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6. Exponential Functions
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United States of AmericaTennessee
Integrated Math 1

6.05 Linear and exponential models

Lesson
Practice
Lesson

Concept summary

Linear, and exponential functions differ in how the output of the function changes with regards to the input x.

We can use two things to determine the change in output: the first difference, and the common ratio.

xf(x)\text{first difference}
-2-4
-1-2-2-(-4)=2
000-(-2)=2
122-0=2
244-2=2
366-4=2

The first difference is the difference between two consecutive outputs.

If the first difference is constant, then the function is linear.

xf(x)\text{common ratio}
-2\dfrac{1}{4}
-1\dfrac{1}{2}\dfrac{1}{2}\div \dfrac{1}{4}=2
011\div \dfrac{1}{2}=2
122\div 1=2
244\div 2=2
388\div 4=2

If the first is not constant, then we look at the common ratio. If the common ratio is constant then the function is exponential.

We can also identify a function by the shape of its graph:

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
  • g\left(x\right) is a linear function
  • h\left(x\right) is an exponential function

Worked examples

Example 1

Find the average rate of change of the function f(x) between x = 2 and x = 6.

-1
1
2
3
4
5
6
7
8
x
1
2
3
4
5
6
7
8
f(x)

Approach

Find the values of f(2) and f(6) using the graph.

Divide the difference between the y-values by the change in x. In this case getting the expression

\frac{f(6)-f(2)}{6-2}

Evalute the expression to find the average rate of change.

Solution

\displaystyle \text{Average rate of change}\displaystyle =\displaystyle \frac{f(6)-f(2)}{6-2}
\displaystyle {}\displaystyle =\displaystyle \frac{5-1}{6-2}Substitute the values in for f(6) and f(2)
\displaystyle {}\displaystyle =\displaystyle \frac{4}{4}Evaluate the numerator and denominator
\displaystyle {}\displaystyle =\displaystyle 1Cancel the common factors in the fraction

Reflection

The average rate of change is the slope of the straight line between the two points.

Example 2

Would a linear, or exponential function best model the number of pieces a cake is cut into, if each piece of cake is cut in half every minute?

Assume you start with a whole cake.

Approach

Construct a table with the number of pieces of cake.

Construct a first differences, second differences and the common ratios table of consecutive terms to identify whether there are any which are constant.

  • If the common ratio is constant for all x, then f(x) is exponential.

  • If the first difference is constant for all x, then f(x) is linear.

Determine whether the situation is best described be a linear, or exponential function.

Solution

We get the following table of values:

\text{Minutes} \\ \text{ passed}\text{Pieces} \\ \text{ of cake}\text{Common} \\ \text{ ratio}\text{First} \\ \text{ difference}
01
1221
2422
3824
41628
532216

The common ratio is constant for all values, therefore the best function to model this scenario is an exponential function.

Outcomes

M1.N.Q.A.1

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

M1.N.Q.A.1.A

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

M1.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.*

M1.N.Q.A.1.D

Choose an appropriate level of accuracy when reporting quantities.*

M1.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

M1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

M1.F.IF.C.6

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

M1.F.IF.C.6.A

Compare properties of two different functions. Functions may be of different types and/or represented in different ways.

M1.F.LE.A.1

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

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

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