# Graphs of Exponential Functions

Graphing and sketching exponential functions: step by step tutorial. The properties such as domain, range, horizontal asymptotes and intercepts of the graphs of these functions are also examined in details. Free graph paper is available.

## Review

We first start with the properties of the graph of the basic exponential function of base a,

f (x) = ax , a > 0 and a not equal to 1.
The domain of function f is the set of all real numbers. The range of f is the interval (0 , +∞).
The graph of f has a horizontal asymptote given by y = 0. Function f has a y intercept at (0 , 1). f is an increasing function if
a is greater than 1 and a decreasing function if a is smaller than 1 .

### Example 1

f is a function given by
f (x) = 2(x - 2)

1. Find the domain and range of f.
2. Find the horizontal asymptote of the graph.
3. Find the x and y intercepts of the graph. of f if there are any.
4. Sketch the graph of f.

1. The domain of f is the set of all real numbers. To find the range of f,we start with
2x > 0
Multiply both sides by 2-2 (a positive number).
2x 2-2 > 0

Use exponential properties to rewrite the above as
2(x- 2) > 0
This last inequality suggests that f(x) > 0. Hence the range of f is given by the interval: (0, + ∞).
2. As x decreases without bound, f(x) = 2(x- 2) approaches 0. The graph of f has a horizontal asymptote at y = 0.
3. To find the x intercept we need to solve the equation
f(x) = 0
2(x - 2)  = 0
This equation does not have a solution, see range above, f(x) > 0. The graph of f does not have an x intercept. The y intercept is given by
(0 , f(0)) = (0,2(0 - 2)) = (0 , 1/4).
4. So far we have the domain, range, y intercept and the horizontal asymptote. We need extra points.
(4 , f(4)) = (4, 2(4 - 2)) = (4 , 22) = (4 , 4)
(-1 , f(-2)) = (-1, 2(-1 - 2)) = (-1 , 2-3) = (-1 , 1/8)
Let us now use all the above information to graph f. Matched Problem to Example1:   f is a function given by

f (x) = 2(x + 2)

1. Find the domain and range of f.
2. Find the horizontal asymptote of the graph of f.
3. Find the x and y intercepts of the graph of f if there are any.
4. Sketch the graph of f.

### Example 2

f is a function given by
f (x) = 3(x + 1) - 2
1. Find the domain and range of f.
2. Find the horizontal asymptote of the graph of f.
3. Find the x and y intercepts of the graph of f if there are any.
4. Sketch the graph of f.

1. The domain of f is the set of all real numbers. To find the range of f, we start with
3x > 0
Multiply both sides by 3 which is positive.
3x3 > 0
Use exponential properties
3(x+ 1) > 0
Subtract 2 to both sides
3(x+ 1) -2 > -2
This last statement suggests that f(x) > -2. The range of f is (-2, +∞).
2. As x decreases without bound, f(x) = 3(x+ 1) -2 approaches -2. The graph of f has a horizontal asymptote y = -2.
3. To find the x intercept we need to solve the equation f(x) = 0 3(x + 1) - 2  = 0
Add 2 to both sides of the equation 3(x + 1) = 2
Rewrite the above equation in Logarithmic form x +1 = log3 2
Solve for x x = log3 2 - 1
The y intercept is given by (0 , f(0)) = (0,3(0 + 1) - 2) = (0 , 1).
4. So far we have the domain, range, x and y intercepts and the horizontal asymptote. We need extra points. (-2 , f(-2)) = (-2, 3(-2 + 1) - 2) = (4 , 1/3-2) = (4 , -1.67)
(-4 , f(-4)) = (-4, 3(-4 + 1) - 2) = (-4 , 2-3) = (-4 , -1.99)
Let us now use all the above information to graph f. Matched Problem to Example2: f is a function given by

f (x) = 2(x - 2) + 1

1. Find the domain and range of f.
2. Find the horizontal asymptote of the graph of f.
3. Find the x and y intercepts of the graph of f if there are any.
4. Sketch the graph of f.

## More References and Links to Exponential Functions and Graphing

Graphing Functions
Exponential Functions - Interactive Tutorial.
Tutorial on Exponential Functions (1)
Tutorial on Exponential Functions (2)
Graphs of Basic Functions.