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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
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
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
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
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
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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.
W. Finch DHS Math Dept
Transformations 3 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
−3
−2
−1
1
2
3
x
y
W. Finch DHS Math Dept
Transformations 4 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Quadratic Function
Parent Quadratic Function (Squaring Function)I Equation f (x) = x2
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
1
2
3
x
y
W. Finch DHS Math Dept
Transformations 5 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
1
2
3
x
y
W. Finch DHS Math Dept
Transformations 6 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Cubic Function
Parent Cubic Function
I Equation f (x) = x3
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
−3
−2
−1
1
2
3
x
y
W. Finch DHS Math Dept
Transformations 7 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
1
2
3
x
y
W. Finch DHS Math Dept
Transformations 8 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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 Symmetry wrt origin
I Decreasing (−∞, 0) and (0,∞)
I Vertical asymptote y -axis
I Horizontal asymptote x-axis
−3 −2 −1 1 2 3
−3
−2
−1
1
2
3
x
y
W. Finch DHS Math Dept
Transformations 9 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Greatest Integer Function
Parent Greatest Integer Function (Step Function
I Equation f (x) = [[x ]]
I Domain (−∞,∞)
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
−4
−2
2
4
x
y
W. Finch DHS Math Dept
Transformations 10 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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) =
W. Finch DHS Math Dept
Transformations 11 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Graph a Piecewise-Defined Function
Sketch a graph offunction f : f (x) =
{−x + 4 x ≤ 1
−(x − 1)2 x > 1
W. Finch DHS Math Dept
Transformations 12 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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 ?
W. Finch DHS Math Dept
Transformations 13 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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 ?
W. Finch DHS Math Dept
Transformations 13 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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 ?
W. Finch DHS Math Dept
Transformations 13 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
−4
−2
2
4
f (x)
g(x)
h(x)
x
y
W. Finch DHS Math Dept
Transformations 14 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Vertical Translations
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.
W. Finch DHS Math Dept
Transformations 15 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Vertical Translations
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.
W. Finch DHS Math Dept
Transformations 15 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Translations
Graph the following function on your calculator as Y1 :
f (x) = |x |
Now add the following functions into Y2 and Y3 :
g(x) = |x + 3|
h(x) = |x − 4|
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 16 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Translations
Graph the following function on your calculator as Y1 :
f (x) = |x |
Now add the following functions into Y2 and Y3 :
g(x) = |x + 3|
h(x) = |x − 4|
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 16 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Translations
Graph the following function on your calculator as Y1 :
f (x) = |x |
Now add the following functions into Y2 and Y3 :
g(x) = |x + 3|
h(x) = |x − 4|
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 16 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Horizontal Translations
f (x) is the parent function.
g(x) is a horizontal shiftleft 3 units.
h(x) is a horizontal shiftright 4 units.
−6 −4 −2 2 4 6
2
4
f (x)
g(x)h(x)
x
y
W. Finch DHS Math Dept
Transformations 17 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Horizontal Translations
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.
W. Finch DHS Math Dept
Transformations 18 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Horizontal Translations
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.
W. Finch DHS Math Dept
Transformations 18 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflections
Graph the following function on your calculator as Y1 :
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 ?
W. Finch DHS Math Dept
Transformations 19 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflections
Graph the following function on your calculator as Y1 :
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 ?
W. Finch DHS Math Dept
Transformations 19 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflections
Graph the following function on your calculator as Y1 :
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 ?
W. Finch DHS Math Dept
Transformations 19 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflection wrt x-axis
f (x) is the parent function.
g(x) is a reflection wrtx-axis.
−4 −2 2 4
−4
−2
2
4
f (x)
g(x)
x
y
W. Finch DHS Math Dept
Transformations 20 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflections
Graph the following function on your calculator as Y1 :
f (x) =√x
Now add the following function into Y2 :
h(x) =√−x
How would you describe the graph of h compared to the graphof f ?
W. Finch DHS Math Dept
Transformations 21 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflections
Graph the following function on your calculator as Y1 :
f (x) =√x
Now add the following function into Y2 :
h(x) =√−x
How would you describe the graph of h compared to the graphof f ?
W. Finch DHS Math Dept
Transformations 21 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflections
Graph the following function on your calculator as Y1 :
f (x) =√x
Now add the following function into Y2 :
h(x) =√−x
How would you describe the graph of h compared to the graphof f ?
W. Finch DHS Math Dept
Transformations 21 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Reflection wrt y -axis
f (x) is the parent function.
h(x) is a reflection wrty-axis.
−6 −4 −2 2 4 6
2
4
f (x)g(x)
x
y
W. Finch DHS Math Dept
Transformations 22 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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.
W. Finch DHS Math Dept
Transformations 23 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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.
W. Finch DHS Math Dept
Transformations 23 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Nonrigid Transformations (Dilations)
Graph the following function on your calculator as Y1 :
f (x) =√x
Now add the following functions into Y2 and Y3 :
g(x) = 3√x
h(x) =1
4
√x
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 24 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Nonrigid Transformations (Dilations)
Graph the following function on your calculator as Y1 :
f (x) =√x
Now add the following functions into Y2 and Y3 :
g(x) = 3√x
h(x) =1
4
√x
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 24 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Nonrigid Transformations (Dilations)
Graph the following function on your calculator as Y1 :
f (x) =√x
Now add the following functions into Y2 and Y3 :
g(x) = 3√x
h(x) =1
4
√x
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 24 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Vertical Stretch/Shrink
f (x) is the parent function.
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
2
4
6
8
f (x)
g(x)
h(x)
x
y
W. Finch DHS Math Dept
Transformations 25 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Vertical Stretch/Shrink
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.
W. Finch DHS Math Dept
Transformations 26 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Vertical Stretch/Shrink
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.
W. Finch DHS Math Dept
Transformations 26 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Nonrigid Transformations (Dilations)
Graph the following function on your calculator as Y1 :
f (x) = |x |
Now add the following functions into Y2 and Y3 :
g(x) = |2x |
h(x) =
∣∣∣∣13x∣∣∣∣
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 27 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Nonrigid Transformations (Dilations)
Graph the following function on your calculator as Y1 :
f (x) = |x |
Now add the following functions into Y2 and Y3 :
g(x) = |2x |
h(x) =
∣∣∣∣13x∣∣∣∣
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 27 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Nonrigid Transformations (Dilations)
Graph the following function on your calculator as Y1 :
f (x) = |x |
Now add the following functions into Y2 and Y3 :
g(x) = |2x |
h(x) =
∣∣∣∣13x∣∣∣∣
How would you describe the graphs of g and h compared tothe graph of f ?
W. Finch DHS Math Dept
Transformations 27 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Horizontal Stretch/Shrink
f (x) is the parent function.
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
2
4
f (x)
g(x)h(x)
x
y
W. Finch DHS Math Dept
Transformations 28 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Horizontal Stretch/Shrink
A nonrigid transformation in the graph of y = f (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.
W. Finch DHS Math Dept
Transformations 29 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
Horizontal Stretch/Shrink
A nonrigid transformation in the graph of y = f (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.
W. Finch DHS Math Dept
Transformations 29 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
−4
−2
2
4
f (x)
x
y
−4 −2 2 4
−4
−2
2
4
g(x)
x
y
W. Finch DHS Math Dept
Transformations 30 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
−4
−2
2
4
f (x)
x
y
−4 −2 2 4
−4
−2
2
4
g(x)
x
y
W. Finch DHS Math Dept
Transformations 31 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
W. Finch DHS Math Dept
Transformations 32 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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|
W. Finch DHS Math Dept
Transformations 33 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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).
1 2 3
1
2
3
x
y
1 2 3
1
2
3
x
y
W. Finch DHS Math Dept
Transformations 34 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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).
1 2 3
1
2
3
x
y
−2 −1 1 2 3
−3
−2
−1
1
x
y
W. Finch DHS Math Dept
Transformations 35 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
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
2
3
x
y
−1 1
−4
−3
−2
−1
1
x
y
W. Finch DHS Math Dept
Transformations 36 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
What You Learned
You can now:
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.
I Do problems Chap 1.5 # 1-15 odd, 19-23 odd, 25, 29,33, 35, 39, 45
W. Finch DHS Math Dept
Transformations 37 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
What You Learned
You can now:
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.
I Do problems Chap 1.5 # 1-15 odd, 19-23 odd, 25, 29,33, 35, 39, 45
W. Finch DHS Math Dept
Transformations 37 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
What You Learned
You can now:
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.
I Do problems Chap 1.5 # 1-15 odd, 19-23 odd, 25, 29,33, 35, 39, 45
W. Finch DHS Math Dept
Transformations 37 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
What You Learned
You can now:
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.
I Do problems Chap 1.5 # 1-15 odd, 19-23 odd, 25, 29,33, 35, 39, 45
W. Finch DHS Math Dept
Transformations 37 / 37
Introduction Function Families Translations Reflections Nonrigid Abs Value Transform Ex Summary
What You Learned
You can now:
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.
I Do problems Chap 1.5 # 1-15 odd, 19-23 odd, 25, 29,33, 35, 39, 45
W. Finch DHS Math Dept
Transformations 37 / 37