company logo review 3.1-3.4 pre-calculus. determine whether the graph of each relation is symmetric...
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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)