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Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Functions are limited to linear, quadratic, absolute value, and exponential.

Transformations of Functions

Today I will...Discover the rules for transformations.Describe the effects of the graphbased on the equation.Create an equation based onverbal description of the function.

Transformations of FunctionsTransformations allow you to move a graph of a function into a new position w/out having to find new ordered pairs to plot on the graph.

Parent Function ­ most simple form of function

Absolute Value: y = |x| Quadratic: y = x2

Square Root: y =  x Cubic: y = x3

Parent Functions with Transformation Options

 ***Knowing these will help you out***

Quadratic   y = x2 y = a(x ­ h)2 + k

Absolute Value y = |x| y = a|x ­ h| + k

Square Root y =  x y = a√x ­ h + k

Cubic y = x3 y = a(x ­ h)3 + k*They all have the same location of the a, h, and k.

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Transformations of Functions

y = f(x) + k shifts f(x) up k unitsy = f(x) ­ k shifts f(x) down k unitsy = f(x ­ h) shifts f(x) right h unitsy = f(x + h) shifts f(x) left h unitsy = ­f(x) reflects f(x) over x­axisy = a(f(x)), a > 1 vertically stretches f(x) y = a(f(x)), 0<a<1 vertically compresses f(x)

Transformations by a/h/k value:

reflected over x­axis ­avertically stretched  a > 1vertically compressed 0 < a < 1

shift left + hshift right ­ h

shift up + kshift down ­ k

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Describe the transformation(s):y = |x + 1|

Describe the transformation(s):y = √x + 2

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Describe the transformation(s):y = 3(x + 4)2

Describe the transformation(s):y = |x ­ 5| + 6

Describe the transformation(s):y = ­2(x + 3)2

Describe the transformation(s):y = ­2|x + 3| ­ 4

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Remember:

Quadratic y = a (x ­ h)2 + k

Absolute Value y = a | x ­ h | + k

Square Root  y = a √ x ­ h  + k

Cubic y = a (x ­ h)3 + k

Write the function given the transformations.

Square root function that is reflected over the x­axis and shifted right 3 units.

Write the function given the transformations.

Quadratic function that is vertically stretchedby a factor of 4 and shifted down 2 units.

Write the function given the transformations.

Cubic function that is reflected over the x­axis, shifted down 5 units, and shifted right 3 units.