3. CONTENTS Parent Function Reflection Across X-Reflection Across X- and Y-axisand Y-axis Vertical Stretch and Vertical Shrink HorizontalHorizontal and Vertical Shiftand Vertical Shift Combination Click this icon at any time to return to this page!

4. The Parent Function Click Here to Continue! This graph is the function from which all exponential functions originate

5. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=0 End Behavior: As x  ∞, y  ∞ As x  -∞, y  0 Interval of Increase: (-∞, ∞) Interval of Decrease: ~NONE~ Interval Where Positive: (1, ∞) Interval Where Negative:~NONE~ Intercepts: X-int.=none Y-int.=1 -No Vertex -The Function, as well as the transformations are neither even or odd!

6. Reflection Across X-axis

7. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=0 End Behavior: As x  ∞, y  -∞ As x  -∞, y  0 Interval of Increase:~NONE~ Interval of Decrease: (-∞, ∞) Interval Where Positive: ~NONE~ Interval Where Negative: (0,-∞) Intercepts: X-int.=none Y-int.=-1 -No Vertex -The Function, as well as the transformations are neither even or odd!

8. So…What Changed?  This Function has been reflected across the x-axis. This transformation caused the y intercept, the place where the line touches the y axis, to reflect to -1, rather than regular1. The asymptote (the value that the line approaches but never meets) is the same (the x-axis), but the graph is approaching it from the other side of the coordinate plane.

9. Reflection Across Y-axis

10. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=0 End Behavior: As x  ∞, y  0 As x  -∞, y  ∞ Interval of Increase: ~NONE~ Interval of Decrease: (∞,0) Interval Where Positive: (1, ∞) Interval Where Negative: (-∞,1) Intercepts: X-int.=none Y-int.=1 -No Vertex -The Function, as well as the transformations are neither even or odd!

11. What Changed?  This function is reflected across the y-axis. All of the x-values are in their opposite place across the y-axis. The y-intercept and asymptotes are the same.

12. Vertical Stretch

13. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=0 End Behavior: As x  ∞, y  ∞ As x  -∞, y  0 Interval of Increase: (-∞, ∞) Interval of Decrease: ~NONE~ Interval Where Positive: (0,∞) Interval Where Negative: ~NONE~ Intercepts: X-int.=none Y-int.=2 -No Vertex -The Function, as well as the transformations are neither even or odd!

14. What Changed?  The graph has is an example of Vertical stretch, meaning the graph is “stretched” so that it reaches higher values faster than the parent function would. The y- intercept is no longer 1, but 2. The asymptote stays the same.

15. Vertical Shrink

16. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=0 End Behavior: As x  ∞, y  0 As x  -∞, y  ∞ Interval of Increase: ~NONE~ Interval of Decrease: (∞,0) Interval Where Positive: (1, ∞) Interval Where Negative: (-∞,1) Intercepts: X-int.=none Y-int.=1 -No Vertex -The Function, as well as the transformations are neither even or odd!

17. What Changed?  This function is the opposite of Vertical Shrink. It reaches higher numbers slower than the parent function will. The y-intercept is now decreased to ½ while the asymptote is still the x-axis.

18. Horizontal Shift

19. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=0 End Behavior: As x  ∞, y  ∞ As x  -∞, y  0 Interval of Increase: (-∞, ∞) Interval of Decrease: ~NONE~ Interval Where Positive: (0,∞) Interval Where Negative: ~NONE~ Intercepts: X-int.=none Y-int.=2 -No Vertex -The Function, as well as the transformations are neither even or odd!

20. What Changed?  The graph has shifted slightly to the left! Opposite of what makes sense, when you add to the exponent, the whole graph is shifted to the left, the “negative side”. The opposite happens when you subtract. The y-intercept has been shifted to 2 while the asymptote is still the same.

21. Vertical Shift

22. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=3 End Behavior: As x  ∞, y  ∞ As x  -∞, y  3 Interval of Increase: (-∞, ∞) Interval of Decrease:~NONE~ Interval Where Positive: (- ∞, ∞) Interval Where Negative: ~NONE~ Intercepts: X-int.=none Y-int.=4 -No Vertex -The Function, as well as the transformations are neither even or odd!

23. What Changed?  It’s quite obvious that the parent function has been shifted up! Adding to the whole function moves the graph up, and vice versa when subtracting. The asymptote is now 3 and the y-intercept is 4.

24. Combination!

25. Characteristics: Domain: All Real Numbers Range: All Real Numbers Asymptote: x=-4 End Behavior: As x  ∞, y  -∞ As x  -∞, y  -4 Interval of Increase:~NONE~ Interval of Decrease: (-∞, ∞) Interval Where Positive: ~NONE~ Interval Where Negative: (-∞, ∞) Intercepts: X-int.=none Y-int.=-5 -No Vertex -The Function, as well as the transformations are neither even or odd!

26. What Changed?  This graph is a combination of two different transformations. The first transformation is a reflection across the x-axis, with the second transformation shifting the whole graph down 4 units. The graph will never EVER touch the 1 st and 2 nd quadrants.

27. Director Producer Writer Graph Artist Mathematician Effects Pictures Presentation Best Math Teacher Ever!!!!! Emanuel Hill Jr. Click the button to go back to the Beginning! Thank You for Watching the Exponent Family.

28. Please Press the Escape button to exit this presentation