ExpoandPower

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Curriculum Ready

Exponential and Power Graphs

Exponential and Power Graphs

Exponen

Exponen

tial

tial

and

and

Power Graphs

Exponential & Power Graphs

Exponential & Power Graphs

_EXPONENTIA

_EXPONENTIA

_{L &}

_{L &}

_{POWER GRAPHS}

_{POWER GRAPHS}

Answer these quesons,

Answer these quesons, beforebefore working through the chapter. working through the chapter.

Answer these quesons,

Answer these quesons, after after  working through the chapter. working through the chapter.

These are graphs which result from equaons that are

These are graphs which result from equaons that are not linear or not linear or quadrac. The exponenalquadrac. The exponenal

graph has the variable as

graph has the variable as the exponent. The power graphs raise the variable the exponent. The power graphs raise the variable to any powerto any powernn..

But now I think:

But now I think:

What do I know now that I didn’t know before? 

What do I know now that I didn’t know before? 

I used to think:

I used to think:

Which of these equaons is for an exponenal graph and which is for a power graph:

Which of these equaons is for an exponenal graph and which is for a power graph:

7

7

 y

 y == xx or or y  y == xx77??

For which value of

For which value of x x is is 22 x x _{equal to zero? equal to zero?}

Is it possible for

Is it possible for _{y  y} == 55_xx44 to  to be negave? be negave? Why?Why?

Which of these equaons is for an exponenal graph and which is for a power graph:

Which of these equaons is for an exponenal graph and which is for a power graph:

7

7

 y

 y == xx or or y  y == xx77??

For which value of

For which value of x x is is 22 x x _{equal to zero? equal to zero?}

Is it possible for

Exponential & Power Graphs

Basics

, 1 2 -^ h ^ 1 2, h , . 1 0 5 -^ h ^ 1 0 5, . h

Exponential Graphs

Exponenal graphs are of funcons with the variable in the exponent of the form y = a x or y a

1 x

=

` j

 where _{a 2}1_.

They have this form:

This could also be wrien as a-x

Here are some important properes about exponenal graphs:

• They always cut the y-axis at

^

0 1,

h

 since _a0 = 1 _{for any value of a}.

• The exponenal graph never cuts the x -axis since a x is never negave or zero if a 2 .0

• The greater the value of a (the base), the steeper the curve.

-2 -1 0 1 2 4 3 2 1  x  y

The y-intercept of ALL exponenal curves is always ^ 0 1, h

No x-intercepts

Sketch the graphs of y = 2x  and y

2 1 x 

=

` j

 on the same set of axes

 x  y  x  y  y 2 1 x =

` j

y=2 x  y = a x y a 1 x =

` j

,a 1 ^ h ^ -1,ah 1 1

Exponential & Power Graphs

Basics

-5 -4 -3 -2 -1 0 1 2 3 4 5 10 9 8 7 6 5 4 3 2 1

The graphs below are of the funcons _{y =}3x 

 and y =2x 

a _{Which is the steeper curve?}

3

 y = x is steeper than y = 2x. This is because 3 2 .2

b

c

d

What is the y-intercept of each curve? Both curves have y-intercept

^

0 1,

h

Why do both curves have the same y-intercept?

Any exponenal curve y = a x will have y-intercept 1 since a0 = 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

2

 y = x or y = 3x is negave or zero.

 x  y  y=3 x y=2 x (Steeper curve) (Gentler curve)

Exponential & Power Graphs

Basics

, 1 2 -^ h 1, 2 -^ h

 What about Negative Graphs?

If there is a minus (-) in front of the exponenal (eg. _{y = -}3x _or

2

 y= - -x) then the graph is reected about the x-axis. Graphically it looks like the graph is ipped upside down.

The same is done for y =-2-x or y

2 1 x =

-` j

:

The graph of _{y = -}3x _{is drawn by ipping the graph of 3}

 y = x about the x-axis. This is like ipping the graph of _y= 3x _{upside down.}

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.

Sketch the graph of y =-2-x  Sketch the graph of y = -3 x 

3  y= x 2  y= -x 3  y= - x  y= -2-x , 1 3 ^ h , 1 -3 ^ h -2 -1 0 1 2 -2 -1 0 1 2 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4  x  y  x  y

Exponential & Power Graphs

Questions

Basics

, 3 ^ d h , 2 ^ c h , 0 ^ a h , 1 ^ bh , 1 -^ e h , 2 -^ f h a c e b d f 

2. Without sketching the graphs, idenfy the y-intercepts of y = 6x  and y = 10x . How do you know this?

3. The two curves below represent y =4x  and y =8x . Idenfy each graph and answer these quesons:

a

b

Idenfy the coordinates of each point:

 A= B=

C = D=

 E = F =

Why is A common on both curves?

 x  y

1. The curve below represents y =2x . Find the missing values in the sketch.

 x  y  A  B  D  E   F -3 -2 -1 0 1 2 3 C 

Exponential & Power Graphs

Questions

Basics

4. The graph below represents y

3 1 x 

=

` j

_.

a

b

Idenfy the coordinates of each point using the equaon:

 A= B=

C = D=

What are the intercepts of the equaon y

3 1 x

=

` j

?

5. The curve below represents y =-2x . Find: a c e b d  x , 3 ^ d h , 2 ^ c h , 0 ^ a h , 1 ^ bh , 1 -^ e h  y  x  y  A  B  D -3 -2 -1 0 1 2 3 C 

Exponential & Power Graphs

Questions

Basics

a b

6. Sketch the graphs of these equaons on the axes below:

 y = 4 x y = 3-x -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1  x  y

Exponential & Power Graphs

Questions

Basics

7. Sketch the graphs of these equaons on the axes below:

a b  y = 2-x 2  y =- -x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 14 13 12 11 10 9 8 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14  x  y

Exponential & Power Graphs

Knowing More

Power graphs are drawn from the equaon y = axn where a is a constant and the exponent n is a posive integer.

Sketching Power Graphs

Here are some examples where a is posive:

( , )  y x a n 2 2 1 = = = ( , )  y x a 1 n 3 3 = = = ( , )  y x a 1 n 2 2 = = = ( , )  y x a n 2 2 4 4 = = =

Can you see a paern? There is generally a paern whena is posive (a 2 0):

• If n is odd: As the graph moves from le to right, the graph moves up from negave, 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 posive, 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.

If n=1, the graph is a straight line If n=2, the graph is a parabolas Inecon point -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4  x  y  x  y  x  y  x  y

Exponential & Power Graphs

Knowing More

Graphs drawn from y = axn whereais negave (a1 0) behave in the opposite way.

Sketching Power Graphs when

a

 is Negative

Here are some examples where a is negave:

( , )  y x a n 2 2 1 =-=- = ( , )  y x a 1 n 3 3 =-=- = ( , )  y x a n 3 3 2 2 =-=- = ( , )  y x a 1 n 4 4 =-=- =

Can you see a paern? There is generally a paern whena is negave (a 1 0):

• If n is odd: As the graph moves from le to right, the graph moves down from posive, 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 negave, touches the origin and then decreases (moved down) in the negave direcon.

• The greater the value of |a| or n, the steeper the curve. The smaller the value of |a| or n, the gentler the curve.

Inecon point -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4  x  y  x  y  x  y  x  y

Exponential & Power Graphs

Knowing More

Here are some examples of how to draw power graphs:

a

b

Sketch the graphs of these equaons  y = 3x4

 y =-2x5

a =3 (posive) and n = 4 (even)

a =-2 (negave) and n= 5 _(odd)

Step 1_{: Plot the points for x =- = 1, x 0, x= 1}.

Step 1_{: Plot the points for x =- = 1, x 0, x= 1}.

Step 2_{: Draw the graph through these points.}

Step 2_{: Draw the graph through these points.} Start here and

move down

Start here and move down

Start here and move down Start here and

move down Move up through here Move up through here Pass through the origin Pass through the origin Pass through the origin Pass through the origin Move down through here Move down through here Inecon point -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4  x  y  x  y  x  y  x  y

Exponential & Power Graphs

Questions

Knowing More

1. Explain the role of a and n in the funcon y = ax n.

2. Idenfya andn and then sketch the graphs of these equaons:

a c b d  y = x2  y = 2x3  y = x2+1  y = 2 x3-4

Exponential & Power Graphs

Questions

Knowing More

a c b d  y = 3x5 2  y = - x8  y = x6  y =-x11 3. Sketch the graphs for these equaons:

Exponential & Power Graphs

Using Our Knowledge

Shifting Power Graphs Vertically 

This happens when the equaon is given as  y = ax n+d  or  y = ax n-d .

• For the case of  y = ax n+d , shi the power graph up d  units. • For the case of  y = ax n-d , shi the power graph down d  units. Here are some examples:

a

Draw the graphs for these equaons:

b _y = - _x3-2

Step 1_{: Draw the graph of  y = 2x}4.

Step 1_{: Draw the graph of y} = -_x3.

Step 2_{: Shi this graph up}3 units.

Step 2_{: Shi this graph down}2 units.

3 units  y = 2 x4+3 2 units -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 8 7 6 5 4 3 2 1 3 2 1 -1 -2 -3 -4 -5 3 2 1 -1 -2 -3 -4 -5 8 7 6 5 4 3 2 1  x  y  x  y  x  y  x  y

Exponential & Power Graphs

Using Our Knowledge

Shifting Exponential Graphs Vertically 

This happens when the equaon is given as  y = a  x+d  or  y = a  x-d 

• For the case of  y = a  x+d , shi the power graph up d  units. • For the case of  y = a  x-d , shi the power graph down d  units. Here are some examples:

a

Draw the graphs for these equaons:

b y = - 3 - x+4

Step 1_{: Draw the graph of y 2} x

= .

Step 1_{: Draw the graph of y= -}3-x_.

Step 2_{: Shi the graph down}3 units.

Step 2_{: Shi this graph up}4 units.  y = 2  x-3 3 units 4 units -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 5 4 3 2 1 -1 -2 -3 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 5 4 3 2 1 -1 -2 -3  x  x  y  x x  y y  y Imagine x-axis shis too

Exponential & Power Graphs

Questions

Using Our Knowledge

1. Sketch the power graphs for these equaons:

a

b

 y = 3 x4-4

Exponential & Power Graphs

Questions

_{Using Our Knowledge}

c

d

 y = 4 x6+2

Exponential & Power Graphs

Questions

Using Our Knowledge

2. Sketch the exponenal graphs for these equaons:

a b  y = 4  x-2 1  y 2 1 x =

` j

+

Exponential & Power Graphs

Questions

_{Using Our Knowledge}

c

d

 y = - 3  x+4

Exponential & Power Graphs

Thinking More

Graphs can also be shied sideways. This happens when the equaon is given as y = a x

^

-k

h

n or  y a x k n

=

^

+

h

.

Shifting Power Graphs Horizontally 

• For the case of  y = a x

^

-k

h

n, shi the power graph of y ax

n

= right k units. • For the case of  y = a x

^

+k

h

n, shi the power graph of y = axn le k units. Here are some examples:

a

Draw the graphs for these equaons

b _y =-

_^

_x+1

_h

3

Step 1_{: Draw the graph of  y = 2x}4.

Step 1_{: Draw the graph of y} = -_x3.

Step 2_{: Shi this graph3 units to the right.}

Step 2_{: Shi this graph1 unit to the le.} 3 units

 y = 2

^

x-3

h

4

1 unit Plus (+) means shi le

Plus (-) means shi right

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

Exponential & Power Graphs

Thinking More

This happens when the equaon is given as  y =a x k

-  or  y

a x k

= +

Shifting Exponential Graphs Horizontally 

• For the case of  y =a x k

-, shi the exponenal graph y = a x to the right k units. • For the case of  y =a x k+ , shi the exponenal graph y = a x to the le k units. Here are some examples:

a

Draw the graphs for these equaons

b _{y = -}3- ^ x-2h

Step 1_{: Draw the graph of  y 2}x

= .

Step 1_{: Draw the graph of y= -}3-x_.

Step 2_{: Shi this graph up}1 unit to the le.

Step 2_{: Shi this graph}2 units to the right.  y = 2 x+1

Plus (+) means shi le

Minus (-) means shi right

2 units 1 unit -4 -3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 2 3 4 5 4 3 2 1 -1 -2 -3 4 3 2 1 -1 -2 -3 -4 4 3 2 1 -1 -2 -3 -4 5 4 3 2 1 -1 -2 -3  x  x  y  x x  y y  y

Exponential & Power Graphs

Questions

Thinking More

1. Sketch the power graphs for these equaons:

a

b

 y =

^

x-3

h

5

3

Exponential & Power Graphs

Questions

_{Thinking More}

2. Sketch the exponenal graphs for these equaons:

a

b

 y = 3 x+4

3

Exponential & Power Graphs

Questions

Thinking More

3. The solid graph below has the equaon y =2x 4:

a

b

c

The doed curve is a vercal transformaon of the solid curve. Find the equaon for the doed curve.

What is the y-intercept of the doed curve? Is this what you expected?

Find the equaon of the dashed line, if it is a horizontal transformaon of the solid curve.

-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5  x  y

Exponential & Power Graphs

Questions

_{Thinking More}

4. The solid graph below has the equaon y =4x :

a

b

c

d

e

The dashed curve is a horizontal transformaon of the solid curve. Find the equaon of this curve.

The doed line is a vercal transformaon of the solid curve. Find the equaon of this curve.

What is the y-intercept of the solid curve?

What is the y-intercept of the doed curve? Is this what you were expecng?

Find the coordinates of the point labelled A.

 A -3 -2 -1 0 1 2 3 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1  x  y

Exponential & Power Graphs

 Answers

4  y= x 3  y= -x

Basics:

Both graphs have the y-intercept at y= 1 as exponenal graphs always intercept the y-axis at(0,1) since _a0 = 1 _{for any value of a}.

a c e b b d f  1. 6. 7. 2. 3. 4. 5.  y =1  y = 4  y 2 1 = y 4 1 =  y = 8  y = 2 a b ,  A =

^

0 1

h

, C  1 8 1 = -

`

j

,  E =

^

1 8

h

 F  , 2 1 2 =

` j

,  B 1 4 1 = -

`

j

,  D =

^

1 4

h

Exponenal graphs always intercept the y-axis at(0, 1) since _a0 = 1 _{for any}

value of a.

The y-intercept is at y= 1 and the graph does not intercept the x- axis

a b (0,1)  A = ( 2,9) C  = -  D (1, ) 3 1 = ( 1, 3)  B = -a c e b d  y =-1  y = -4  y 2 1 = - y = -8  y =-2

Basics:

a a b  y=2-x 2  y=- -x

Exponential & Power Graphs

 Answers

Knowing More:

Knowing More:

1.

2.

2.

3. whena is negave(a< 0):

• If n is odd: As the graph moves from le to right, the graph moves up from negave, 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 posive, touches the origin and then increases as it moves to the right. whena is posive(a > 0):

• If n is odd: As the graph moves from le to right, the graph moves down from posive, 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 negave, touches the origin and then decreases in the negave direcon. The greater the value of a or n, the steeper the curve. The smaller the value of a or n, the gentler the curve.

a b a= 1, n= 2 a = 1, n= 2 a= 2, n= 3 a= 2, n= 3 c d a _y = 3_x5

Exponential & Power Graphs

 Answers

b _y = _x6

Knowing More:

Using Our Knowledge:

3. 1. c d 2  y = - x8  y =-x11 d _y = - _x7-3 a _y = 3 _x4-4 b _y = - 2 _x5+5 c _y = 4 _x6+2

Exponential & Power Graphs

 Answers

Using Our Knowledge:

Using Our Knowledge:

Thinking More:

2. 2. 1. a b  y = 4  x-2 1  y 2 1 x =

` j

+ c d  y = - 3  x+4  y = - 2 - x-2 a b  y =

^

x-3

h

5 3  y = -

^

x+4

h

4

Exponential & Power Graphs

 Answers

Thinking More:

3. 2. 4. a a b c 2 5  y = x4

-The y-intercept is at _y =-5_{. This is}

expected as it is5 units down from the  y-intercept of y = 2x4 2 ( 3)  y = x- 4 b c d e 4 3  y = x+

The y-intercept of the solid curve is y= 1

The y-intercept of the doed curve is y=4

The coordinates of A are(-2, 1) 4  y = (  x+2) a b  y = 3 x+4 3  y= x-5