Overview
This lecture explains how functions are transformed by shifting, stretching, shrinking, and reflecting their graphs, including examples with parent functions and step-by-step graphing using transformations.
Types of Transformations
- Adding a constant outside the function (f(x) + a) shifts the graph vertically up (if a > 0) or down (if a < 0).
- Adding or subtracting inside the function (f(x - a)) shifts the graph horizontally right (if a > 0) or left (if a < 0).
- A negative outside the function (-f(x)) reflects the graph over the x-axis.
- A negative inside the function (f(-x)) reflects the graph over the y-axis.
- A negative both inside and outside (-f(-x)) reflects the graph over the origin.
- Multiplying the function by a constant (a·f(x), a > 1) stretches vertically; 0 < a < 1 shrinks vertically.
- Multiplying inside the function (f(bx), b > 1) shrinks horizontally; 0 < b < 1 stretches horizontally.
Examples with Parent Functions
- For f(x) = x², f(x) + 3 is a parabola shifted up 3 units; f(x) - 2 shifts down 2 units.
- For y = |x|, y = |x + 2| shifts 2 units left; y = |x - 3| shifts 3 units right.
- For y = √x, y = -√x reflects over x-axis; y = √(-x) reflects over y-axis; y = -√(-x) reflects over the origin.
Stretches and Shrinks
- y = 2|x| is a vertical stretch (y-values doubled).
- y = (1/2)|x| is a vertical shrink (y-values halved).
- y = √(2x) produces a horizontal shrink (x-values are halved for the same y).
- y = √(½x) produces a horizontal stretch (x-values are doubled for the same y).
Graphing Using Transformations
- Use parent functions: y = x, y = x², y = x³, y = √x, y = ³√x, y = |x|.
- To graph y = (x - 2)² + 3, shift the parabola right 2 and up 3.
- To graph y = 3 - (x + 2)², rewrite as y = - (x + 2)² + 3; shift left 2, up 3, reflect over x-axis (opens downward).
- For y = 4 - √(3 - x), shift right 3, up 4, and reflect over both axes (over the origin).
Key Terms & Definitions
- Vertical shift — moves the graph up or down.
- Horizontal shift — moves the graph left or right.
- Reflection — flips the graph over a specified axis.
- Vertical stretch/shrink — multiplies y-values, making the graph taller or shorter.
- Horizontal stretch/shrink — multiplies x-values, making the graph wider or narrower.
- Parent function — the simplest form of a function used as a base for transformations.
Action Items / Next Steps
- Review and memorize the main parent functions and their graphs.
- Practice graphing transformations using step-by-step shifts, reflections, and stretches.
- Complete any assigned homework on function transformations.