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