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 , 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 .
• By starting with the equations and then visualizing the graphs. For example, they may see that has a constant term of -2, so it must correspond to C, and that Graph B intercepts the -axis at a higher point than Graph A, so B must correspond to .

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 -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 identifying the coefficient and constant term 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 using technology. They then experiment with adding positive and negative constant values to and multiplying it by positive and negative coefficients. They generalize their observations afterward. 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 to critique the reasoning of others (MP3).

In the digital version of the activity, students use an applet to explore how different values affect the graph of a quadratic function. By using the applet to experiment with different values, students can dynamically see how the graphs change as the values change. The digital version is preferable, if it is available, so that students can actively see the effect that different values have on the graph.

This activity is optional because it goes beyond the requirements of the standards. It gives additional insight into the reason that graphs change when modifying the quadratic coefficient or the constant term in a quadratic expression.

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

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 closely the different equations, graphs, and structures and make connections (MP2, MP7) and to justify their decisions as they practice constructing logical arguments (MP3).

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.