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.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their observations about the effect of the different modifications to . Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. 

If it does not come up, tell students that people often describe the shape when is positive as a parabola that “opens upward” and the shape when is negative as a parabola that “opens downward,” and add these phrases to the reference.

For each change to the expression 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 moves the graph down?”

Discuss questions such as:

• “The points and are three points on the graph representing . When we add 3 to how do the values for , , and change?” (Their values increase by 3: .) “What about when we subtract 3 from ?” (They decrease by 3: .)
• “How do the values change when you multiply by a positive number, say, 3?” (The values for triple those of . For , and , the points will be and .)
• “How do the tripled values affect the graph?” (They stretch the graph upward 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.

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).

Invite students to share their analyses of how the values in the tables relate to 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 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.

Some students may think that a factor such as relates to an -intercept of . In earlier lessons students learned the the zeros of the function give the -coordinates of the -intercepts. Show students the equations and , and ask them to solve each equation and relate the solutions back to making the expression equal 0. Some students may benefit from seeing the expression written as to further emphasize that the expression takes the value 0 when 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 -intercept of the graph helps to find the equation in standard form (or vice versa).
• The -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.