ExpoandPower
Short Description
Exponents and Power Mathematics at IGCSE and O Level...
Description
Exponential and Power Graphs
Exponential and Exponential Power Graphs
Curriculum Ready
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Exponential & Power EXPONENTIAL EXPONENTIA L & Graphs POWER
GRAPHS
These are graphs which result from equaons that are not linear or quadrac. The exponenal graph has the variable as the exponent. The power graphs raise the variable to any powern power n.
Answer these quesons, before working through the chapter.
I used to think: Which of these equaons is for an exponenal graph and which is for a power graph: 7 x y = 7 or y = x ?
For which value of x of x is is 2 x equal to zero?
Is it possible for y
= 5x
4
to be negave? Why?
Answer these quesons, after working working through the chapter.
But now I think: Which of these equaons is for an exponenal graph and which is for a power graph: 7 x y = 7 or y = x ?
For which value of x of x is is 2 x equal to zero?
Is it possible for y
= 5x
4
to be negave? Why?
What do I know now that I didn’t know before?
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1
Basics
Exponential & Power Graphs Exponential Graphs
Exponenal graphs are of funcons with the variable in the exponent of the form y = a x or y =
` 1a j where a 2 1 . x
They have this form: This could also be x
y =
y = a
y
^1, ah
1
` 1a j
x
^-1, ah
wrien as
-x
a
y
1
x
x
Here are some important properes about exponenal graphs: •
They always cut the y-axis at ^0, 1h since a0
•
The exponenal graph never cuts the x -axis since
•
The greater the value of a (the base), the steeper the curve.
Sketch the graphs of y
=2
x
and y
=
= 1 for a
any value of a. x
is never negave or zero if a 2
0.
` 12 j on the same set of axes x
y
4
y =
` 12 j
x
x
y= 2
3
^1, 2h
2
^-1, 2h
1
The y-intercept of ALL exponenal curves is always ^0, 1h
^-1, 0.5 h
^1, 0.5 h x
-2
2
K
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-1
0
1
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2
No x-intercepts
Basics
Exponential & Power Graphs
The graphs below are of the funcons y
=3
x
and y
=2
x
y x
x
y = 2
y = 3
(Steeper curve)
10
(Gentler curve)
9
8
7
6
5
4
3
2
1
x -5
a
-3
-2
-1
0
1
2
3
4
5
Which is the steeper curve? y = 3
b
-4
x
is steeper than y =
2
x
. This is because
3 2 2.
What is the y-intercept of each curve? Both curves have y-intercept ^0, 1h
c
Why do both curves have the same y-intercept? Any exponenal curve y = a x will have y-intercept 1 since a0
d
= 1.
Do either of the curves ever touch the x-axis? No, the curves get very close to the x-axis but never touch. This is because there is no value for x such that x x y = 2 or y = 3 is negave or zero.
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Basics
Exponential & Power Graphs What about Negative Graphs?
If there is a minus (-) in front of the exponenal (eg. y = -3 x or y = -2-x ) then the graph is reected about the x-axis. Graphically it looks like the graph is ipped upside down. Sketch the graph of y = -3 x y
4
The graph of y = -3 x is drawn by ipping the graph of y = 3 x about the x-axis. This is like ipping the graph of y = 3 x upside down.
^1, 3h
3
2 y = 3
x
1
-2
-1
0
1
2
x
-1 y = -3
x
-2
^1, -3h
-3
-4
The same is done for y
-x
= -2
or y
=
-x
` -12 j
:
Sketch the graph of y = -2-x y 4
The graph of y = -2-x is drawn by ipping the graph of y = 2-x about the x-axis. This is like ipping the graph of y = 2-x upside down.
3
^-1, 2h
2
1 -2
-1
0 -1
^-1, 2h
-x
y = 2
x 1
2
-x
y = -2
-2 -3 -4
4
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Questions
Exponential & Power Graphs
1. The curve below represents y
=2
x
Basics
. Find the missing values in the sketch.
y a
b
c
d
e
f
^3, d h
^2, c h ^1, b h ^-1, e h ^-2, f h
^0, a h x
2. Without sketching the graphs, idenfy the y-intercepts of y
3. The two curves below represent y a
= 4
x
and y
=8
x
=6
x
and y = 10 x . How do you know this?
. Idenfy each graph and answer these quesons:
Idenfy the coordinates of each point: A =
B=
C =
D=
y
E
D
E =
F= F A
B
b
-3
Why is A common on both curves?
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-2
C
-1
0
x 1
2
K
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TOPIC
3
5
Questions
Exponential & Power Graphs 4. The graph below represents y = 1
Basics
x
`3j .
a
y
Idenfy the coordinates of each point using the equaon: C
A =
B=
C =
D= B
A D -3 b
What are the intercepts of the equaon y =
5. The curve below represents y a
= -2
x
-2
-1
0
x 1
2
3
` 13 j ? x
. Find: y
b
x
^-1, e h
^0, a h ^1, b h ^2, c h
c
d
^3, d h
e
6
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Questions
Exponential & Power Graphs
Basics
6. Sketch the graphs of these equaons on the axes below: a
x
y = 4
-x
y =3
b
y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2
1 -13 -12 -11 -10 -9
-8
-7
-6
-5
-4
-3
-2
-1 0
x 1
2
3
4
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5
6
7
8
9
10
11
12
13
K
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TOPIC
7
Questions
Exponential & Power Graphs
Basics
7. Sketch the graphs of these equaons on the axes below: y
-x
a
y = 2
b
y = -2
14
-x
13 12 11 10 9 8 7 6 5 4 3 2
1
-6
-5
-4
-3
-2
-1
0 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 -11 -12 -13 -14
8
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x 1
2
3
4
5
6
Knowing More
Exponential & Power Graphs Sketching Power Graphs
Power graphs are drawn from the equaon y = ax n where a is a constant and the exponent n is a posive integer. Here are some examples where a is posive: y
-4
-3
-2
= 2x
y
y
y
4
4
3
3
2
2
1
1
x
0
-1
1
2
3
4
-4
-3
-2
-1
x 1
2
-4
=
x
3
y
4 = 2x
(a = 2, n = 4)
y
y
4
4
3
3
2
2
1
1
0 -1
4
If n = 2 , the graph is a parabolas
-3
(a = 1, n = 3)
-1
3
-2
If n = 1 , the graph is a straight line
-4
y
0 -1
-3
-2
2
(a = 1, n = 2)
-2
-3
x
(a = 2, n = 1)
-1
-4
=
1
2 3 4 Inecon point
x
-4
-3
-2
-1
0
1
2
3
4
x
-1
-2
-2
-3
-3
-4
-4
Can you see a paern? There is generally a paern when a is posive ( a 2 0 ): •
If n is odd: As the graph moves from le to right, the graph moves up from negave, through the origin and then increases as it moves to the right.
•
If n is even: As the graph moves from le to right, the graph moves down from posive, touches the origin and then increases as it moves to the right.
•
The greater the value of a or n, the steeper the curve. The smaller the value of a or n, the gentler the curve.
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Knowing More
Exponential & Power Graphs Sketching Power Graphs when a is Negative Graphs drawn from y
= ax
n
where a is negave ( a
1 0 ) behave
in the opposite way.
Here are some examples where a is negave: y
-4
-3
=-2x
(a =-3, n = 2)
y
y
-2
-1
4
4
3
3
2
2
1
1
0
x 1
2
3
4
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
=-x
3
y
=-x
1
(a =-1, n = 4)
y
y
4
4
3
3
2
2
-2
-1
0
x 2
3
3
4
2
3
4
x
1
Inecon point 1
2
4
(a =-1, n = 3)
1 -3
2 =-3x
(a =-2, n = 1)
y
-4
y
4
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
x 1
Can you see a paern? There is generally a paern when a is negave ( a 1 0 ): •
If n is odd: As the graph moves from le to right, the graph moves down from posive, through the origin, and then decreases as it moves right.
•
If n is even: As the graph moves from le to right, the graph moves up from negave, touches the origin and then decreases (moved down) in the negave direcon.
•
The greater the value of |a| or n, the steeper the curve. The smaller the value of |a| or n, the gentler the curve.
10
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Knowing More
Exponential & Power Graphs
Here are some examples of how to draw power graphs: Sketch the graphs of these equaons a
4
y = 3x
a
= 3 (posive) and
n
= 4 (even)
Step 2: Draw the graph through these points.
Step 1: Plot the points for x =-1, x = 0, x = 1 . y
y
4
Start here and move down
-4
-3
-2
4
Move up through here
3
-1
2
1
1
0
x 1
2
3
4
-4
-3
-2
-1
0
x 1
2
-1
Pass through the origin
-2
Move up through here
3
2
-1
b
Start here and move down
-3
-4
-4
4
Pass through the origin
-2
-3
3
5
y = -2x
a
= -2 (negave)
and n = 5 (odd) Step 2: Draw the graph through these points.
Step 1: Plot the points for x =-1, x = 0, x = 1 . y
-3
-2
4
4
3
3
2
Start here and move down
-4
y
Pass through the origin
1 -1
0
2
3
4
-4
-3
-3
-2
Inecon point
Pass through the origin
1
x 1
-1 -2
2
Start here and move down
-1
0
x 1
2
Move down through here
-3
-4
4
-1 -2
Move down through here
3
-4
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Exponential & Power Graphs
Questions
1. Explain the role of a and n in the funcon y = ax n .
2. Idenfy a and n and then sketch the graphs of these equaons: 2
a
y = x
c
y = 2x
12
3
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2
b
y = x + 1
d
y = 2x - 4
3
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Knowing More
Exponential & Power Graphs
Questions
Knowing More
3. Sketch the graphs for these equaons: 5
a
y = 3x
c
y = -2x
8
6
b
y = x
d
y = -x
11
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13
Using Our Knowledge
Exponential & Power Graphs Shifting Power Graphs Vertically This happens when the equaon is given as y = ax n + d or y
=
ax n
•
For the case of y
=
ax n + d , shi the power graph up d units.
•
For the case of y
=
ax n
-
-
d .
d , shi the power graph down d units.
Here are some examples: Draw the graphs for these equaons: a
4
y = 2x + 3
Step 1: Draw the graph of y = y
-4
b
-3
-2
-1
4
Step 2: Shi this graph up 3 units.
2x .
y
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
x 1
2
3
4
-4
-3
-2
-1
x 1
2
3
4
3
y = -x - 2
Step 1: Draw the graph of y = -x3 .
Step 2: Shi this graph down 2 units.
y
-4
14
0
3 units
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-3
-2
-1
y
3
3
2
2
1
1
0 -1
x 1
2
3
4
-4
-3
-2
-1
0 -1
-2
-2
-3
-3
-4
-4
-5
-5
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x 1
2
3 4 2 units
Using Our Knowledge
Exponential & Power Graphs Shifting Exponential Graphs Vertically This happens when the equaon is given as y
=
a x + d or y
=
a x
•
For the case of y
=
a x + d , shi the power graph up d units.
•
For the case of y
=
a x
-
-
d
d , shi the power graph down d units.
Here are some examples: Draw the graphs for these equaons: a
x
y = 2 - 3
Step 1: Draw the graph of y = y
-4
-3
-2
-1
Step 2: Shi the graph down 3 units.
x
2 .
y
5
5
4
4
3
3
2
2
1
1
0
x 1
2
3
4
-4
-3
-2
-1
-1
0 -1
x 1
2
3
4
3 units -2
-2
-3
-3
Imagine x-axis shis too b
- x
y = -3
+4
Step 1: Draw the graph of y = -3-x .
Step 2: Shi this graph up 4 units.
y
-4
-3
-2
-1
y
4
4
3
3
2
2
1
1
0
x 1
2
3
4
-4
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
-4
-4
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4 units
x 1
2
3
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4
15
Exponential & Power Graphs
Questions
1. Sketch the power graphs for these equaons: 4
a
y = 3x - 4
b
y = -2x + 5
16
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TOPIC
5
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Using Our Knowledge
Exponential & Power Graphs
Questions
Using Our Knowledge
6
c
y = 4x + 2
d
y = -x - 3
7
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17
Exponential & Power Graphs
Questions
2. Sketch the exponenal graphs for these equaons: x
a
y = 4 - 2
b
y =
18
` 12 j
x
+1
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Using Our Knowledge
Exponential & Power Graphs
Questions
Using Our Knowledge
x
c
y = -3 + 4
d
y = -2
- x
-2
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Thinking More
Exponential & Power Graphs Shifting Power Graphs Horizontally Graphs can also be shied sideways. This happens when the equaon is given as y
a^ x
-
k hn or y
=
a ^ x + k hn .
k hn , shi the power graph of y = ax n right k units.
•
For the case of y
=
a^ x
•
For the case of y
=
a ^ x + k hn , shi the power graph of y = ax n le k units.
-
=
Here are some examples: Draw the graphs for these equaons a
4
y = 2^ x - 3 h
Plus (-) means shi right
Step 1: Draw the graph of y = y
-4
b
-3
-2
-1
4
Step 2: Shi this graph 3 units to the right.
2x .
y
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
x 1
2
3
4
-4
-3
-2
-1
0
3 units
x 1
2
3
4
3
4
3
y = -^ x + 1 h
Plus (+) means shi le
Step 1: Draw the graph of y = -x3 .
Step 2: Shi this graph 1 unit to the le.
y
-4
-3
-2
-1
y
3
3
2
2
1
1
0 -1
x 1
2
3
4
-4
-3
-2
-1
0 -1
-2
-2
-3
-3
-4
-4
-5
-5
x 1
2
1 unit
20
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Thinking More
Exponential & Power Graphs Shifting Exponential Graphs Horizontally This happens when the equaon is given as y
a x
-
k
or y
=
a x + k
•
For the case of y
=
a x
•
For the case of y
=
a x + k , shi the exponenal graph y =
-
k
=
, shi the exponenal graph y = a x to the right k units. x
a
to the le k units.
Here are some examples: Draw the graphs for these equaons a
x + 1
y = 2
Plus (+) means shi le
Step 1: Draw the graph of y = y
Step 2: Shi this graph up 1 unit to the le.
x
2 .
y
5
5
4
4
3
3
2
2
1
1
1 unit
-4
b
-3
-2
-1
x
0 -1
0 -1
-2
-2
-3
-3
1
2
3
4
-4
-3
-2
-1
x 1
2
3
4
-^ x - 2h
y = -3
Minus (-) means shi right
Step 1: Draw the graph of y = -3-x .
Step 2: Shi this graph 2 units to the right.
y
-2
-1
y
4
4
3
3
2
2
1
1
0
x 1
2
3
4
5
6
-2
-1
-1
0
x 1
2
3
4
5
6
-1 -2
-2
2 units -3
-3
-4
-4
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21
Exponential & Power Graphs
Questions
1. Sketch the power graphs for these equaons: 5
a
y = ^ x - 3h
b
y = -3 ^ x + 4h
22
4
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Thinking More
Exponential & Power Graphs
Questions
Thinking More
2. Sketch the exponenal graphs for these equaons: x + 4
a
y = 3
b
y = 3
x-5
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Questions
Exponential & Power Graphs
3. The solid graph below has the equaon y
4
= 2x
Thinking More
: y 5 4 3 2
1
-5
-4
-3
-2
-1
0
x 1
2
3
4
5
-1 -2 -3 -4 -5
a
The doed curve is a vercal transformaon of the solid curve. Find the equaon for the doed curve.
b
What is the y-intercept of the doed curve? Is this what you expected?
c
Find the equaon of the dashed line, if it is a horizontal transformaon of the solid curve.
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Questions
Exponential & Power Graphs
4. The solid graph below has the equaon y a
= 4
x
Thinking More
: y
The dashed curve is a horizontal transformaon of the solid curve. Find the equaon of this curve.
16 15 14 13 12 11
b
The doed line is a vercal transformaon of the solid curve. Find the equaon of this curve.
10 9 8 7 6 5
c
What is the y-intercept of the solid curve?
4 3 2
A
-3
d
What is the y-intercept of the doed curve? Is this what you were expecng?
e
Find the coordinates of the point labelled A.
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-2
1 -1
0
1
2
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3
x
25
Answers
Exponential & Power Graphs
Basics: 1.
Basics:
a
y
=1
b
y
= 2
c
y
= 4
d
y
=8
e
y
=
1 2
f
y
=
6. b
a
-x
y = 3
1 4
2. Both graphs have the y-intercept at y = 1 as exponenal graphs always intercept the y-axis at (0,1) since a0 = 1 for any value of a.
3.
a
B = -1, 1
`
A = ^0, 1h C = -1, 1
`
8
j
E = ^1, 8h
4.
5.
26
4
j
D = ^1, 4h F = 1 , 2
`2 j
b
Exponenal graphs always intercept the y-axis at (0, 1) since a0 = 1 for any value of a.
a
A = (0,1)
B = (- 1, 3)
C = (-2,9)
D = (1, 1 ) 3
b
The y-intercept is at y = 1 and the graph does not intercept the x- axis
a
y
= -1
b
y
= -2
c
y
= -4
d
y
= -8
e
y
=
7.
-x
a
y = 2
b
y =-2
-1 2
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-x
y = 4
x
Answers
Exponential & Power Graphs
Knowing More:
Knowing More:
1. when a is posive (a > 0):
2.
•
If n is odd: As the graph moves from le to right, the graph moves up from negave, through the origin and then increases as it moves to the right.
•
If n is even: As the graph moves from le to right, the graph moves down from posive, touches the origin and then increases as it moves to the right.
when a is negave (a < 0): •
If n is odd: As the graph moves from le to right, the graph moves down from posive, through the origin and then decreases as it moves to the right.
•
If n is even: As the graph moves from le to right, the graph moves up from negave, touches the origin and then decreases in the negave direcon.
The greater the value of a or n, the steeper the curve. The smaller the value of a or n, the gentler the curve.
2.
a
b
a = 1, n = 2
c
a = 2, n = 3
d
a = 2, n = 3
a
y = 3x
a = 1, n = 2
3.
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27
Answers
Exponential & Power Graphs
Knowing More: 3.
28
Using Our Knowledge:
6
1.
b
y= x
c
y = -2x
d
y = -x
8
11
K
11
SERIES
TOPIC
4
a
y = 3x - 4
b
y = -2x + 5
c
y = 4x + 2
d
y = -x - 3
5
6
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7
Answers
Exponential & Power Graphs
Using Our Knowledge: 2.
Using Our Knowledge:
x
a
y = 4 - 2
b
y =
` 12 j
x
2.
- x
y = -2
-2
+1
Thinking More: 1.
c
d
5
a
y = ^ x - 3h
b
y = -3^ x + 4h
x
y = -3 + 4
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Exponential & Power Graphs
Thinking More: 2.
a
b
3.
a b
4.
30
x + 4
y = 3
y = 3
x-5
4
y = 2x - 5
The y-intercept is at y = -5 . This is expected as it is 5 units down from the y-intercept of y = 2x4
c
y
4 = 2 (x - 3)
a
y
( x + 2) =4
b
y = 4 + 3
c
The y-intercept of the solid curve is y = 1
d
The y-intercept of the doed curve is y =4
e
The coordinates of A are (-2, 1)
x
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100% Exponental and Power Graphs Mathletics 100%
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Notes
Exponential & Power Graphs
100% Exponental and Power Graphs Mathletics 100%
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K
11
SERIES
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31
Notes
Exponential & Power Graphs
32
K
11
SERIES
TOPIC
100% Exponental and Power Graphs Mathletics 100%
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