Graphing and Manipulating Exponential Functions (Lesson 4.5)
Unit 4 Day 7
CED Topic(s): 2.4
Graph functions of the form y=b^x and identify key characteristics including end behavior, concavity, domain and range, and key points.
Determine the growth factor of an exponential function from its graph, including when the function has been transformed.
Apply knowledge of transformations to exponential functions.
Explain using exponent properties and transformations why two exponential functions are equivalent.
Quick Lesson Plan
Today students explore the graphs of exponential functions as well as their transformations in a card sort activity. To prepare for this activity, print and cut the card sort found under “Additional Materials.” Each group will need one set of cards. Alternatively, you can use the virtual card sort using this Desmos link.
In this card sort, students work through two sets of graphs and match them with their equations. In the first set, students focus primarily on exponential growth and decay and use key points on the graph to determine the “b” value. Only one graph has negative outputs, and students should identify this graph as depicting a reflection across the x-axis. Additionally students determine whether the function is increasing or decreasing and concave up or concave down. A unique feature of exponential functions is that they are always either only increasing or only decreasing and either concave up or concave down. Getting students to explain why this is the case is a great extension question!
We expect students will come in with some familiarity with graphing exponential functions, especially the parent exponential functions. If this is not the case for the students at your school, we suggest doing this Algebra 2 lesson before today’s AP Precalculus lesson.
In question 3, students will reason with equivalent equations and create informal arguments for why 2^-x is equivalent to (½)^x. A more formal proof is offered in the margin notes as you debrief the lesson. In the second set of graphs, students work with horizontal and vertical shifts and think about how a horizontal shift and vertical stretch could both represent the same graph. In question 6, most students will claim that Petra is correct in her reasoning as they are familiar with identifying horizontal shifts from an equation. Play devil’s advocate and ask them to consider why Pierre might have thought it was a vertical stretch (without indicating if Pierre is correct or incorrect). Get them to see that at every x-value, the output is three times bigger than it would be on the parent function.
What transformation occurred based on the equation? What effect will that have on the graph?
How can an exponential function be decreasing but still have an integer as its base?
Does dividing two consecutive terms always give the base? Why or why not?
What does it mean to raise a number to a negative exponent?
Why are all of these functions either always increasing or always decreasing?
Why are all of these functions either always concave up or always concave down?
The goal of today’s lesson is for students to recognize the effect that various transformations have on the graph of the parent exponential function and to be able to explain algebraically and graphically why two different transformations could have the same effect. Identifying and generating equivalent forms of a function is one of the key skills of the course!
The debrief should include a strategy share for how students were able to match the equation with the graph, with a focus on how the transformation would affect the range and the shape of the function. Additionally, question 6 should include a discussion of why 3^(x+1) is equivalent to 3*3^x. Invite students to give convincing arguments and open the floor for debate. The idea that multiplying by 3 adds one additional factor of 3 and thus increases the exponent by 1 can be brought out verbally or algebraically.
Although knowledge of exponent properties can be used to see some of the connections between equivalent equations, emphasis should be placed on why they work instead of rules to be memorized.
In the QuickNotes section, have students share out in groups whether a certain transformation would affect the domain, range, or horizontal asymptote (end behavior) of an exponential function and why. Students should understand through exploration, not explicit telling, that no transformation affects the domain, that vertical shifts and reflections affect the range and that only vertical shifts affect the horizontal asymptote. Furthermore, end behavior can be explored on the left and right side of the graph.