Transformations of Functions and Graphs

1. Transformations of Functions and Graphs We will be looking at simple functions and seeing how various modifications to the functions transform them. Transformations Transformations Transformations Transformations

2. Above is the graph of As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function. VERTICAL TRANSLATIONS What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them). What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).

3. Vertical Translation Transformation UP y = f(x) + k k units y = f(x) ̶ k Down k units

4. VERTICAL TRANSLATIONS

5. Practice what is the transformation? Parent function Up 10 units y = f(x) +10 y = f(x) Down 9 units Up 5 units y = f(x) Down 7 units

6. HORIZONTAL TRANSLATIONS Above is the graph of As you can see, a number added or subtracted from the x will cause a horizontal shift or translation in the function but opposite way of the sign of the number. What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function). What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).

7. Horizontal Translation For h>0, and Transformation left h units y = f(x + h) Right y = f(x ̶ h) h units

8. HORIZONTAL TRANSLATIONS shift left 1 shift right 3

9. Practice what is the transformation? Parent function Left 10 units y = f(x+10) y = f(x) Right 9 units Left 5 units y = f(x) Right 7 units

10. What would the graph of look like? We could have a function that is transformed or translated both vertically AND horizontally. up 3 left 2 Above is the graph of

11. More Practice what is the transformation? Parent function Left 1 and down 6 y = f(x+1)-6 y = f(x) Right 3 and up 2 Left 5 and up 7 y = f(x) y = f(x-8)-1 Right 8 and down 1

12. and DILATION: If we multiply a function by a non-zero real number it has the effect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number. Let's try some functions multiplied by non-zero real numbers to see this.

13. The bigger ais. The narrower the graph is. So the graph af(x), whereais any real number GREATER THAN 1, is the graph of f(x) Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value. Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value. vertically Above is the graph of stretchedby a factor of a. What would2f(x) look like? What would4f(x) look like?

14. What if the value of a was positive but less than 1? The smaller ais. The wider the graph is. So the graph af(x), whereais 0 < a < 1, is the graph of f(x) Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value. Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value. vertically Compressed by a factor of a. Above is the graph of What would1/2 f(x) look like? What would1/4 f(x) look like?

15. Vertically Stretched Transformation y = af(x) Vertically Stretched a>1 By factor of a Compressed Vertically 0<a<1 By factor of a

16. VERTICAL TRANSLATIONS

17. Procedure: Multiple Transformations (From left to right) 1. Reflecting 2. Stretching or shrinking 3. Horizontal translation 4. Vertical translation

18. Practice what is the transformation? y = 5f(x+10)-6 vertically stretched by factor of 5, Left 10, down 6 vertically compressed by factor of ¼ , Right 7, up 2

19. Practice what is the transformation? vertically compressed by factor of 1/5, Left 6, down 7 vertically stretched by factor of ¼ , Right 9, up 2

20. What if the value of a was negative? So the graph -f(x) is a reflection about the x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the x-axis) Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value. Above is the graph of What would- f(x) look like?

21. There is one last transformation we want to look at. So the graph f(-x) is a reflection about the y-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the y-axis) Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value. Above is the graph of What would f(-x) look like? (This means we are going to take the negative of x before putting in the function)

22. Summary of Transformations So Far **Do reflections and dilations BEFORE vertical and horizontal translations** If a > 1, vertically stretched by a factor of a If 0 < a < 1, vertically compressed by a factor of a reflected across x-axis -f (x) k>0 Up k units k<0 Down k units reflected across y-axis f(-x) h >0 Left h units h <0 Right h units (opposite sign of number with the x)

23. We know what the graph would look like if it wasfrom our library of functions. Graph using transformations moves up 1 reflects about the x -axis moves right 2