Overview
This lecture covers how to graphically transform functions by applying vertical and horizontal shifts, stretches, compressions, and reflections, using basic quadratic and related functions as examples.
Vertical and Horizontal Shifts
- Adding a number outside the function (e.g., ( x^2 + 2 )) shifts the graph vertically.
- Vertical shifts move the graph up (positive) or down (negative) by that number of units.
- Adding or subtracting inside the function’s variable (e.g., ( (x + 2)^2 )) shifts the graph horizontally.
- A positive value inside shifts left; a negative value shifts right.
Stretches and Compressions
- Multiplying the function by a coefficient outside (e.g., ( 2x^2 )) causes a vertical stretch if greater than 1, making the graph narrower.
- If the coefficient is between 0 and 1 (e.g., ( \frac{1}{2}x^2 )), the graph is vertically compressed and becomes wider.
- Multiplying inside the variable (e.g., ( (2x)^2 )) causes a horizontal compression if greater than 1, making the graph narrower horizontally.
- A coefficient inside between 0 and 1 horizontally stretches the graph.
Reflections
- Multiplying the whole function by –1 (e.g., ( -x^2 )) reflects the graph over the x-axis (flips it vertically).
- Negating the variable inside (e.g., ( (-x)^2 )) would reflect over the y-axis; for even powers, this does not change the graph, but for odd powers, it does.
Combined Transformations
- When multiple transformations appear (e.g., ( -2(x - 3)^2 + 4 )), apply each step one at a time: shift, stretch/compress, and reflect as indicated.
- The order of transformations does not impact the final shape as long as each is accounted for properly.
Application to Other Functions
- The same transformation principles apply to absolute value, cubic, square root, and other common functions.
Key Terms & Definitions
- Vertical Shift — Moving the graph up or down by adding a constant outside the function.
- Horizontal Shift — Moving the graph left or right by adding/subtracting inside the variable.
- Vertical Stretch/Compression — Changing the graph’s width by multiplying the output by a constant.
- Horizontal Stretch/Compression — Changing the graph’s width by multiplying the input by a constant.
- Reflection — Flipping the graph over the x-axis (vertical) or y-axis (horizontal).
Action Items / Next Steps
- Review the table summarizing transformation types provided in class.
- Practice applying transformations to different base functions.
- Prepare for comprehension check on these concepts.