Exponential vs. Linear: Growth and Decay

Exponential vs. Linear Lesson: Growth and Decay

I can determine if a function is modeling growth or decay.

I can create a table for a function.

I can evaluate a function.

Review: How Do Linear Functions Change?

Linear functions increase or decrease according to their rate of change, or slope (rise over run).

Positive Rate of Change

• Previously, we've discussed how linear functions are increasing when they go upward from left to right on a graph

• We also know a linear function if its m-value (slope) is positive

Negative Rate of Change

• Previously, we've discussed how linear functions are decreasing when they go downward from left to right on a graph

• We also know a linear function if its m-value (slope) is negative

Poll

Exponential functions use the terms "growth" and "decay" instead. Which of the two graphs do you think is "growing" from left to right, and could be labeled "growth"? Click on your choice.

Multiple Choice

Which of the graphs is exponential decay?

Now I know when an exponential graph is modeling "growth" or "decay", but how can I tell just from looking at a function?

Let's investigate!

Multiple Choice

Use desmos.com/calculator to determine which table matches the function 

 $f\left(x\right)=2\left(3\right)^x$  

Multiple Choice

Is

 $f\left(x\right)=2\left(3\right)^x$  growth or decay? How do you know?

• Growth because it's going downwards from left to right.

• Growth because it's going upwards from left to right.

• Growth because the table of values is decreasing, showing a pattern of division.

• Growth because all exponential functions grow.

Here are more examples of exponential functions that model "growth":

•  $f\left(x\right)=4^x$  

•  $g\left(x\right)=\frac{1}{2}\left(2\right)^x$  

•  $y=12\left(50\right)^x$  

•  $f\left(x\right)=2\left(1.5\right)^x$  

Open Ended

What do you notice about the examples of exponential growth functions? Is there a way you can determine an exponential function is "growing" instead of "decaying"?

Multiple Choice

Use desmos.com/calculator to determine which table matches the function 

 $h\left(x\right)=12\left(\frac{1}{3}\right)^x$  

Multiple Choice

Is

 $h\left(x\right)=12\left(\frac{1}{3}\right)^x$  growth or decay? How do you know?

• Decay because it's going downwards from left to right.

• Decay because it's going upwards from left to right.

• Decay because the table of values is increasing, showing a pattern of multiplication.

• Decay because all exponential functions decrease.

Here are more examples of exponential functions that model "decay":

•  $f\left(x\right)=\left(\frac{1}{4}\right)^x$  

•  $g\left(x\right)=2\left(\frac{2}{3}\right)^x$  

•  $y=-12\left(0.9\right)^x$  

•  $y=10\left(\frac{1}{2}\right)^x$  

Open Ended

Closure: Respond to each question with a complete sentence. What do you notice is the same about "growth" functions and "decay" functions? What do you notice is different about "growth" functions and "decay" functions? How can you tell a function is "decaying" or "growing", without seeing the graph?