Unit 3 – Transformations Transformations of Functions ...€¦ · I Identify and graph basic...

Unit 3 – Transformations

Transformations of Functions (Unit 3.1)

William (Bill) Finch

Mathematics DepartmentDenton High School

Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary

Lesson Goals

When you have completed this lesson you will:

I Identify and graph basic parent functions.

I Use translations to sketch the graph of a function.

I Use reflections to sketch the graph of a function.

I Use non-rigid transformations to sketch the graph of afunction.

W. Finch DHS Math Dept

Transformations 2 / 37

Lesson Goals

Family of Functions

A family of functions shares a certain set of characteristicswith each other.

A parent function is the simplest form for a function family.

Linear Functions

Parent Linear Function (Identity Function)

I Equation f (x) = x

I Domain (−∞,∞)

I Range (−∞,∞)

I x-int (0, 0)

I y -int (0, 0)

I Increasing (−∞,∞)

I Odd function

I Symmetry wrt origin

−3 −2 −1 1 2 3

Quadratic Function

Parent Quadratic Function (Squaring Function)I Equation f (x) = x2

I Range [0,∞)

I x-int (0, 0)

I y -int (0, 0)

I Decreasing (−∞, 0)

I Increasing (0,∞)

I Minimum (0, 0)

I Even function

I Symmetry wrt y -axis

−3 −2 −1 1 2 3

Absolute Value Function

Parent Absolute Value FunctionI Equation f (x) = |x |I Domain (−∞,∞)

I Range [0,∞)

I x-int (0, 0)

I y -int (0, 0)

I Decreasing (−∞, 0)

I Increasing (0,∞)

I Minimum (0, 0)

I Even function

I Symmetry wrt y -axis

−3 −2 −1 1 2 3

Cubic Function

Parent Cubic Function

I Equation f (x) = x3

I Range (−∞,∞)

I x-int (0, 0)

I y -int (0, 0)

I Increasing (−∞,∞)

I Odd function

−3 −2 −1 1 2 3

Square Root Function

Parent Square Root Function

I Equation f (x) =√x

I Domain [0,∞)

I Range [0,∞)

I x-int (0, 0)

I y -int (0, 0)

I Increasing (0,∞)1 2 3

Reciprocal Function

Parent Reciprocal Function

I Equation f (x) =1

xI Domain (−∞, 0) ∪ (0,∞)

I Range (−∞, 0) ∪ (0,∞)

I No intercepts

I Odd function

I Decreasing (−∞, 0) and (0,∞)

I Vertical asymptote y -axis

I Horizontal asymptote x-axis

−3 −2 −1 1 2 3

Greatest Integer Function

Parent Greatest Integer Function (Step Function

I Equation f (x) = [[x ]]

I Range Set of all Integers

I x-intercept [0, 1)

I y -intercept (0, 0)

I Each step is constant

I Steps one unit vertical jump

−4 −3 −2 −1 1 2 3 4

Evaluate the Greatest Integer Function

The greatest integer function f (x) = [[x ]] returns thelargest integer less than or equal to x .

a) f (3.2) =

b) f (3.9) =

c) f (−1.6) =

d) f (−1.2) =

e) f (5) =

Graph a Piecewise-Defined Function

Sketch a graph offunction f : f (x) =

{−x + 4 x ≤ 1

−(x − 1)2 x > 1

Translations

Graph the following function on your calculator as Y1 :

f (x) = x2

Now add the following functions into Y2 and Y3 :

g(x) = x2 + 3

h(x) = x2 − 4

How would you describe the graphs of g and h compared tothe graph of f ?

Translations

f (x) = x2

g(x) = x2 + 3

h(x) = x2 − 4

Translations

f (x) = x2

g(x) = x2 + 3

h(x) = x2 − 4

Vertical Translations

f (x) is the parent function.

g(x) is a vertical shiftup 3 units.

h(x) is a vertical shiftdown 4 units.

−4 −2 2 4

Let c be a positive real number. A vertical shift in the graphof y = f (x) :

I c units up: h(x) = f (x) + c

I c units down: h(x) = f (x)− c

In other words . . . if you add or subtract c units to theoutput of the function, the resulting graph shows the outputtranslated up or down c units.

Let c be a positive real number. A vertical shift in the graphof y = f (x) :

I c units up: h(x) = f (x) + c

I c units down: h(x) = f (x)− c

In other words . . . if you add or subtract c units to theoutput of the function, the resulting graph shows the outputtranslated up or down c units.

Translations

f (x) = |x |

g(x) = |x + 3|

h(x) = |x − 4|

Translations

f (x) = |x |

g(x) = |x + 3|

h(x) = |x − 4|

Translations

f (x) = |x |

g(x) = |x + 3|

h(x) = |x − 4|

Horizontal Translations

g(x) is a horizontal shiftleft 3 units.

h(x) is a horizontal shiftright 4 units.

−6 −4 −2 2 4 6

g(x)h(x)

Let c be a positive real number. A horizontal shift in thegraph of y = f (x) :

I c units left: h(x) = f (x + c)

I c units right: h(x) = f (x − c)

In other words . . . if you add or subtract c units to the inputof the function, the resulting graph shows the outputtranslated left or right c units.

Let c be a positive real number. A horizontal shift in thegraph of y = f (x) :

I c units left: h(x) = f (x + c)

I c units right: h(x) = f (x − c)

In other words . . . if you add or subtract c units to the inputof the function, the resulting graph shows the outputtranslated left or right c units.

Reflections

f (x) = x2

Now add the following function into Y2 :

g(x) = −x2

How would you describe the graph of g compared to the graphof f ?

Reflections

f (x) = x2

g(x) = −x2

Reflections

f (x) = x2

g(x) = −x2

Reflection wrt x-axis

g(x) is a reflection wrtx-axis.

−4 −2 2 4

Reflections

f (x) =√x

h(x) =√−x

How would you describe the graph of h compared to the graphof f ?

Reflections

f (x) =√x

h(x) =√−x

Reflections

f (x) =√x

h(x) =√−x

Reflection wrt y -axis

h(x) is a reflection wrty-axis.

−6 −4 −2 2 4 6

f (x)g(x)

Reflections

A reflection in the graph of y = f (x) :

I Reflection wrt x-axis: h(x) = −f (x)

I Reflection wrt y -axis: h(x) = f (−x)

In other words . . . if you take the opposite of the output, orthe opposite of the input the resulting graph is a reflectionwith respect to one of the axes.

Reflections

A reflection in the graph of y = f (x) :

I Reflection wrt x-axis: h(x) = −f (x)

I Reflection wrt y -axis: h(x) = f (−x)

In other words . . . if you take the opposite of the output, orthe opposite of the input the resulting graph is a reflectionwith respect to one of the axes.

Nonrigid Transformations (Dilations)

f (x) =√x

g(x) = 3√x

h(x) =1

f (x) =√x

g(x) = 3√x

h(x) =1

f (x) =√x

g(x) = 3√x

h(x) =1

Vertical Stretch/Shrink

g(x) is a vertical stretch by afactor of 3.

h(x) is a vertical shrink by afactor of 1/4.

2 4 6 8

A nonrigid transformation in the graph of y = f (x) :

I vertical stretch when a > 1 : h(x) = a ·f (x)

I vertical shrink when 0 < a < 1 : h(x) = a ·f (x)

In other words . . . if you multiply the output of a function bya constant the resulting stretch or shrink is in the verticaldirection.

I vertical stretch when a > 1 : h(x) = a ·f (x)

I vertical shrink when 0 < a < 1 : h(x) = a ·f (x)

In other words . . . if you multiply the output of a function bya constant the resulting stretch or shrink is in the verticaldirection.

f (x) = |x |

g(x) = |2x |

h(x) =

∣∣∣∣13x∣∣∣∣

f (x) = |x |

g(x) = |2x |

h(x) =

∣∣∣∣13x∣∣∣∣

f (x) = |x |

g(x) = |2x |

h(x) =

∣∣∣∣13x∣∣∣∣

Horizontal Stretch/Shrink

g(x) is a horizontal shrink bya factor of 2.

h(x) is a horizontal stretchby a factor of 1/3.

−6 −4 −2 2 4 6

g(x)h(x)

I horizontal shrink when a > 1 : h(x) = f (a·x)

I horizontal stretch when 0 < a < 1 : h(x) = f (a·x)

In other words . . . if you multiply the input of a function by aconstant the resulting stretch or shrink is in the horizontaldirection.

I horizontal shrink when a > 1 : h(x) = f (a·x)

I horizontal stretch when 0 < a < 1 : h(x) = f (a·x)

In other words . . . if you multiply the input of a function by aconstant the resulting stretch or shrink is in the horizontaldirection.

Absolute Value Transformation

g(x) = |f (x)|

This transformation reflectsany portion of the graph off (x) that is below the x-axisso that it is above the x-axis.

−4 −2 2 4

Absolute Value Transformation

g(x) = f (|x |)

This transformation results inthe portion of the graph left ofthe y -axis being replaced by areflection of the portion off (x) to the right of the y -axis.

−4 −2 2 4

Example

Use the graph of f (x) =1

xto graph each function.

a) g(x) =1

x− 2

b) g(x) =1

x − 1

c) g(x) =1

x + 3+ 1

Example

Use the graph of f (x) = x2 to graph each function.

a) g(x) = 3x2

b) g(x) = −(x2 + 4)

c) g(x) = |x2 − 5|

Example

Describe how the graphs of f (x) =√x (on the left) and g(x)

(on the right) are related. Then write an equation for g(x).

Example

Describe how the graphs of f (x) =√x (on the left) and g(x)

(on the right) are related. Then write an equation for g(x).

−2 −1 1 2 3

Example

Describe how the graphs of f (x) = |x | (on the left) and g(x)(on the right) are related. Then write an equation for g(x).

−3 −2 −1 1 2 3

−1 1

What You Learned

You can now:

I Do problems Chap 1.5 # 1-15 odd, 19-23 odd, 25, 29,33, 35, 39, 45

What You Learned

You can now:

What You Learned

You can now:

What You Learned

You can now:

What You Learned

You can now:

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