Overview
This lecture explains how to graph function transformations, including vertical and horizontal shifts, stretches, compressions, and reflections, focusing on quadratic functions as examples.
Vertical Shifts
- Adding a constant to a function (e.g., ( f(x) = x^2 + 2 )) shifts the graph vertically.
- If the constant is positive, the graph shifts up; if negative, it shifts down.
- The vertex of ( f(x) = x^2 + c ) moves to (0, c).
Horizontal Shifts
- Adding or subtracting inside the function argument (e.g., ( f(x) = (x + 2)^2 )) shifts the graph horizontally.
- ( f(x) = (x + a)^2 ) shifts left by ( a ); if the sign is negative, it shifts right.
- The vertex of ( f(x) = (x - a)^2 ) is at (a, 0).
Vertical Stretches and Compressions
- Multiplying the function by a constant ( a ) (e.g., ( f(x) = 2x^2 )), stretches the graph vertically by that factor.
- If ( a > 1 ), the graph is narrower (vertical stretch); if ( 0 < a < 1 ), it is wider (vertical compression).
Horizontal Stretches and Compressions
- Multiplying ( x ) inside the function by a constant (e.g., ( f(x) = (2x)^2 )) affects the graph horizontally.
- If the factor is greater than 1, the graph compresses horizontally; if less than 1, the graph stretches.
Reflections
- A negative sign in front of the function (e.g., ( f(x) = -x^2 )) reflects the graph over the x-axis.
- A negative inside the variable (e.g., ( f(x) = (-x)^2 )) reflects over the y-axis; for even functions like ( x^2 ), this has no visible effect, but matters for other functions.
Combining Transformations
- Complicated functions (e.g., ( f(x) = -2(x-3)^2 + 4 )) require applying multiple transformations in order: vertical shift, horizontal shift, stretch/compression, and reflection.
- These transformation rules apply to other basic functions, like absolute value, cube, or square root.
Key Terms & Definitions
- Vertical Shift — Moves the graph up or down by adding/subtracting a constant outside the function.
- Horizontal Shift — Moves the graph left or right by adding/subtracting inside the argument.
- Vertical Stretch/Compression — Multiplies the function by a constant, changing its steepness.
- Horizontal Stretch/Compression — Multiplies the input variable by a constant, compressing or stretching the graph.
- Reflection — Flips the graph across the x-axis (negative outside) or y-axis (negative inside).
Action Items / Next Steps
- Review the provided summary table of transformations.
- Practice graphing different transformations of basic functions.
- Prepare for comprehension check on this material.