Today students will see one use for solving equations. In the first lesson of this unit students identified key features of exponential and logarithmic graphs. I now want students to use equations to find x and y intercepts.
Students come into class with a question that requires students to really think about an exponential function. At first, my students often say that an exponential cannot have an x-intercept because its asymptote is at y=0. I begin asking questions such as:
Students work with others in their groups to attempt to write an equation. When students ask me if their idea is correct I direct students to use a calculator to see if the graph is correct.
This activity requires some students to really persevere. By not telling students if their equation is correct I allow students to do more problem solving. When each group is sure they have an equation I have one person in the group put the answer on the board. I have some groups that find the answer very quickly I have this group wait (2-3 minutes) to share the result. This allows more groups to persevere and determine an equation. For the students who are really confused seeing other groups results will help them determine an equation.
I now ask students to find a logarithmic equation that has both intercepts. Students determine the equations and share the results. Here students should see shifting the graph to the left or reflecting across the y and shifting to the right will result in the graph with both intercepts.
After the equations are shared I ask student to explain why these equations have both intercepts. Students discuss how shifting the graph down gives both intercepts. Some students have reflected the graph and shifted the graph up which is another way to make the graph have both intercepts.
Now that my students understand that an exponential or a logarithmic function can have both an x- and a y-intercept, I am ready to help them learn how to determine the intercepts. As we pursue this goal, we will also review finding the other key features (domain, range, asymptote).
I'll start by putting an exponential function on the board. We discuss how this graph is a transformation of the parent function. Students realize the graph has a vertical shift of -3. Students now find the domain and range of the function. As they find these they are also able to identify the asymptote. When we need to determine the intercepts I ask students, "How are we going to find the x-intercept? What do we know about the x-intercept?" My students typically say that the value of y is zero for the x-intercept. Many will set the equation equal to zero and solve. In this case, the answer we get is ln(3) which is not a rational number. In order to help students interpret this result correctly, I have the students write the exact x-intercept as an ordered pair. Writing the ordered pair helps students understand that we are talking about a point whose y-value is zero.
Once the x-intercept most students find the y-intercept by letting x=0. I again have students write the intercept as an ordered pair. Once the key features are found we make a sketch of the graph when we make a sketch the students approximate the value of ln 3. We identify the asymptote and plot the intercepts.
The class next analyzes a logarithmic function. This function has a vertical shift of -4 so the function has both x- and y-intercepts. Students work on this problem in groups for a few minutes and then share what they have determined. I ask the students to explain how they determined each feature. To make sure students understand how to find the intercepts we put the process on the board. Again we estimate the value of any irrational values so we can draw a sketch of the function.
Students need time to problem solve as groups or individuals. I could continue putting problems on the board but letting students work at their own pace and ask questions will help students more than doing board work. I give students the Analyzing Graphs worksheet. I also give each student some graph paper to use for sketching their graphs. This practice time allows me to work with students who are struggling. It also gives students a chance to complete part of the assignment in class, a good incentive for students to use their time wisely.
As class ends I ask the students to answer questions about the intercepts. Students struggle with these questions until they remember how they find the x and y intercepts. Students should realize that an exponential function always has x=0 in its domain which means it will always cross the y-axis. Similarly a logarithmic function will always have y=0 in the range so it will always have an x-intercept.
After a few minutes of discussing I ask students to put their answers and reasoning on a sticky note (no names necessary.) I have on the board a line for each function. The line has yes on one end and no on the other end. The middle is where students put unsure. Each student places their sticky note on the line.
After the notes are placed I take notes for each side and the middle and share the responses. Students discuss the comments on the sticky notes. After a few comments students begin to ask if they can change their mind about the answer.
Using sticky notes is a way to get students to respond without worrying about any consequences since no names are placed on the notes. This gives me a quick assessment of understanding. When we discuss the answers students are able to hear reasons and adjust their decisions.