# Lesson 4: Exploring the Properties of Exponential Functions

```Unit 4
MCR3U1
Lesson 4: Exploring the Properties of Exponential Functions
Using a graphing calculator, graph each of the following exponential functions and complete the
table below.
Function
Sketch
x-intercept /
y-intercept
Asymptote
Domain /
Range
y  2x
1
y 
2
x
y  2x  1
y  2x 1
x
1
y    1
2
Page 1
Unit 4
MCR3U1
Function
Sketch
x-intercept /
y-intercept
Asymptote
Domain /
Range
x
1
y    1
2
y  2 x
1
y 
2
x
y  2 x 1
y  2 x1  1
Page 2
Unit 4
MCR3U1
Graphing Exponential Functions
The graph of y  b x can be illustrated on a graph. For example, we can graph the function
y  2x (b = 2) using a table of values.
x
y
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0.13
0.18
0.25
0.35
0.50
0.71
1.00
1.41
2.00
2.83
4.00
5.668
8.00
9
8
y  2x
7
6
5
4
3
2
1
0
-4
-2
0
2
4
Exponential functions have the same properties as linear and quadratic functions. These
properties can be illustrated as follows.
y-intercept: Substitute x = 0 into y  2 x .
y  20
1
Therefore, the y-intercept is y = 1
x-intercept: Substitute y = 0 into y  2 x .
0  2x
This equation has no real solutions since 2 x  0 for all real values of x. So, there is no x-intercept.
Domain
Since we can define
2x
for all real values of x, the domain is
x  R.
Range
Since we can find a value of x for all positive real values of y, the range is
y  0.
Asymptotes
Recall that an asymptote is a line that a curve approaches more and more closely but never
touches. The asymptote for the graph of y  2 x is the line y = 0, or the x-axis.
Page 3
Unit 4
MCR3U1
How would the graph change if b 
x
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
1
?
2
9
y
8
5.66
4
2.83
2
1.41
1
0.71
0.5
0.35
0.25
0.18
0.13
8
7
6
5
4
1
y 
2
x
3
2
1
0
-4
-2
2
0
4
The y-intercept for both graphs is y = 1. The domain and range of both graphs are the same.
The only difference between the two graphs is that when b  1 , the graph of y  b x is
increasing. When 0  b  1,the graph of y  b x is decreasing.
How would the graph change if a  1.5 ?
x
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
4
y
0.296
0.363
0.444
0.544
0.667
0.816
1.00
1.225
1.50
1.837
2.25
2.756
3.375
-4
The graph is increasing. However, it is
not as close to the y-axis, so it is increasing slower.
4
3
3
2
2
1
1
0
-2
0
2
4
Page 4
```