15.1: Which One Doesn’t Belong: Four Graphs (10 minutes)
This warm-up prompts students to carefully analyze and compare properties of graphs, particularly the locations of the intercepts. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and how they talk about characteristics of the graph of a quadratic function. These images were chosen to draw attention to the location of the intercepts, whether the graph of a quadratic function opens upward or downwards, and the location of the vertex of the graph of a quadratic function.
Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn’t belong.
Each graph represents either a linear or quadratic function. Which one doesn’t belong?
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as intercept, vertex, zero (of a function), opens upward, or opens downward. Also, press students on unsubstantiated claims.
15.2: Shift That Line Around (15 minutes)
This activity gives students an opportunity to use a graphing calculator to do the same things they will in the associated Algebra 1 lesson, and to draw the same sorts of connections between changing the parameters of a function and a graph representing the function. In this activity, though, the functions are linear instead of quadratic.
Provide access to graphing technology.
Using graphing technology, graph \(y = 2x\).
- Keep that graph, and add a second graph with each of the following changes to the equation. See how it compares to the graph of \(y=2x\). Complete the table. For the last two rows, think about how the original equation should be modified so that the graph has the specified \(x\)-intercept or \(y\)-intercept.
changes to \(y=2x\) \(x\)-intercept \(y\)-intercept no change \((0,0)\) \((0,0)\) add 5 to \(2x\): \(y=2x + 5\) subtract 8 from \(2x\): \(y=2x-8\)
add or subtract a number of your choice to \(2x\).
write the new equation:
- Experiment with each of the following changes to the equation and see how it affects the graph. Complete the table. For the last two rows, think about how the original equation should be modified so that the graph has the specified \(x\)-intercept or \(y\)-intercept.
changes to \(y=2x\) \(x\)-intercept \(y\)-intercept no change \((0,0)\) \((0,0)\) add 5 to \(x\) before it is multiplied by 2: \(y=2(x+5)\) subtract 8 from \(x\) before it is multiplied by 2: \(y=2(x-8)\)
add to or subtract from \(x\) another number of your choice before it is multiplied by 2.
write the new equation:
- Elena says the graph of \(y=2(x-3)+5\) is the graph of \(y=2x\) shifted 3 units to the right and up 5 units. Clare says that means the \(y\)-intercept is \((0,5)\) and the \(x\)-intercept is \((3,0)\). Do you agree with either or both of them, and why?
Possible questions for discussion:
- Can you summarize how adding or subtracting a constant term to \(2x\) affected the graph?
- Can you summarize how adding or subtracting a constant to \(x\) (before multiplying by 2) affected the graph?
- Is there a pattern in how the \(x\)- and \(y\)-intercepts are related to the equation? What is the relationship?
- We can write the equation as a sum or difference (ex. \(y=2x +10\)) or as a product (ex. \(y=2(x+5)\)). How is each form helpful or unhelpful for graphing?
15.3: Finding Equivalent Expressions (15 minutes)
In the first part of the activity, students apply what they noticed in the previous activity. In the second part, they practice identifying equivalent expressions. This is in preparation for the associated Algebra 1 lesson, when they will need to rewrite expressions in equivalent forms, including expanding expressions like \((x-5)^2\). When they identify equivalent expressions, students notice and make use of structure (MP7).
Consider helping students set up a graphic organizer on scratch paper or on their desks, to provide three regions for placing equivalent expressions and showing any work.
- Without graphing, predict the location of the \(x\)- and \(y\)-intercepts of the graphs of these equations. Then, check using graphing technology.
- \(y = 4x + 8\)
- \(y = 4(x + 8)\)
- \(y =5x-10\)
- \(y = 5(x-10)\)
- Sort these expressions into three groups of equivalent expressions. Be prepared to explain how you know they are equivalent.
- \(2x^2 - 10x\)
- \(2(x^2 - 5x)\)
- \(\text-2x^2 - 10x\)
- \(2x(\text-x - 5)\)
Ask students to share any strategies they used for identifying equivalent expressions. Ask a few students to demonstrate, encouraging them to identify when they are using the distributive property. If students struggled to rewrite any particular expressions, invite them to share their work and identify any errors.