = (x - 3)² + 2 to standard form
f(x) = (x - 3)² + 2
f(x) = (x² - 6x + 9) + 2 ← Expand (x-3)²
f(x) = x² - 6x + 11
3Properties of Parabolas
If a > 0 (Positive)
∪
Opens UPWARD
- • Vertex is the MINIMUM point
- • Function decreases then increases
- • Smiling parabola
If a < 0 (Negative)
∩
Opens DOWNWARD
- • Vertex is the MAXIMUM point
- • Function increases then decreases
- • Frowning parabola
Effect of |a| (Absolute Value)
|a| > 1: Narrower parabola
Example: f(x) = 3x² is narrower than f(x) = x²
|a| < 1: Wider parabola
Example: f(x) = 0.5x² is wider than f(x) = x²
Key Features
Vertex: The highest or lowest point (h, k)
Axis of Symmetry: Vertical line x = h through the vertex
Y-intercept: Where the parabola crosses y-axis (0, c)
X-intercepts (Roots): Where the parabola crosses x-axis (if any)
4Finding the Vertex
From Standard Form (ax² + bx + c)
h = -b / 2a
k = f(h)
Vertex = (h, k)
Example: Find vertex of f(x) = x² - 4x + 3
a = 1, b = -4, c = 3
h = -b/2a = -(-4)/2(1) = 4/2 = 2
k = f(2) = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1
Vertex: (2, -1)
Example 2: Find vertex of f(x) = -2x² + 8x - 5
a = -2, b = 8, c = -5
h = -8/2(-2) = -8/(-4) = 2
k = -2(2)² + 8(2) - 5 = -8 + 16 - 5 = 3
Vertex: (2, 3)
Since a < 0, this is a maximum
From Vertex Form (a(x - h)² + k)
The vertex is simply (h, k) - just read it directly!
f(x) = 3(x - 2)² + 5 → Vertex: (2, 5)
f(x) = -(x + 1)² - 3 → Vertex: (-1, -3)
Note: (x + 1) = (x - (-1)), so h = -1
5Graphing Quadratic Functions
Steps to Graph
- Determine if parabola opens up (a> 0) or down (a< 0)
- Find and plot the vertex (h, k)
- Draw the axis of symmetry: x = h
- Find and plot the y-intercept (0, c)
- Use symmetry to plot the mirror point
- Find x-intercepts (if they exist) by solving f(x) = 0
- Connect points with a smooth U-shaped curve
Example: Graph f(x) = x² - 2x - 3
1. a = 1 > 0, so opens upward
2. Vertex: h = -(-2)/2(1) = 1, k = 1 - 2 - 3 = -4 → (1, -4)
3. Axis of symmetry: x = 1
4. Y-intercept: (0, -3)
5. Mirror of (0, -3) across x=1 is (2, -3)
6. X-intercepts: x² - 2x - 3 = 0 → (x-3)(x+1) = 0 → (-1, 0) and (3, 0)
Quick Graphing Tips
• Always plot at least 5 points for accuracy
• Use symmetry! Points equidistant from axis have same y-value
• The vertex is always on the axis of symmetry
• Check your graph: parabola should be symmetric