1.6 PreCalculus Parent Functions Graphing Techniques.

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Presentation transcript:

1.6 PreCalculus Parent Functions Graphing Techniques

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

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

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

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

Quadratic f(x) = x 2 Abs Value f(x) = |x| Square Rt. f(x) = Translate Up Translate Down Translate Left Translate Right g(x) = x 2 + A g(x) = |x| + A g(x) = + A g(x) = x 2 − A g(x) = (x + A) 2 g(x) = (x − A) 2 g(x) = |x| − A g(x) = |x + A| g(x) = |x − A| g(x) = − A g(x) = Assume that A is a positive, real number!

Quadratic f(x) = x 2 Abs Value f(x) = |x| Square Rt. f(x) = Vertical Stretch Vertical Compression Horizontal Stretch Horizontal Compression g(x) =| 1 A x| ( 1 A x) 2 g(x) = Ax 2 g(x) = (Ax) 2 g(x) = A|x| g(x) = |Ax| g(x) = A Assume that A is a positive, real number!

Quadratic f(x) = x 2 Abs Value f(x) = |x| Square Rt. f(x) = Reflection over x-axis Reflection over y-axis Assume that A is a positive, real number! g(x) = −x 2 g(x) = −|x|g(x) = − g(x) = (-x) 2 g(x) = |-x|

Rational Functions Translate Up Stretch Translate Down Compression Translate Left Reflection over x-axis Translate Right Reflection over y-axis

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

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

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

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

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

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

Find the function that is finally graphed after the following three transformations are applied to the graph of y = |x|. 1.Shift left 2 units. 2.Shift up 3 units. 3.Reflect about the y-axis.

Find the function that is finally graphed after the following three transformations are applied to the graph of 1.Shift down 5 units. 2.Shift right 2 units. 3.Reflect about the x-axis.

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

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

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

Graphing Techniques f(x) = -x 3 (reflect over x-axis) x y 1. Graph f(x) = x 3. 2.Reflect all points over the x-axis.