1.6 PreCalculus Parent Functions Graphing Techniques

1. 1.6 PreCalculusParent FunctionsGraphing Techniques

2. Transformations Vertical Translations Horizontal Translations Graph stays the same, but moves up or down. Graph stays the same, but moves left or right.

3. Transformations Vertical Stretch Horizontal Stretch Width stays the same, but height increases. Height stays the same, but width increases.

4. Transformations Vertical Compression Horizontal Compression Width stays the same, but height decreases. Height stays the same, but width decreases.

5. Transformations Reflection Over the x-axis Graph “flips” up-side down. Reflection Over the y-axis Graph “flips” side-ways.

6. g(x) = − A g(x) = g(x) = + A g(x) = g(x) = |x| + A g(x) = x2 + A g(x) = x2 − A g(x) = |x| − A g(x) = (x + A)2 g(x) = |x + A| g(x) = (x − A)2 g(x) = |x − A| Assume that A is a positive, real number!

7. g(x) = A 1 1 2 g(x) = g(x) = | ( x x | ) A A g(x) = Ax2 g(x) = A|x| g(x) = (Ax)2 g(x) = |Ax| Assume that A is a positive, real number!

8. g(x) = − g(x) = −x2 g(x) = −|x| g(x) = (-x)2 g(x) = |-x| Assume that A is a positive, real number!

9. Rational Functions

10. 1 2 g(x) = ( x ) 2 Identify each transformation from the parent graph f(x) = x2. down 2 g(x) = x2 + 5 up 5 g(x) = x2 – 2 g(x) = (x – 3)2 right 3 g(x) = (x + 1)2 left 1 g(x) = −x2 reflection over x-axis g(x) = (-x)2 reflection over y-axis vertical comp. factor of ½ vertical stretch factor of 2 g(x) = 2x2 Horiz. stretch Factor of 2 Horiz. Comp. Factor of ½ g(x) = (2x)2

11. Identify each transformation from the parent graph f(x) = x2. g(x) = -2x2 + 5 up 5 vertical stretch factor of 2 reflection over x-axis g(x) = (x – 3)2 − 2 down 2 right 3 g(x) = -(x + 1)2 reflection over x-axis left 1 g(x) = (-2x)2 Horiz. Comp. Factor of ½ reflection over y-axis

12. 1 g(x) = | x | 2 Identify each transformation from the parent graph f(x) = |x|. down 10 g(x) = |x| + 3 up 3 g(x) = |x| – 10 g(x) = |x – 2| right 2 g(x) = |x + 5| left 5 g(x) = −|x| reflection over x-axis g(x) = |-x| reflection over y-axis vertical comp. factor of ½ vertical stretch factor of 2 g(x) = 2|x| Horiz. stretch Factor of 2 Horiz. Comp. Factor of ½ g(x) = |2x|

13. Identify each transformation from the parent graph f(x) = |x|. g(x) = 5|x| − 4 down 4 vertical stretch factor of 5 vertical stretch factor of 2 g(x) = 2|x – 5| - 3 down 3 right 5 g(x) = -|x| + 3 reflection over x-axis up 3 g(x) = |-3x| Horiz. Comp. Factor of ⅓ reflection over y-axis

14. f(x) = g(x) = 2 g(x) = − 2 g(x) = g(x) = + 3 g(x) = Identify each transformation from the parent graph up 3 down 2 right 4 left 2 vertical comp. factor of ½ vertical stretch factor of 2 horiz. stretch factor of 2 horiz. Comp. factor of ½ reflection over x-axis reflection over y-axis

15. f(x) = g(x) = Identify each transformation from the parent graph up 1 left 4 vertical stretch factor of 2 right 5 vertical comp. factor of ½ reflection over x-axis horiz. Comp. factor of ⅓ reflection over y-axis down 4

16. Find the function that is finally graphed after the following three transformations are applied to the graph of y = |x|. • Shift left 2 units. • Shift up 3 units. • Reflect about the y-axis.

17. Find the function that is finally graphed after the following three transformations are applied to the graph of • Shift down 5 units. • Shift right 2 units. • Reflect about the x-axis.

18. y x Graphing Techniques f(x) = x2 – 4 (down 4) 1. Graph f(x) = x2. 2. Shift all of the points down 4 units.

19. y x Graphing Techniques f(x) = (x – 3)3 (right 3) 1. Graph f(x) = x3. 2. Shift all of the points right 3 units.

20. y x Graphing Techniques f(x) = |x - 2| + 3 (right 2, up 3) 1. Graph f(x) = |x|. 2. Shift all of the points right 2 and up 3.

21. y x Graphing Techniques f(x) = -x3 (reflect over x-axis) 1. Graph f(x) = x3. • Reflect all points • over the x-axis.