Review 3.1-3.4

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Review 3.1-3.4

Pre-Calculus

Determine whether the graph of each relation is symmetric with respect to the x-axis, the y-axis, the line y = x, the line y = -x, and/or the origin.

Find a point that works: (9, 2)

Now test to see if each point exists according to the chart:

(a, b)

y-axis (-a, b)

x-axis (a, -b)

origin (-a, -b)

y = x (b, a)

y = -x (-b, -a)

Does (-9, 2) exists? NO

Does (9, -2) exists? YES

Does (-9, -2) exists? NO

Does (2, 9) exists? NO

Does (-2, -9) exists? NO

So this graph is symmetric w/ respect to the x-axis

Determine whether the graph of each relation is symmetric with respect to the x-axis, the y-axis, the line y = x, the line y = -x, and/or the origin.

I know this is a ellipse because it has two squared terms with two different coefficients.

It has a center (0, 0)

So this graph is symmetric w/ respect to thex-axis, y-axis, and origin.

Find a point that works: (1, 5)

Now test to see if each point exists according to the chart:

(a, b)

y-axis (-a, b)

x-axis (a, -b)

origin (-a, -b)

y = x (b, a)

y = -x (-b, -a)

Does (-1, 5) exists? NO

Does (1, -5) exists? NO

Does (-1, -5) exists? YES

Does (5, 1) exists? NO

Does (-5, -1) exists? NO

So this graph is symmetric w/ respect to the origin

Determine whether the graph of each relation is symmetric with respect to the x-axis, the y-axis, the line y = x, the line y = -x, and/or the origin.

Determine whether each function is even, odd or neither.

If all the signs are opposite, then the function is EVEN

Figure out f(-x) and –f(x)

Determine whether each function is even, odd or neither.

If all the signs are opposite and the same, then the function is NEITHER even or odd.

Figure out f(-x) and –f(x)

Determine whether each function is even, odd or neither.

If all the signs are the same,then it is ODD

Figure out f(-x) and –f(x)

Describe the transformations that has taken place in each family graph.

Right 5 units

Up 3 units

More Narrow

More Narrow, and left 2 units

Describe the transformations that has taken place in each family graph.

More Wide, and right 4 unitsRight 3 units, and up 10 units

More Narrow

Reflected over x-axis, and moved right 5 units

Describe the transformations that has taken place in each family graph.

Reflect over x-axis, and up 2 units

Reflected over y-axis

Right 2 units

FINDING INVERSE FUNCTIONSFINDING INVERSE FUNCTIONS

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of ,

y x 2

x y 2

x y2

y x

f 1(x) x , x 0

f (x) x 2

x 0

FINDING INVERSE FUNCTIONSFINDING INVERSE FUNCTIONS

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of f (x) = 4x + 5

y 4x 5

x 4y 5

x 5 4y

x 54

y

f 1(x) x 5

4

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of f (x) = 2x3 - 1

f 1(x) x 1

23

y 2x 3 1

x 2y 3 1

x 12y 3

x 1

2y 3

y x12

3

STEPS

Replace f (x) with y

Interchange the roles of x and y

Solve for y

Replace y with f -1(x)

Find the inverse of

Find the inverse of Steps for findingan inverse.

1. solve for x

2. exchange x’sand y’s

3. replace y with f-1

Let’s consider the function and compute some values and graph them.

3xxf

x f (x)

-2 -8-1 -1 0 0 1 1 2 8

Is this a function? Yes

What will “undo” a cube? A cube root

31 xxf

This means “inverse function”

x f -1(x)

-8 -2-1 -1 0 0 1 1 8 2

Let’s take the values we got out

of the function and put them into the inverse function

and plot them

These functions are reflections of each other about

the line y = x

3xxf

31 xxf

(2,8)

(8,2)

(-8,-2)

(-2,-8)

Graph then function and it’s inverse of the same graph.

Parabola shifted 4 units left, and 1 unit down

Now to graph the inverse, just take each point and switch the x and y value and graph

the new points.

Ex: (-4, -1) becomes (-1, -4)

Finally CHECK yourself by sketching the line y = x and make sure

your graphs are symmetric with that

line.

Graph then function and it’s inverse of the same graph.

Cubic graph shifted 5 units to the left

Now to graph the inverse, just take each point and switch the x and y value and graph

the new points.

Ex: (-5, 0) becomes (0, -5)

Finally CHECK yourself by sketching the line y = x and make sure

your graphs are symmetric with that

line.

Graph then function and it’s inverse of the same graph.

Parabola shifted down 2 units

Now to graph the inverse, just take each point and switch the x and y value and graph

the new points.

Ex: (0, -2) becomes (-2, 0)

Finally CHECK yourself by sketching the line y = x and make sure

your graphs are symmetric with that

line.

Graph

Vert asymp:x2-4=0x2=4x=2 & x=-2

Horiz asymp:(degrees are =)y=3/1 or y=3

4

32

2

x

xy

x y

4 4

3 5.4

1 -1

0 0

-1 -1

-3 5.4

-4 4

left of x=-2 asymp.

Between the 2 asymp.

right of x=2 asymp.

Domain: all real #’s except -2 & 2

Range: all real #’s except 0<y<3

Find the horizontal asymptote:

x

. f xx

2 11

2

x. f x

x

3

2

12

x

. f xx x2

23

20

H.A. : y 2

H.A. : none

H.A. : y 0

Exponents are the same; divide the coefficients

Bigger on Top; None

Bigger on Bottom; y=0

Find the domain. Excluded values are where your vertical asymptotes are.

6

62

xx

xR

062 xx

023 xx

2,3 so xx

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find horizontal or oblique asymptote by comparing degrees

degree of the top = 0

0xremember x0

= 1

degree of the bottom = 2

If the degree of the top is less than the degree of the bottom the x axis is a horizontal asymptote.

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find some points on either side of each vertical asymptote

x R(x)

Choose an x on the left side of the

vertical asymptote.

-4

4.014

6

644

64 2

R

0.41

16

6

611

61 2

R

-1

Choose an x in between the vertical asymptotes.

Choose an x on the right side of the vertical

asymptote.

4

16

6

644

64 2

R

1

6

62

xx

xR

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Connect points and head towards asymptotes.

There should be a piece of the graph on each side of the vertical asymptotes.

Pass through the point and head

towards asymptotes

Pass through the points and head towards asymptotes. Can’t go up or it would cross the x axis and there

are no x intercepts there.

Pass through the point and head

towards asymptotes

Go to a function grapher or your graphing calculator and see how we

did.

Find the domain. Excluded values are where your vertical asymptotes are.

9

342

2

x

xxxR

092 x

033 xx3,3 so xx

Let's try another with a bit of a "twist":

But notice that the top of the fraction will factor and the fraction can then be

reduced.

33

13

xx

xx

We will not then have a vertical asymptote at x = -3, It will be a HOLE at x = -3

vertical asymptote from this factor only since other factor cancelled.

3

1

x

xxS

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

Find horizontal or oblique asymptote by comparing degrees

degree of the top = 1

degree of the bottom = 1

If the degree of the top equals the degree of the bottom then there is a horizontal

asymptote at y = leading coefficient of top over leading coefficient of bottom.

11

1y

1

1

3

1

x

xxS

Find some points on either side of each vertical asymptote

x S(x)

4 5

Let's choose a couple of x's on the right side of the vertical asymptote.

51

5

34

144

S

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

We already have some points on the left side of the vertical asymptote so we can

see where the function goes there

3.23

7

36

166

S

6 2.3

3

1

x

xxS

Connect points and head towards asymptotes.

There should be a piece of the graph on each side of

the vertical asymptote.

Pass through the points and head

towards asymptotes

Pass through the point and head

towards asymptotes

Go to a function grapher or your graphing calculator and see how we

did.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

REMEMBER that x -3 so find the point on the graph where x is -3 and make a "hole" there since it

is an excluded value.

3

1

33

133

S

Find the equations of the horizontal asymptotes of:

3xf x

x 4

f x

4

3x

x

y 3

2

2

x 2xf x

3 4x

2

2

2xf x

3

x

4x

1y

4

4 2

2

x 2x 1f x

x x 1

24

2

2x

1

x 1f x

xx

none

1f x

2x f x

x

1

2 y 0

3

3

4xf x

x 1

3

3f

4xx

x 1

y 4

4

3

3x 4f x

x 3x

4

3

4f x

3

3x

x x

none

1f x

x

Vertical Asymptotes: x 0Horizontal Asymptotes: y 0

Holes: none

Intercepts: none

x 2

f xx 2 x 2

Vertical Asymptotes: x 2Horizontal Asymptotes: y 0

Holes: 1

2,4

Intercepts:

1

x 2

10,

2

10,

2

3x 12

xf x

1

Vertical Asymptotes: x 1Horizontal Asymptotes: y 1

Holes: none

Intercepts: 0,1

3x 10 2

1 x

3x 1

21 x

2 2x 3x 1

x 1

1, 0

2

2

x 5x 6f x

x 2x 3

Vertical Asymptotes: x 1Horizontal Asymptotes: y 1

Holes: 13,

4

Intercepts: 0, 2

2, 0

x 2 x 3

x 1 x 3

2

3x 9Simplify :

x 9

3 x 3

x 3 x 3

3

x 3

Extension: The graph contains an hole at x = -3

Note: Cancelled and eliminated

Extension: The graph contains an asymptote at x = 3

Note: not eliminated

Vertical Asymptotes: x 3Horizontal Asymptotes: y 0

Holes: 13,

2

Intercepts: 0, 1

2

3x 9f x

x 9

3 x 3

x 3 x 3

3

x 3

24x 8xSimplify :

12x 24

4x x 2

12 x 2

x

3

Extension: The graph contains a hole at x = -2

Note: cancelled and eliminated

Vertical Asymptotes: none

Horizontal Asymptotes: none

Holes: 22,

3

Intercepts: 0, 0

24x 8x

f x12x 24

4x x 2

12 x 2

x

3

Graph the rational function which has the following characteristics

Vert Asymp at x = 1, x = -3

Horz Asymp at y = 1

Intercepts (-2, 0), (3, 0), (0, 2)

Passes through (-5, 2)

Graph the rational function which has the following characteristics

Vert Asymp at x = 1, x = -1

Horz Asymp at y = 0

Intercepts (0, 0)

Passes through (-0.7, 1), (0.7, -1), (-2, -0.5), (2, 0.5)