Graphing Exponential Functions and Its Inverse
MHF4U
1. Using a table of values, graph the function 𝑦 = 2𝑥 .
𝑥
𝑦 = 2𝑥
−3
−2
−1
0
1
2
3
4
2. Complete the following table.
𝑦 = 2𝑥
𝑥-intercept
𝑦-intercept
Domain
Range
Asymptote Eqns
Increasing or
Decreasing
End Behaviours
3. Using a table of values, graph the functions 𝑦 = 3𝑥 and 𝑦 = 5𝑥 on grid above.
𝑥
−3
−2
−1
0
1
2
3
4
𝑦 = 3𝑥
𝑥
−3
−2
−1
0
1
2
3
4
𝑦 = 5𝑥
4. Discuss the similarities between the functions 𝑦 = 2𝑥 , 𝑦 = 3𝑥 and 𝑦 = 5𝑥 .
1 𝑥
5. Using a table of values, graph the function 𝑦 = (2) on the same grid as 𝑦 = 2𝑥 .
𝑥
1 𝑥
𝑦 = (2)
−3
−2
−1
0
1
2
3
4
6. Complete the table.
1 𝑥
𝑦=( )
2
𝑥-intercept
𝑦-intercept
Domain
Range
Asymptote Eqns
Increasing or
Decreasing
End Behaviours
1 𝑥
7.
Discuss the differences and similarities between the functions 𝑦 = 2𝑥 and 𝑦 = (2) .
8.
What point is common to all of these exponential functions? __________ Why?
9.
Using a table of values (no decimals), graph the function 𝑦 = 2𝑥 on the grid below.
𝑥
−3
−2
−1
0
1
2
3
4
𝑦 = 2𝑥
12. Graph the inverse of 𝑦 = 2𝑥 on the same grid. (Recall: the inverse is a reflection over the
line _________ )
13. For the inverse function above, state:
Inverse of 𝑦 = 2𝑥
𝑥-intercept
𝑦-intercept
Domain
Range
Asymptote Eqns
Increasing or
Decreasing
End Behaviours
1 𝑥
14. Using a table of values (no decimals), graph the function 𝑦 = (2) on the grid below.
𝑥
1 𝑥
𝑦=( )
2
−3
−2
−1
0
1
2
3
4
1 𝑥
15. Graph the inverse of 𝑦 = (2) on the same grid.
16. For the inverse function state its:
1 𝑥
Inverse of 𝑦 = ( )
2
𝑥-intercept
𝑦-intercept
Domain
Range
Asymptote Eqns
Increasing or
Decreasing
End Behaviours
Summary Chart
Sketch
𝑦 = 𝑏𝑥
𝑏 > 0, 𝑏 ≠ 1
Inverse of 𝑦 = 𝑏 𝑥
𝑏>1
𝑏>1
0<𝑏<1
0<𝑏<1
Domain
Range
𝑥-intercept
𝑦-intercept
Asymptotes
When is the
function
increasing?
When is the
function
decreasing?
Example 1
Determine the equation for the data below.
𝒙
𝒚
0
3
1
9
2
27
3
81
4
243
5
729
Example 2
a)
Determine the equation for each relation.
b)
c)
𝒙
𝒚
𝒙
𝒚
𝒙
𝒚
0
2
0
5
1
0
1
8
1
6
3
1
2
32
2
9
9
2
3
128
3
14
27
3
4
256
4
21
81
4
Example 3
Describe the transformations that can be applied to the base function to
obtain the graph of each function. Rewrite the equation if necessary.
1
a) 𝑦 = − 2 (3𝑥−2 ) + 4
𝑥
b) 𝑦 =
1 −2−2
(2)
−3
2
1
Example 4
The graph of 𝑓(𝑥) = 27 (92𝑥+1 ) can be obtained by applying a vertical stretch or
compression to the graph of 𝑦 = 3𝑥 . Determine the stretch/compression factor.
Example 5
A culture of 15 bacteria divides in half every day. Graph its growth for 5 days.
a) Write an equation where 𝑡 represents the number of days and 𝐵 represents the
amount of bacteria.
b) Find the number of bacteria after 15 days.
c) Determine the average rate of change between days 3 and 4.
d) Determine the instant rate of change at 5 days.
e) How long will it take to have 1000 bacteria?
Thinking
The graph of the function 𝑦 = 𝑓(𝑥) is obtained by applying a horizontal stretch by a factor of 4,
followed by a horizontal shift 3 units left, to the base graph of 𝑦 = 2𝑥 . Find (𝑓(𝑥))(𝑓(2 − 𝑥)).