Topics
5. Exponential Functions
topic badge
United States of AmericaTennessee
Algebra 1

5.02 Characteristics of exponential functions

Lesson
Practice
Lesson

Concept summary

To draw the graph of an exponential function we can use a variety of strategies including:

  • Filling out a table of values for the function and drawing the curve through the points found
  • Using technology, such as a physical or online graphing calculator
  • By identifying key features from the equation:
\displaystyle f\left(x\right)=ab^x
\bm{a}
The initial value gives us the the value of the y-intercept
\bm{b}
We can use the constant factor, or common ratio, to identify other points on the curve

It is important to choose an appropriate scale for the axes of the graph as we want to see all the key features.

The constant factor, b, can be found by finding the common ratio.

We can determine the key features of an exponential function from its graph:

1
2
x
3
6
9
12
15
18
21
24
27
y
  • The graph is increasing
  • y approaches a minimum value of 0
  • The domain is -\infty<x<\infty
  • The range is 0<y
  • The y-intercept is at \left(0,\, 3\right)
  • The common ratio is 4
  • The horizontal asymptote is y=0
-4
-3
-2
-1
1
2
3
4
x
5
10
15
20
25
y
  • The graph is decreasing
  • y approaches a minimum value of 0
  • The domain is -\infty<x<\infty
  • The range is 0<y
  • The y-intercept is at \left(0,\, 10\right)
  • The common ratio is \dfrac{1}{2}
  • The horizontal asymptote is y=0
Asymptote

A line that a curve or graph approaches as it heads toward positive or negative infinity.

x
y

Worked examples

Example 1

Consider the exponential function y=2.5\left(4\right)^x.

a

Draw the graph of the function by first finding the common ratio and the y-intercept.

Approach

We can identify both the y-intercept and common ratio (constant factor) from the equation since it is of the form y=ab^x. Using these two key features, we can plot other points to the left and right of the y-intercept and connect them with a smooth curve.

Solution

The function has a common ratio of 4 since that is the base of the exponent, and a y-intercept at \left(0,2.5\right) as that is the coefficient.

We can start by plotting the y-intercept and choosing an appropriate scale.

-2
-1
1
2
x
5
10
15
20
25
30
35
40
45
y

We can see that the function will always be positive, so we don't need any negative y-values.

We know that the common ratio is 4. If we want to go from -2 to 2 on the x-axis, we need to go up to at least 2.5(4)^2=40 on the y-axis.

This would be an appropriate scale as it doesn't have too many labels or tick marks and is easy to read.

-2
-1
1
2
x
5
10
15
20
25
30
35
40
45
y

Either using a table of values or using the common ratio, we can plot another three points to get a good shape for the graph. Then we can connect the points with a smooth curve.

Reflection

When drawing the graphs of exponential functions we want to be sure the y-intercept is clearly displayed and that the exponential curve is also visible. Be sure to choose a scale for the y-axis that will show all important characteristics. In this case, we chose to scale by 5s which allows us to read both the y-intercept at 2.5, a second point at \left(1,10\right) , the horizontal asymptote at y=0 and the steep slope that all exponential functions have.

b

Check the graph from part (a) using technology.

Approach

When using a graphing calculator, there will typically be an input bar for functions. We just need to type out the function in the input bar. We need to be careful that we format our input correctly by using the correct buttons.

The GeoGebra calculator has a math input keyboard, but we can also use a computer keyboard.

  • Math input keyboard
  • Computer keyboard

Solution

The graph should look the same as in part (a). We may need to change the axes or zoom settings to check.

Example 2

Consider the table of values for the function y = 2\left(\dfrac{1}{3}\right)^{ x }.

x-5-4-3-2-101234510
y486162541862\dfrac{2}{3}\dfrac{2}{9}\dfrac{2}{27}\dfrac{2}{81}\dfrac{2}{243}\dfrac{2}{59\,049}
a

Describe the behavior of the function as x increases.

Approach

We want to identify if the values of y are increasing or decreasing as x increases.

Solution

As x increases, the function decreases at a slower and slower rate.

Reflection

We can see that the equation has a constant factor that is less than 1. This is why the function is decreasing.

b

Determine the y-intercept of the function.

Approach

The y-intercept occurs when x=0. We can read these coordinates from the table.

Solution

\left(0,\,2\right)

Reflection

We can see that the equation has an initial value of 2. This is the value of the y-intercept, and the result of substituting x=0 into the equation.

c

State the domain of the function.

Approach

The domain is the complete set of possible values for x. For exponential functions, the graph extends indefinitely in both horizontal directions.

Solution

All real x.

Reflection

All exponential equations of the form y=ab^x have a domain of all real x.

d

State the range of the function.

Approach

The range is the complete set of possible values for y. We can see graph extends indefinitely up towards the left but it approaches an asymptote at y=0 towards the right.

Solution

y>0

Reflection

All exponential equations of the form y=ab^x have a range of y>0 for positive values of a.

Outcomes

A1.N.Q.A.1

Use units as a way to understand real-world problems.*

A1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

A1.A.REI.D.5

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A1.F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*

A1.F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the context of the function it models. *

A1.F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.*

A1.F.IF.C.7

Graph functions expressed algebraically and show key features of the graph by hand and using technology.*

A1.F.IF.C.9

Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.*

A1.F.LE.A.1

Distinguish between situations that can be modeled with linear functions and with exponential functions.*

A1.F.LE.A.1.C

Recognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another.

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP6

Attend to precision.

A1.MP7

Look for and make use of structure.

A1.MP8

Look for and express regularity in repeated reasoning.

What is Mathspace

About Mathspace