Unit 5, Lesson 6
Writing Equations for Exponential Functions
Algebra 2
Lesson Synthesis
The purpose of this discussion is for students to understand how they can use the properties of exponents to calculate the and values for an exponential function starting from two known points on the graph of the function. Display this graph for all to see:
Here are some questions for discussion:
- “Where on a graph do we see the initial value, , of the function?” ( is the function value when the input is 0, so is where the graph of intersects the -axis, 300.)
- “What do we know about the function when ?” ()
- “How can we use to calculate the number ?” (Dividing each side by 300, we have that the decay factor over 1.5 hours is , which means . Using the properties of exponents, , so because .)
- “What equation can we write for this function?” ()
Student Lesson Summary
Equations are helpful for communicating how quantities are changing. We can write equations from descriptions or from graphs.
Sometimes, the information on how a quantity is changing is given in a graph instead of in words. We can find an equation for an exponential function using two points on its graph, just as we’ve done in the past with linear functions. Let’s say we want to find a function of the form , whose graph contains and .
Because , the -intercept of the graph is . In this example, the intercept is given as , so we know that .
Using the second given point, , we know . This means that . Solving this equation we have:
To determine the exact value of , let’s use the properties of exponents. Since is positive, we can show that because
We can now write an equation defining : .