Topics
5. Exponential Functions
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United States of AmericaFL
Grade 912

5.04 Characteristics of exponential functions

Lesson
Practice
Lesson

Concept summary

To draw the graph of an exponential function we can fill out a table of values for the function and draw the curve through the points found. We can also identify 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 to identify other points on the curve

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:

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x
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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
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-2
-1
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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

Draw a graph of y=2.5\left(4\right)^x by first finding the common ratio and the y-intercept.

Approach

The function has a common ratio of 4, and a y-intercept at \left(0,2.5\right). We can find other points on the curve using a table of values.

x-2-10123
y0.156250.6252.51040160

Solution

-2
-1
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y

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.

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

MA.912.AR.5.6

Given a table, equation or written description of an exponential function, graph that function and determine its key features.

MA.912.F.1.2

Given a function represented in function notation, evaluate the function for an input in its domain. For a real-world context, interpret the output.

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