relations and functionscontent.njctl.org/courses/math/archived-coursesunits...2009/05/13 ·...

Relations and Functions

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Table of Contents

Midpoint and Distance FormulaCirclesDomain and RangeDiscrete v ContinuousRelations and Functionsy = a f( b(x + c)) + d Transformations

Operations with FunctionsCompositionsInverses

Vertex Form of a Parabola

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Midpoint and DistanceFormula

Return to Tableof Contents

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The Midpoint FormulaGive points A(x1,y1) and B (x2,y2), the point midway between A and B is

Examples: Find the midpoint of the segment with the given endpoints.

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1 Find the midpoint of K(1,8) & L(5,2).

A (2,3)

B (3,5)

C (-2,-3)

D (-3,-5)

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2 Find the midpoint of H(-4 , 8) & L(6, 10).

A (2,8)

B (-5,-2)

C (-2,-8)

D (5,2)

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3 Given the midpoint of a segment is (4 , 9) and one endpoint is (-3 , 10), find the other midpoint.

A (-10 , 8)

B (11 , 8)

C (-10 , 11)

D (.5 , 9.5)

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Example: If the distance between (3, -2) and (8, y) is 6, find the possible values of y.

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4 What is the distance between (2, 4) and (-1, 8)?

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5 What is the distance between (0, 7) and (5, -5)?

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6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x.

A -7

B -5

C -3

D -1

E 0

F 1

G 3

H 5

I 7

J 9

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Circles

Return to Tableof Contents

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7 Write the equation of the circle with center (5 , 2) and radius 6

A

B

C

D

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8 Write the equation of the circle with center (-5 , 0) and radius 7

A

B

C

D

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9 Write the equation of the circle with center (-2 , 1) and radius

A

B

C

D

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10 What is the center and radius of the following equation?

A

B

C

D

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12 What is the center and radius of the following equation?

A

B

C

D

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13 What is eccentricity of a circle?

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Write the equation of the circle in standard form that meets the following criteria:

Complete the square for the x's

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14 What is the equation of the circle that has a diameter with endpoints (0 , 0) and (16 , 12)?

A

B

C

D

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15 What is the equation of the circle with center (-3 , 5) and contains point (1, 3)?

A

B

C

D

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16 What is the equation of the circle with center (7 , -3) and tangent to the x-axis?

A

B

C

D

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Domain and Range

Return toTable of Contents

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Domain is the values of x that work for a given relation or equation.

What is the domain for each of the following?

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234

7-3 8

-2 3-5

412

589

x y x y x y

{2, 3, 4} {1, 2}{-2, 3, -5}

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X Y

1 2

2 3

3 2

4 3

5 2

X Y

1 3

2 4

5 -5

3 9

4 7

X Y

-3 4

-1 5

0 8

-1 9

3 11

{1, 2, 3, 4, 5} {1, 2, 3, 4, 5} {-3, -1, 0, 3}

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{-2, 0, 2, 3} {reals} {-3 < x < -1}

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19 What is the domain of the following?

A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x<0

I x>0

J Reals

{(3,1), (2,-1), (1,1)}

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x<0

I x>0

J Reals

-103

-2-1 0

x y

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x<0

I x>0

J Reals

X Y

-2 3

0 2

-1 -1

3 2

-2 0

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x<0

I x>0

J Reals

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x<0

I x>0

J Reals

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x<0

I x>0

J Reals

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H x<0

I x>0

J Reals

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Range is the values of y that work for a given relation or equation.

What is the range for each of the following?

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234

7-3 8

-2 3-5

412

589

x y x y x y

{-3, 7, 8} {5, 8, 9}{4}



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X Y

1 2

2 3

3 2

4 3

5 2

X Y

1 3

2 4

5 -5

3 9

4 7

X Y

-3 4

-1 5

0 8

-1 9

3 11

{-5, 3, 4, 7, 9} {2, 3} {4, 5, 8, 9, 11}



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{-3, -2, -1, 2, 3} {2} {-3 < y < 4}



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26 What is the range of the following?

A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y<0

I y>0

J Reals

{(3,1), (2,-1), (1,1)}

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y<0

I y>0

J Reals

-103

-2-1 0

x y

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y<0

I y>0

J Reals

X Y

-2 3

0 2

-1 -1

3 2

-2 0

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y<0

I y>0

J Reals

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y<0

I y>0

J Reals

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y<0

I y>0

J Reals

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A -3

B -2

C -1

D 0

E 1

F 2

G 3

H y<0

I y>0

J Reals

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Discrete v Continuous


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A relation is discrete if it is made up separate points.

For example, you go to a bakery to buy donuts. How many can you buy? 0, 1, 2, 3, . . . These are separate values. 1.2, 1.375, 1.5899 do not have meaning.

A relation is continuous if the points are not separate.

For example, the repairman says he will be to your home between 1 and 5. What time could he show up? 1, 2, 3, 4, 5. Do the values between 1 and 2 have meaning?

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Are the following relations discrete or continuous?

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33 Is the given relation discrete or continuous?

A Discrete

B Continuous

{(3,1), (2,-1), (1,1)}

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A Discrete

B Continuous

-103

-2-1 0

x y

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A Discrete

B Continuous

X Y

-2 3

0 2

-1 -1

3 2

-2 0

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A Discrete

B Continuous

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A Discrete

B Continuous

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A Discrete

B Continuous

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A Discrete

B Continuous

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Relations and Functions


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A function is a relation that has each value in the domain has exactly one value in the range. In other words, x does not repeat.

Is each of the following relations a function?

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234

7-3 8

-2 3-5

412

589

x y x y x y



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X Y

1 2

2 3

3 2

4 3

5 2

X Y

1 3

2 4

5 -5

3 9

4 7

X Y

-3 4

-1 5

0 8

-1 9

3 11



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On a graph, a function does not have a point above another point. This is called the Vertical Line Test.


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An equation is a function only if when an x is substituted in there is only 1 y-value.


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40 Is the following relation a function?

Yes

No

{(3,1), (2,-1), (1,1)}

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Yes

No

-103

-2-1 0

x y

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Yes

No

X Y

-2 3

0 2

-1 -1

3 2

-2 0

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Yes

No

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Yes

No

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Yes

No

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Yes

No

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y = a f( b(x + c)) + d

Return toTable ofContents

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Vertical Shifts

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Vertical Shifts occur when a constant is added to a function.

The parent function y = f(x) is slid:upward if c > 0

downward if c < 0

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Let the graph of f(x) be

Graph y = f(x) + 2

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Graph y = f(x) - 3

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Graph y = f(x) + 1

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Graph y = f(x) - 1

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Consider the graph y = x2 and the rules for vertical shifts,

Graph

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Consider the graph y = x2 and the rules for vertical shifts,

Graph

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Consider the graph y = |x| and the rules for vertical shifts,

Graph

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Consider the graph y = |x| and the rules for vertical shifts,

Graph

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Consider the graph y = #x and the rules for vertical shifts,

Graph

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47 Given the graph of h(x), which of the following graphs is y = h(x) +1 ?

A B

C D

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48 Given the graph of h(x), which of the following graphs is y = h(x) - 1 ?

A B

C D

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Horizontal Shifts

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Horizontal Shifts occur when a constant is added to a function.

The parent function y = f(x) is slid:left if c > 0

right if c < 0

Notice the direction is opposite the sign of c.

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Graph y = f(x + 2)

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Graph y = f(x - 3)

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Graph y = f(x + 1)

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Graph y = f(x - 1)

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Consider the graph y = x2 and the rules for horizontal shifts,

Graph

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Consider the graph y = x2 and the rules for horizontal shifts,

Graph

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Consider the graph y = |x| and the rules for horizontal shifts,

Graph

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Consider the graph y = |x| and the rules for horizontal shifts,

Graph

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Consider the graph y = #x and the rules for horizontal shifts,

Graph

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Consider the graph y = #x and the rules for horizontal shifts,

Graph

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49 Given the graph of h(x), which of the following graphs is y = h(x + 1) ?

A B

C D

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50 Given the graph of h(x), which of the following graphs is y = h(x - 1)?

A B

C D

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Reflections

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Reflection over the x-axis:

Reflection over the y-axis:

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Graph y = f(-x)

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Graph y = - f(x)

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Consider the graph y = x2 and the rules for reflections,

Graph

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Consider the graph y = |x| and the rules for reflections,

Graph

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Consider the graph y = #x and the rules for reflections,

Graph

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Consider the graph y = #x and the rules for reflections,

Graph

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51 Given the graph of h(x), which of the following graphs is y = -h(x) ?

A B

C D

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52 Given the graph of h(x), which of the following graphs is y = h(-x) ?

A B

C D

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53 Given the graph of h(x), which of the following graphs is y = -h(-x) ?

A B

C D

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Vertical Stretch & Shrink

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Vertical Stretches and Shrinks occur when a constant is multiplied to a

function.

The parent function y = f(x) is :streched if |c| > 1

shrunk if 0 < |c| < 1

Stretches and shrinks are the first transformation that do not yield

congruent figures.

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Graph y = 2f(x)

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Graph y =(1/3 )f(x)

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Graph y = 3f(x)

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Graph y = (1/2)f(x)

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Consider the graph y = x2 and the rules for stretches and shrinks,

Graph

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Graph

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Consider the graph y = |x| and the rules for stretches and shrinks,

Graph

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Graph

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Consider the graph y = #x and the rules for stretches and shrinks,

Graph

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Graph

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54 Given the graph of h(x), which of the following graphs is y = 2h(x)?

A B

C D

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55 Given the graph of h(x), which of the following graphs is y =1/2 h(x)?

A B

C D

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Horizontal Stretch & Shrink

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Horizontal Stretches and Shrinks occur when a constant is multiplied to

x in a function.

The parent function y = f(x) is:shrunk if |c| > 1

stretched if 0 < |c| < 0

Horizontal stretches and shrinks will also impact the vertical stretches and shrinks.

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Graph y = f(2x)

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Graph y = f(3x)

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Graph y = f(.5x)

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Graph y = f(.75x)

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Graph

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Graph

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Graph

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Graph

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Graph

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Graph

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56 Given the graph of h(x), which of the following graphs is y = h(2x)?

A B

C D

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57 Given the graph of h(x), which of the following graphs is y = h(1/2x)?

A B

C D

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Combining Transformations

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Graph y = 2f(.5x+1) - 2

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Graph y =(-1/3 )f(2x + 1) + 2

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Graph y = 3f(-.5x - 2) + 1

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Graph y = (-1/2)f(-x + 2) +1

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Graph

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Graph

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Graph

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58 Given the graph of h(x), which of the following graphs is y = 2h(-x+1) - 3?

A B

C D

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59 Given the graph of h(x), which of the following graphs is y = -0.5h(2x - 1) + 2?

A B

C D

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Vertex Form of a

ParabolaReturn toTable ofContents

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Vertex FormA quadratic can be written in vertex form:

Vertex form shows the location of the vertex (h , k).The a still tells the direction of the openness.

Example: Find the vertex, direction of openness and the axis of symmetry of:

Vertex: (6 , 4)

Direction of openness is up

Axis of Symmetry: x = 6

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Find the vertex, direction of openness and the axis of symmetry of:

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60 What is the vertex of

A

B

C

D

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61 What is direction of openness of

A up

B down

C left

D right

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62 The axis of symmetry of is x =

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A up

B down

C left

D right

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A up

B down

C left

D right

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Converting from Standard Form to Vertex FormTo convert, use completing the square.

Step 1 - Write the equation in the form y = x2 + bx +__+ c - __

Step 2 - Find (b ÷ 2)2

Step 3 - Complete the square by adding (b ÷ 2)2 to the first blank and subtracting it in the second blank.

Step 4 - Factor the perfect square trinomial.

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Let's look at an example to solve:

y= x2 + 14x +__+ 15 -__ Step 1

(14 ÷ 2)2 = 49 Step 2 - Find (b÷2)2

y= x2 + 14x + 49 + 15 - 49 Step 3 - Complete the Square

y= (x + 7)2 - 34 Step 4 - Factor and simplify

y= x2 + 14x + 15

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y= x2 - 12x +__- 16 -__ Step 1

(12 ÷ 2)2 = 36 Step 2 - Find (b÷2)2

y= x2 - 12x + 36 - 16 - 36 Step 3 - Complete the Square

y= (x + 6)2 - 52 Step 4 - Factor and simplify

y= x2 - 12x - 16

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72 What is the vertex form of:

A

B

C

D

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A

B

C

D

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A

B

C

D

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Converting from Standard Form to Vertex FormWhat if "a" does not equal 1?

Step 1 - Write the equation in the form y = ax2 + bx +__+ c - __

Step 2 - Factor: y = a(x2 + (b/a)x +__)+ c - __

Step 3 - Find (b/a ÷ 2)2

Step 4 - Complete the square by adding (b/a ÷ 2)2 to the first blank and subtracting a(b/a ÷ 2)2 in the second blank.

Step 5 - Factor the perfect square trinomial.

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y= 3x2 + 12x +__- 10 -__ Step 1

y= 3(x2 + 4x + __) - 10 - __ Step 2 - Factor out "a"

(4 ÷ 2)2 = 4 Step 3 - Find (b÷2)2

y= 3(x2 + 4x + 4) - 10 - 12 Step 4 - Complete the Square

y= 3(x + 2)2 - 22 Step 5 - Factor and simplify

y= 3x2 + 12x - 10

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y= -4x2 - 32x +__+ 45 -__ Step 1

y= -4(x2 + 8x + __) + 45 - __ Step 2 - Factor out "a"

(8 ÷ 2)2 = 16 Step 3 - Find (b÷2)2

y= -4(x2 - 8x + 16) + 45 - -64 Step 4 - Complete the Square

y= -4(x - 4)2 + 109 Step 5 - Factor and simplify

y= -4x2 - 32x + 45

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Operations with Functions


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Functions can be combined to make other functions.Given: andFind if:

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78 Given and , find h(x) if

A

B

C

D

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A

B

C

D

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A

B

C

D

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A

B

C

D

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Compositions


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Composite functions are when functions is substituted into a second function.

There are 2 ways of writing a composite:

Method #1:

Method #2:

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To simplify a composite of functions, substitute into the "inner" function, simplify, and then substitute that value in for the variable in the "outer" function.Examples: and

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82 Find the value of

A

B

C

D

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A

B

C

D

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A

B

C

D

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A

B

C

D

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A

B

C

D

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Inverses


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87 What is ?

A undefined

B 0

C 2

D infinite solutions

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88 What is ?

A undefined

B x

C 2x

D infinite solutions

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The graph of an inverse is the reflection of the function over y=x.

Can you make a conjecture about the x- and y-values of the function and it's inverse?

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Draw the inverse of the given function.

Is the inverse a function? Why or why not?

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Draw the inverse of the given function.

Is the inverse a function? Why or why not?

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To determine if an inverse is going to be a function, use the Horizontal Line Test. That is, "Does every horizontal line drawn touch the function no more than once?" It only takes one horizontal line to touch more than once for the function to fail the horizontal line test and there for it's inverse would not be a function.

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89 Will the inverse of the given function be a function?

Yes

No

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90 Which graph is the inverse of the given function?

A

B

C D

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A

B

C D

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A

B

C D

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93 Will the inverse of f(x)= 2x2 - 23x +4 be a function?

Yes

No

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The inverse of a function is the reflection over y=x which results in the swapping of x- and y- values.

Example: Find the inverse of f(x)= {(1,2),(3,5),(-7,6)}

f-1(x)={(2,1),(5,3),(6,-7)

Example: Find the inverse of X Y

3 2

4 4

5 -5

6 7

X Y

2 3

4 4

-5 5

7 6

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94 What is the inverse of {(1,4), (5,3),(2,-1)}?

A {(4,1), (3,5), (2,-1)}

B {(-1,-4), (-5,-3), (-2,1)}

C {(4,1), (3,5),(-1,2)}

D {(-4,-1), (-3,-5), (1,-2 )}

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95 If the inverse of a function is {(1,0), (3,3),(-4,-5)}, what was the original function?A {(0,1), (3,3), (-5,-4)}

B {(-1,-4), (-5,-3), (-2,1)}

C {(0,-1), (-3,-3),(4,5)}

D {(0,1), (3,3), (-4,-5)}

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96 What is the inverse of:

A B C DX Y

4 3

0 1

3 -2

7 4

X Y

4 3

1 0

2 3

4 7

X Y

4 3

0 0

-2 3

7 4

X Y

-3 -4

-1 0

-2 -3

-4 -7

X Y

3 4

1 0

-2 3

4 7

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97 Will the inverse of be a function?

Yes No

X Y

1 2

2 3

3 4

5 2

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Knowing that the inverse of a function switches x- and y-values, this fact can be used when given an equation.

Given:

Swap x and y's:Solve for y:

So the inverse of is

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Find the inverse of the following functions.

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98 Which of the following choices is the inverse of f(x) if

A

B

C

D

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A

B

C

D

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A

B

C

D

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101 Find the inverse of

A

B

C

D

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A

B

C

D

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103 What is the inverse of y = 3?

A y = 1/3

B x = 3

C x = 1/3

D not possible

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A

B

C

D

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