Relations and Functions

Relations and Functions Linear Equations

Vocabulary:RelationDomainRange

FunctionFunction Notation

Evaluating Functions Slope-intercept Form Point-slope form:

Standard Form.No fractions!No decimals!

Slope

Transformation of f(x) Translation: Vertical (k > 0) Translation: Horizontal (h > 0)

Stretch or compression:

Vertical by a factor of a

Reflection

𝑓 (𝑥 )+𝑘Up k units

Down k units 𝑓 (𝑥 ) −𝑘𝑓 (𝑥−h )Right h units

Left h units 𝑓 (𝑥+h )

𝑎>1Stretch:Compression:

𝑎⋅ 𝑓 (𝑥 ) − 𝑓 (𝑥 )Across x-axis

Across y-axis 𝑓 (−𝑥 )0<𝑎<1

Absolute Value Function f(x) = |x|

0 when ,

0 when ,)(

xxx

xxxxf

-6 -4 -2 2 4 6-6

-4

-2

2

4

6

x

y

VertexAxis of symmetry

x = 0

Graphing1. Graph the vertex2. Graph right branch3. Use symmetry to graph left branch

Vertex: (h, k)Slope of right branch: ±a

khxay

Equation from graph1. Identify the vertex2. Find the slope of the right branch

Pg. 127 #1-20, 23-26, 31-37

Functions, Equations and GraphsFunctions Linear Equations Transformations Absolute Values

More Linear Equations

10 10 10 10 10

20 20 20 20 20

30 30 30 30 30

40 40 40 40 40

50 50 50 50 50

x

y

Is the relation a function?

Answer

A. {(1, 2); (2,3); (1,4); (3,5)}

B. {(1, 5); (2,5); (3,5)}

C.

x

y

Is the relation a function?

A. {(1, 2); (2,3); (1,4); (3,5)}

B. {(1, 5); (2,5); (3,5)}

C.

No

Yes

Yes

What is the domain and range?

Answer

A. {(1, 2); (2,3); (1,4); (3,5)}

B. {(1, 5); (2,5); (3,5)}

What is the domain and range?

A. {(1, 2); (2,3); (1,4); (3,5)}

B. {(1, 5); (2,5); (3,5)}

Domain:{1, 2, 3}Range: {2, 3, 4, 5}

Domain:{1, 2, 3}Range: {5}

Find the values

Answer

𝑓 (𝑥 )=−3 𝑥+4

𝑓 (5 )

𝑓 (−2 )

Find the values𝑓 (𝑥 )=−3 𝑥+4

𝑓 (5 )=−3 (5 )+4=− 11

𝑓 (−2 )=−3 (− 2 )+4=10

Find the value of

Answer

𝑓 (𝑥 )=7− 4 𝑥 𝑔 (𝑥 )=|4 −7 𝑥|+2

Find the value of

𝑓 (𝑥 )=7− 4 𝑥 𝑔 (𝑥 )=|4 −7 𝑥|+2

𝑓 (−3 )=7 − 4 (−3)𝑓 (−3 )=19

𝑔 (2 )=|4 − 7(2)|+2

𝑔 (2 )=12

𝑓 (−3 )+𝑔 (2 )=19+12¿31

Answer

Find the value of

𝑓 (𝑥 )=7− 4 𝑥 𝑔 (𝑥 )=|4 −7 𝑥|+2

Find the value of

𝑓 (𝑥 )=7− 4 𝑥 𝑔 (𝑥 )=|4 −7 𝑥|+2

𝑓 (5 )=7 − 4(5)𝑓 (5 )=−13

𝑔 (− 6 )=|4 −7 (− 6)|+2

𝑔 (− 6 )=48

𝑓 (5 ) −𝑔 (− 6 )=−13 − 48¿− 61

Find the slope between the points:

Answer

and and

Find the slope between the points: and and

𝑚=4 −(− 4)

3 − 2

¿81¿8

𝑚=(−1)− 33−(− 4 )

¿− 47

Write and equation in slope-intercept form:

Answer

and y-intercept and

Write and equation in slope-intercept form:

and y-intercept and

𝑦=−75𝑥+5

𝑦− 3=−12(𝑥+4)

𝑦− 3=−12𝑥−2

𝑦=−12𝑥+1

Write an equation in point-slope form:

Answer

and and

Write an equation in point-slope form: and and

𝑦− 3=−25(𝑥−5)

𝑦+4=8(𝑥− 2)

𝑦− 4=8(𝑥− 3)

𝑚=4 −(− 4)

3 − 2=8

1=8

Write an equation in standard form

Answer

and and

Write an equation in standard form and and

𝑦−1=−34(𝑥− 8)

4 𝑦− 4=− 3(𝑥− 8)4 𝑦− 4=− 3𝑥+24

3 𝑥+4 𝑦=28

𝑦−7=14(𝑥− 1)

4 𝑦− 28=(𝑥−1)4 𝑦− 28=𝑥− 1

−𝑥+4 𝑦=27

𝑚=6− 7

− 3−1=

− 1− 4

=14

-5 5

-5

5

x

y

Write an equation in point-slope form for the graph:

Answer

Write an equation in point-slope form for the graph:

-5 5

-5

5

x

y

𝑦− 4=−23(𝑥−2)

𝑦− 2=−23(𝑥− 5)

-5 5

-5

5

x

y

Describe the transformation or write the transformation in function, then sketch a graph of the transformation.

Answer

𝑓 (𝑥−3 )+2Function notation:

Description:

𝒇 (𝒙 )


𝑓 (𝑥−3 )+2Function notation:

Description:

-5 5

-5

5

x

y

Right 3Up 2

Function notation:

-5 5

-5

5

x

y

Answer


Description:

𝒇 (𝒙 )Stretch by 2Down 2


Function notation:

Description:

-5 5

-5

5

x

y2

Stretch by 2Down 2

-5 5

-5

5

x

y

Answer


Function notation:

Description:

𝒇 (𝒙 )12𝑓 (𝑥+2 ) −3


Function notation:

Description:

12𝑓 (𝑥+2 ) −3

-5 5

-5

5

x

y

Compress by ½ Left 2Down 3

-5 5

-5

5

x

y

Answer


Function notation:

Description:

𝒇 (𝒙 )Reflect across x-axisStretch by 2Right 1Up 4


−2 𝑓 (𝑥−1 )+4Function notation:

Description: Reflect across x-axisStretch by 2Right 1Up 4

-5 5

-5

5

x

y

Answer

Describe transformations of f(x) that produces g(x).

𝑓 (𝑥 )=2𝑥2 𝑔 (𝑥 )=− (𝑥−3 )2+4

Describe transformations of f(x) that produces g(x).

𝑓 (𝑥 )=2𝑥2 𝑔 (𝑥 )=− (𝑥−3 )2+4

−𝟏𝟐𝑓 (𝑥−𝟑 )+𝟒=−

𝟏𝟐

∙ 2 (𝑥−𝟑 )2+𝟒

−𝟏𝟐𝑓 (𝑥−𝟑 )+𝟒=− (𝑥−3 )2+4=𝑔 (𝑥)

Reflect across x-axis; compress by ½; right 3; up 4

-5 5

-5

5

x

y

Graph

Answer

Graph

Equation:

Vertex: (0, 0)Slope of right arm: 1

-5 5

-5

5

x

y

Graph

Answer

Equation:

Vertex: (−3, −4)Slope of right arm: 2

Graph

-5 5

-5

5

x

y

-5 5

-5

5

x

y

Write equation for the graph

Answer

Vertex: (−1, 2)Slope of right arm:

Equation:-5 5

-5

5

x

y

Write equation for the graph

𝑦=−23|𝑥+1|+2

Graph

Answer

Equation:

Vertex: (-2, 3)Slope of right arm: − ½

Graph

-5 5

-5

5

x

y

Graph

Answer

Equation:

Vertex: (3, −3)Slope of right arm: 2

Graph

-5 5

-5

5

x

y

Write an equation of line in standard form that is parallel to the given line and goes through the

given point.

Answer

Point:5 𝑥−2 𝑦=− 4

Write an equation of line in standard form that is parallel to the given line and goes through the

given point.

Point:5 𝑥−2 𝑦=− 4

−2 𝑦=− 5𝑥−4

𝑦=52𝑥+2

𝑦+4=52(𝑥− 1)

2 𝑦+8=5(𝑥− 1)2 𝑦+8=5𝑥−5

−5 𝑥+2 𝑦=−13

Answer

Write an equation of line in standard form that is perpendicular to the given line and goes through

the given point.Point:5 𝑥−2 𝑦=− 4

Write an equation of line in standard form that is perpendicular to the given line and goes through

the given point.Point:5 𝑥−2 𝑦=− 4

−2 𝑦=− 5𝑥−4

𝑦=52𝑥+2

𝑦+4=−25(𝑥−1)

5 𝑦+20=−2(𝑥− 1)5 𝑦+20=−2 𝑥+2

2 𝑥+5 𝑦=− 18

-5 5

-5

5

x

y

Graph the equation:

Answer

𝑦=−45𝑥−4

Graph the equation:

𝑦=−45𝑥−4

-5 5

-5

5

x

y

Graph the equation:

Answer

𝑦− 2=23(𝑥+4 )

Graph the equation:

𝑦− 2=23(𝑥+4 )

-5 5

-5

5

x

y

Answer

Graph the equation:

2 𝑥+3 𝑦=6

Graph the equation:

2 𝑥+3 𝑦=6

-5 5

-5

5

x

y

Relations and Functions

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Transcript of Relations and Functions