Lesson 12

This warm-up activates students’ prior knowledge about how the parameters of a linear expression are visible on its graph, preparing students to make similar observations about quadratic expressions and their graphs.

Students may approach the matching task in different ways:

• By starting with the graphs and thinking about corresponding equations. For example, they may notice that Graph A has a negative slope and must therefore correspond to \(y=3-x\), the only equation whose linear term has a negative coefficient. They may notice that the slope of C is greater than that of B, so C must correspond to \(y=3x-2\).
• By starting with the equations and then visualizing the graphs. For example, they may see that \(y=3x-2\) has a constant term of -2, so it must correspond to C, and that Graph B intercepts the \(y\)-axis at a higher point than Graph A, so B must correspond to \(y=2x+4\).

Invite students with contrasting approaches to share during discussion.

Select students to share how they matched the equations and the graphs. As students refer to the numbers that represent the slope and \(y\)-intercept in the equations, encourage students to use the words “coefficient” and “constant term” in their explanations.

To support students vocabulary development, and to prepare them for the lesson, consider writing the equations from the warm up for all to see and identify the coefficient and constant terms in each equation.

Highlight that the equations and the graphs are connected in more than one way, so there are different ways to know what a graph would look like given its equation, or what an equation would entail given its graph.

Tell students that we will look at such connections between the expressions and graphs that represent quadratic functions.

This activity enables students to see how the coefficient of the squared term and the constant term in a quadratic expression in standard form can be seen on the graph. Students start by graphing \(y=x^2\) using technology. They then experiment with adding positive and negative constant values to \(x^2\) and multiplying it by positive and negative coefficients. They generalize their observations afterwards. Along the way, students practice looking for and expressing regularity through repeated reasoning (MP8).

If working with a partner, students will take turns using the graphing technology and recording observations. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Provide access to devices that can run Desmos or other graphing technology. Consider arranging students in groups of 2. Ask one partner to operate the graphing technology and the other to record the group’s observations, and then to switch roles halfway.

If time is limited, consider arranging students in groups of 4 and asking each group member to experiment with one of the four changes listed in the activity and then reporting the results to their group.

Remind students to adjust their graphing window as needed. If using Desmos, instruct students that creating a slider might be a helpful tool for this activity. For students who need reminding, they can create sliders by typing a letter to represent a parameter, such as \(f(x)=x^2+a\) to change the constant term.

Invite students to share their observations, and if possible, demonstrate their experiments for all to see. Tell students that people often describe the shape when \(a\) is positive as a parabola that “opens upward” and the shape when \(a\) is negative as a parabola that “opens downward.”

For each change to the expression (for example, adding a constant, or multiplying \(x^2\) by a positive number) and the observed change on the graph, solicit students’ ideas about why the graph transformed that way. For example, ask: “Why do you think subtracting a number from \(x^2\) moves the graph down?”

Discuss questions such as:

• “The points \((1,1), (2, 4)\) and \((3,9)\) are three points on the graph representing \(x^2\). When we add 3 to \(x^2\) how do the \(y\) values for \(x=1\), \(x=2\), and \(x=3\) change?” (Their \(y\) values increase by 3: \((1,4), (2, 7), (3, 12)\).) “What about when we subtract 3 from \(x^2\)?” (They decrease by 3: \((1,\text-2), (2,1), (3,6)\).)
• “How do the \(y\) values change when you multiply \(x^2\) by a positive number, say, 3?” (The \(y\) values for \(3x^2\) triple those of \(x^2\). For \(x=1, x=2\), and \(x=3\), the points will be \((1,3), (2, 12)\) and \((3,27)\).)
• “How do the tripled \(y\) values affect the graph? (They stretch the graph up vertically, making the graph appear narrower.)

To help students make stronger connections between the parameters of a quadratic expression and the features of its graph, consider the optional activity included in this lesson.

Earlier, students made a series of observations about how changing the parameters of quadratic expressions transformed the graphs. They may not have fully recognized why the graphs changed the way they did. This optional activity prompts students to explain their earlier observations and further understand the behaviors of the graphs in relation to the quadratic expressions.

Students evaluate quadratic expressions with different coefficients and constant terms and compare the values to those expressions to the values of \(x^2\). Upon studying the table of values, they see, for example, that adding a constant term increases the value of \(x^2\) by that number, moving the corresponding points on the graph up by that amount. They see that multiplying the squared term by 2 or \(\frac12\) changes the output values by a factor of 2 or \(\frac12\), which moves the points for \(x^2\) to a position twice or half as high on the graph.

Making a spreadsheet tool available gives students an opportunity to choose appropriate tools strategically (MP5).

If students have access to a spreadsheet, suggest that it might be a helpful tool in this activity to speed up the calculation process.

Invite students to share their analyses on how the values in the tables relate the behaviors of the graphs they saw earlier. Consider plotting the points on a dynamic graphing tool to make explicit the connections between the values in the tables and the graphs.

In this activity, students apply what they learned about the connections between quadratic expressions and the graphs representing them. They also practice identifying equivalent quadratic expressions in standard and factored forms. Students are given a set of cards containing equations and graphs. They sort them into sets of 3 cards wherein each set contains two equivalent equations and a graph that all represent the same quadratic function. They also explain to a partner how they know the cards belong together. A sorting task gives students opportunities to analyze the different equations, graphs and structures closely and make connections (MP2, MP7) and to justify their decisions as they practice constructing logical arguments (MP3).

Here are the equations and graphs for reference and planning:

As students work, monitor how they go about making the matches. Some students may begin by studying features of the graph and then relate them to the equations. Others may work the other way around. Identify students with contrasting approaches so they can share in the discussion.

Arrange students in groups of 2. Give each group a set of pre-printed cards. Tell students to take turns sorting the cards into sets that represent the same quadratic function. The person whose turn it is to compile a set should explain how they know the cards belong together. (The person who has the last turn should also explain why the cards belong together, aside from the fact that they are the last remaining cards.) The partner should listen and ask for clarification or discuss any disagreement. Once all the cards are sorted, ask students to record their findings in the given graphic organizer.

If time permits, before partners record anything, ask them to compare their sorted sets with another group of students and discuss any disagreements.

Some students may think a factor such as \((x-1)\) relates to an \(x\)-intercept of \((\text-1,0)\). In earlier lessons students learned the the zeros of the function give the \(x\)-coordinates of the \(x\)-intercepts. Show students the equations \(x-1=0\) and \(x+1=0\) and ask them to solve each equation and relate the solutions back to making the expression \((x+1)(x-1)\) equal 0. Some students may benefit from seeing the expression \(x-1\) written as \(x+\text-1\) to further emphasize that the expression takes the value 0 when \(x\) is the opposite. Students will continue this work when solving quadratic equations in the next unit.

Invite students to share how they found pairs of equivalent equations and how they matched the equations to the graphs.

Highlight these explanations:

• The \(y\)-intercept of the graph helps to find the equation in standard form (or vice versa).
• The \(x\)-intercepts of the graph help find the equation in factored form (or vice versa).
• The quadratic expression in factored form can be expanded to find the equivalent expression in standard form.