In this section, you will learn to: identify unit graphs of various functions transform a unit graph...
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Transcript of In this section, you will learn to: identify unit graphs of various functions transform a unit graph...
In this section, you will learn to:identify unit graphs of various
functionstransform a unit graph by
stretching, shifting and reflectingwrite the equation of a transformed
graph using the sketch of the graph
Common Unit Graphs:1) Constant Function:
3f x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
x
y
3f x
1 2 3 4 5-1-2-3-4-5
1
2
-1
-2
-3
-4
-5
x
y
Common Unit Graphs:2) Linear Function:
f x x
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
-3
-4
x
y
Common Unit Graphs:3) Absolute Value Function:
f x x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
x
y
Common Unit Graphs:3) Absolute Value Function:
f x x
1 2 3 4-1-2-3-4
1
2
3
4
-1
x
y
Common Unit Graphs:4) Quadratic Function:
2f x x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
x
y
Common Unit Graphs:4) Quadratic Function:
2f x x
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
x
y
Common Unit Graphs:5) Square Root Function:
f x x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
x
y
Common Unit Graphs:5) Square Root Function:
f x x
1 2 3 4-1-2
1
2
3
4
-1
-2
x
y
Common Unit Graphs:6) Cubic Function:
3f x x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Common Unit Graphs:6) Cubic Function:
3f x x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Common Unit Graphs:7) Rational Function:
1f x
x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Common Unit Graphs:7) Rational Function:
1f x
x
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
-3
-4
x
y
Summary of Graphing:Rigid Transformations: Shape/size do not
change a) Vertical shift c units upward: h x f x c
2f x x 2 2f x x
1 2 3-1-2-3
1
2
3
4
-1
x
y
1 2 3-1-2-3
1
2
3
4
5
6
-1
x
y
Summary of Graphing:Rigid Transformations: b) Vertical shift c units downward:
h x f x c
2f x x 2 2f x x
1 2 3-1-2-3
1
2
3
4
-1
x
y
1 2 3-1-2-3
1
2
-1
-2
-3
x
y
Summary of Graphing:Rigid Transformations: c) Horizontal shift c units to the
right: h x f x c
2f x x 22f x x
1 2 3-1-2-3
1
2
3
4
-1
x
y
1 2 3 4 5-1
1
2
3
4
-1
x
y
Summary of Graphing:Rigid Transformations: d) Horizontal shift c units to the left:
h x f x c
2f x x 22f x x
1 2 3-1-2-3
1
2
3
4
-1
x
y
1-1-2-3-4-5
1
2
3
4
-1
x
y
Summary of Graphing:Rigid Transformations: e) Reflection across the x-axis:
h x f x
2f x x 2f x x
1 2 3-1-2-3
1
2
3
4
-1
x
y
1 2 3-1-2-3
1
-1
-2
-3
-4
-5
x
y
Summary of Graphing:Rigid Transformations: f) Reflection across the y-axis:
f x x f x x
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
x
y
1 2 3 4-1-2-3-4
1
2
3
4
-1
-2
x
y
h x f x
Summary of Graphing:Non-Rigid Transformations:
Shape/size will change a) Vertical stretch by c units if c >
1 : h x c f x 2f x x 22f x x
1 2 3-1-2-3
1
2
3
4
-1
x
y
1 2 3-1-2-3
1
2
3
4
-1
x
y
Summary of Graphing:Non-Rigid Transformations: b) Vertical shrink by c units if 0 < c <
1: h x c f x
2f x x 21
2f x x
1 2 3-1-2-3
1
2
3
4
-1
x
y
1 2 3-1-2-3
1
2
3
4
-1
x
y
Graphing Examples:Describe the transformation of the
followingFunction:
This is an absolute value function shifteda) 4 units to the rightb) 6 units upc) reflection across the x-axisd) vertical shrink
14 6
2f x x
Graphing Examples:
14 6
2f x x f x x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
x
y
2 4 6 8 10 12-2-4
1
2
3
4
5
6
-1
-2
x
y
Y-Axis Reflection Graphing Examples:
2 4 1f x x f x x
2 4 1f x x
1 2 3 4 5 6-1-2-3-4
1
2
-1
-2
-3
-4
-5
-6
x
y
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Graphing Examples:Write a cubic equation with the
followingtransformations:
a) 3 units to the leftb) 2 units downc) reflection across the x-axisd) vertical stretch 33 3 2f x x
Graphing Examples:
33 3 2f x x 3f x x
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
1 2-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Writing an Equation: Write the equation of the graph
below.
1 2 3 4 5 6 7 8 9 10-1
1
2
3
4
-1
-2
-3
-4
x
y
Writing an Equation: Write the equation of the graph
below.
f x x
1 2 3 4 5 6 7 8 9 10-1
1
2
3
4
-1
-2
-3
-4
x
y
Writing an Equation: Write down the transformations.
3 units right 2 units up x-axis reflection Use (4,1) as a point on the graph
1 4 3 2
1 2
1 3 2
y a x h k
a
a
a y x
Writing an Equation: Write the quadratic equation of the
graph below.
1 2 3 4 5 6 7-1-2-3-4-5
1
2
3
-1
-2
-3
-4
-5
x
y
Writing an Equation: The vertex has been translated 1
unit to the right and 1 unit up. This represents (h,k).
The graph has been reflected across the x-axis.
Use one point on the graph, the vertex and solve for the value of a for the quadratic equation
2 .y a x h k
Writing an Equation: The vertex is (h,k) which is (1,1). One point on the graph is Solve for a.
2
21 3 1 1
2 4
1
2
y a x h k
a
a
a
3, 1 .
21Therefore, 1 1
2y x
Writing an Equation: Sketch the graph of f(x+1) given
the following function.
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y
Writing an Equation: Sketch the graph of f(x)-3 given the
following function.
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y
Writing an Equation: Sketch the graph of f(-x) given the
following function.
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y
Writing an Equation: Sketch the graph of - f(x)+1 given the
following function.
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y
Writing an Equation: Sketch the graph of 2f(x)-1 given the
following function.
1 2 3 4 5 6-1-2-3-4-5-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
x
y