Graphs of Exponential Functions (Lesson 3.2)
Unit 3 - Day 2
Graph functions of the form y=b^x and identify horizontal asymptotes, domain, range, and key points
Describe transformations of an exponential function from graphs and equations
Reason about equivalent forms of exponential functions
Quick Lesson Plan
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.
You can print and cut out the cards or you can use the virtual card sort using this Desmos link.
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.
The formalization 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 Important Ideas section, have students share out in groups whether a transformation would affect the domain, range, or horizontal asymptote of an exponential function and why, before writing the conclusion statement.