Lesson 40: Transformations of Quadratic Functions
Quadratic functions can be transformed by shifting, stretching, compressing, and reflecting. Understanding transformations allows you to quickly sketch graphs and identify key features.
Standard Form and Vertex Form
Quadratic function in vertex form: f(x) = a(x − h)² + k
- (h, k) = vertex
- a > 0 → parabola opens upward; a < 0 → downward
- |a| > 1 → vertical stretch; 0 < |a| < 1 → vertical compression
Types of Transformations
- Vertical Shift: f(x) + k moves graph up (k > 0) or down (k < 0)
- Horizontal Shift: f(x − h) moves graph right (h > 0) or left (h < 0)
- Reflection: −f(x) reflects across x-axis
- Vertical Stretch/Compression: a*f(x), changes “width†of parabola
Examples
Example 1: f(x) = (x − 3)² + 2
- Vertex: (3, 2)
- Parabola opens upward (a = 1)
- Shifted right 3 units, up 2 units
Example 2: f(x) = −2(x + 1)² + 4
- Vertex: (−1, 4)
- Parabola opens downward (−2)
- Vertical stretch by factor of 2, shifted left 1 and up 4
Example 3: f(x) = 0.5(x − 2)² − 3
- Vertex: (2, −3)
- Parabola opens upward
- Vertical compression by factor of 0.5, shifted right 2 and down 3
Practice Problems
- Describe the transformations for f(x) = (x + 4)² − 5
- Describe the transformations for f(x) = −3(x − 2)² + 1
- Graph f(x) = 0.5(x + 1)² + 3 and identify vertex
- Graph f(x) = −(x − 5)² − 2 and identify vertex
- Write f(x) = x² shifted left 3, up 4, reflected over x-axis