Exponential Growth or Decay

Exponential Growth

or Decay

​Objectives: Today I can

1. Determine if a function is modeling growth or decay.

2. Model exponential functions related to real world applications.​

3. Determine if a function is exponential, linear or neither.

Let's Analyze the Graph!

1. What doe the x-line represent

2. How is the x-line labeled? (hint Increments)​

3. What does the y-line represent? Notice how it is labeled.

4. Does this Model represent a decay or growth? Explain why.

5. Summarize in your own words what this graph is demonstrating.

6. What social media do we have today?​

Open Ended

What other real world scenario can be represented by an exponential function?

State if it would represent a growth or decay.

Review: How Do Linear Functions Change?

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

​

Consider the Relationship ​represented by this table:

​x	​y
​-2	​-1
1​	​1
​4	3​
​7	5​

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

Linear and exponential relationships differ in the way the y-values change when the x-values increase by a constant amount:

• In a linear relationship, the y-values have equal differences.

• In an exponential relationship, the y-values have equal ratios.

Exponential vs. Linear Growth

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

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 as x was increasing, f(x) was increasing by a constant ratio.

• 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":

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

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 b is 1/3 at it is greater than zero and less than 1.

• 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":

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?