Mathematics for Commerce
Standard Term 1
Lecture 12: Graphs of Exponential Functions
Investigation:
Using a table of values to plot the graph of y = 2 x .
x
−2
−1
0
1
3
2
y
y
10
8
6
4
2
-3
-2
-1
1
2
3
x
Notice some key features of the graph:
• As x → ∞ , y → ∞ .
• As x → −∞ , y → 0 (the negative x-axis is an asymptote).
•
y > 0 for all x values.
These features extend to any graph of this type, referred to as an exponential graph.
Mathematics for Commerce
Standard Term 1
Features of the Graphs of an Exponential Function
y = k ax
( a > 0)
y = k ax
(a < 0)
y
y
x
x
For a > 0 , as x → −∞ , k ax → 0
For a < 0 , as x → +∞ , k ax → 0
•
Graph is continuous for all values of x.
•
Domain ……………………..
•
Range ……………………….
•
Horizontal asymptote …………………………….
•
y-intercept ……………….
•
For a > 0 the graph is increasing for all x.
For a < 0 the graph is decreasing for all x.
Mathematics for Commerce
Standard Term 1
Transformations of Exponential Graphs
Examples
1.
Use transformation methods to sketch the graph of:
(a)
=
y 2x + 1
(b)
=
y 2x − 3
(c)
y= 3 × 2x
(d)
y=
1 x
×2
3
y y0 k ax + b have the following features:
In general, exponential functions of the form=
•
Domain ………………………………………………
•
Horizontal asymptote ………………………………..
•
y-intercept ……………………………………………
Mathematics for Commerce
Standard Term 1
Examples
2.
Sketch the graphs of:
(a)
y = 1 − 2x
(c) =
y 2e −3 x − 4
Suggested Exercises: Tutorial Guide 2-3
(b)
y= 3 + 2e − x