The document describes various transformations that can be applied to graphs of functions, including: vertical and horizontal translations that shift the graph along the y- or x-axis respectively; vertical and horizontal stretches/compressions that enlarge or reduce the graph along the y- or x-axis by a scale factor; and reflections about the x- or y-axis that flip the graph across that axis. Each transformation is defined symbolically by an algebraic modification to the original function, and its geometric effect on changing the x- and y-coordinates of points on the graph is also provided symbolically.

1. Transformations Horizontal/ Vertical Translation Vertical Stretch/Compression Horizontal Stretch/Compression Reflection about x-axis/y-axis

2. Vertical Translation For any function y=f(x), adding any number k to the function will cause the graph of the function to translate (shift) vertically k units. Symbolically…. y = f(x) + k For each point on the graph, the y-value will change by k units while the x-value remains the same. Symbolically… (x, y + k)

3. Vertical Translation y = f(x) + 2

4. Vertical Translation y = f(x) - 2

5. Horizontal Translation For any function y=f(x), replacing x with (x – h) in the function will cause the graph of the function to translate (shift) horizontally h units. Symbolically…. y = f(x - h) For each point on the graph, the x-value will change by h units while the y-value remains the same. Symbolically… (x + h, y)

6. Horizontal Translation y = f(x-3)

7. Horizontal Translation y = f(x+3)

8. Vertical Stretch/Compression For any function y=f(x), multiplying any number a to the function will cause the graph of the function to stretch/compress vertically by a factor of a. Symbolically…. y = a f(x) For each point on the graph, the y-value will be multiplied by a while the x-value remains the same. Symbolically… ( x , a y )

9. Vertical Stretchy = 2 f(x)

10. Vertical Compressiony = 1/2 f(x)

11. Horizontal Stretch/Compression For any function y=f(x), replacing the x with bx in the function will cause the graph of the function to stretch/compress horizontally by a factor of (1/b). Symbolically…. y = f( bx ) For each point on the graph, the x-value will be multiplied by (1/b) while the y-value remains the same. Symbolically… ( (1/b) x , y )

12. Horizontal Compressiony = f ( 2x )

13. Horizontal Stretchy = f((1/2)x)

14. Reflection about x-axis For any function y = f(x), if the function expression is multiplied by -1, the graph of the function will be a reflection over the x-axis. Symbolically… y = - f(x) For each point on the graph, the y-value will be multiplied by -1. Symbolically… ( x, -1 y )

15. Reflection about x-axisy = - f(x)

16. Reflection about y-axis For any function y = f(x), if the x-value is multiplied by -1, the graph of the function will be a reflection over the y-axis. Symbolically… y = f(-x) For each point on the graph, the x-value will be multiplied by -1. Symbolically… (-1 x, y )

17. Reflection about y-axisy = f(-x)

18. Assignment Worksheet – “Move the Monster”