Lesson 12

Graphing the Standard Form (Part 1)

12.1: Matching Graphs to Linear Equations (5 minutes)

Warm-up

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.

Student Facing

Graphs A, B, and C represent 3 linear equations: \(y = 2x+4\), \(y = 3-x\), and \(y =3x-2\). Which graph corresponds to which equation? Explain your reasoning.

Coordinate plane, 3 linear graphs. Line A through negative 2 comma 5 and 3 comma 0. Line B through negative 2 comma 0 and 3 comma 10. Line C through negative 1 comma negative 5 and 4 comma 10.

Student Response

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Activity Synthesis

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.

12.2: Quadratic Graphs Galore (15 minutes)

Activity

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).

Launch

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.

Representing, Conversing: MLR2 Collect and Display. To support small-group discussion, invite students to share what they notice about how the graphs change or stay the same depending on how they change the function. Listen for and amplify language students use to describe the features of the graph such as “opens upward”, “opens downward”, “steeper”, “wider”, etc. Post the collected language in the front of the room so that students can refer to it throughout the rest of the activity and lesson.
Design Principle(s): Maximize meta-awareness; Support sense-making 
Representation: Internalize Comprehension. Represent the same information through different modalities by using individual sketches of each function. Provide students with a graphic organizer that provides space to include sketches and observations for each equation. Some students may benefit from additional support to learn what types of details are helpful in a sketch of a graph. 
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Using graphing technology, graph \(y=x^2\), and then experiment with each of the following changes to the function. Record your observations (include sketches, if helpful).

1. Adding different constant terms to \(x^2\) (for example: \(x^2 + 5\), \(x^2+10\), \(x^2-3\), etc.)

2. Multiplying \(x^2\) by different positive coefficients greater than 1 (for example: \(3x^2\), \(7.5x^2\), etc.)

3. Multiplying \(x^2\) by different negative coefficients less than or equal to -1 (for example: \(\text- x^2\), \(\text-4x^2\), etc.)

4. Multiplying \(x^2\) by different coefficients between -1 and 1 (for example: \(\frac{1}{2}x^2\), \(\text- 0.25x^2\), etc.)

Student Response

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Student Facing

Are you ready for more?

3 parabolas on x and y axes. All are symmetric about y-axis. 2 open up, have positive y-intercepts, and 1 is wider than other. Third parabola opens down, is the widest, and has negative y-intercept.

Here are the graphs of three quadratic functions. What can you say about the coefficients of \(x^2\) in the expressions that define \(f\) (in black at the top), \(g\) (in blue in the middle), and \(h\) (in yellow at the bottom)? Can you identify them? How do they compare?

 

Student Response

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Activity Synthesis

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.

12.3: What Do These Tables Reveal? (10 minutes)

Optional activity

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).

Launch

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

Student Facing

    1. Complete the table with values of \(x^2+10\) and \(x^2-3\) at different values of \(x\). (You may also use a spreadsheet tool, if available.)
      \(x\) -3 -2 -1 0 1 2 3
      \(x^2\) 9 4 1 0 1 4 9
      \(x^2+10\)                                                                 
      \(x^2-3\)
    2. Earlier, you observed the effects on the graph of adding or subtracting a constant term from \(x^2\). Study the values in the table. Use them to explain why the graphs changed they way they did when a constant term is added or subtracted.
    1. Complete the table with values of \(2x^2\), \(\frac12x^2\), and \(\text-2x^2\) at different values of \(x\). (You may also use a spreadsheet tool, if available.)
      \(x\) -3 -2 -1 0 1 2 3
      \(x^2\) 9 4 1 0 1 4 9
      \(2x^2\)                                                              
      \(\frac12 x^2\)
      \(\text -2x^2\)
    2. You also observed the effects on the graph of multiplying \(x^2\) by different coefficients. Study the values in the table. Use them to explain why the graphs changed they way they did when \(x^2\) is multiplied by a number greater than 1, by a negative number less than or equal to -1, and by numbers between -1 and 1.

Student Response

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Activity Synthesis

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.

3 graphs of different colored plotted points on coordinate plane.

12.4: Card Sort: Representations of Quadratic Functions (15 minutes)

Activity

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:

\(y=x^2-1\)

\(y=x^2-4x\)

\(y=x^2-4x+4\)

\(y=x^2-5x+4\)

\(y=x(x-4)\)

\(y=(x+1)(x-1)\)

\(y=(x-1)(x-4)\)

\(y=(x-2)^2\)

Card sort images.

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.

Launch

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.

Student Facing

Your teacher will give your group a set of cards. Each card contains a graph or an equation.

  • Take turns with your partner to sort the cards into sets so that each set contains two equations and a graph that all represent the same quadratic function.
  • For each set of cards that you put together, explain to your partner how you know they belong together.
  • For each set that your partner puts together, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
  • Once all the cards are sorted and discussed, record the equivalent equations, sketch the corresponding graph, and write a brief note or explanation about why the representations were grouped together.

Standard form:

Factored form:

Blank coordinate plane, x, negative 4 to 4 by twos, y, negative 6 to 6 by twos, origin O.

Explanation:

Standard form:

Factored form:

Blank coordinate plane, x, negative 4 to 4 by twos, y, negative 6 to 6 by twos, origin O.

Explanation:

Standard form:

Factored form:

Blank coordinate plane, x, negative 4 to 4 by twos, y, negative 6 to 6 by twos, origin O.

Explanation:

Standard form:

Factored form:

Blank coordinate plane, x, negative 4 to 4 by twos, y, negative 6 to 6 by twos, origin O.

Explanation:

 

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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.

Lesson Synthesis

Lesson Synthesis

To help students consolidate their new insights, consider connecting them to what they already know about the graphs representing linear functions. Discuss questions such as:

  • “How is the constant term in a quadratic equation in standard form—the \(c\) in \(y=ax^2+bx+c\)—like or unlike the constant term in a linear equation in slope-intercept form—the \(b\) in \(y=mx+b\)?” (They both tell us about the \(y\)-intercept. Increasing the value of each constant term moves the graph up, and decreasing it moves the graph down.)
  • “Is the coefficient of the squared term in a quadratic equation in standard form (the \(a\) in \(y=ax^2+bx+c\)) like the coefficient of the linear term in slope-intercept form (the \(m\) in \(y=mx+b\))? Why or why not?” (They both affect the behavior of the graph in similar ways:
    • In linear equations of the form \(y=mx+b\), the greater \(m\) is, the steeper the line gets. In quadratic equations in standard form, the greater \(a\) is, the steeper or narrower the U-shaped graph gets.
    • A negative \(m\) results in a graph that is a downward-sloping line. A negative \(a\) gives a graph that opens downward.)

Encourage students to reflect on the advantages of using expressions in different forms to anticipate the graphs representing quadratic functions. Ask students questions such as:

  • “What information about the graph can you easily obtain from an expression in standard form?” (whether the graph opens up or down, the \(y\)-intercept)
  • “What information can we easily obtain from the factored form?” (the \(x\)-intercepts, the \(x\)-coordinate of the vertex)

An upcoming optional lesson looks at the impact of the coefficient of the linear term. If you are planning on using that lesson, you might tell students that we will look at the linear term, \(b\) in \(y=ax^2+bx+c\), in an upcoming lesson.

12.5: Cool-down - Matching Equations and Graphs (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Remember that the graph representing any quadratic function is a shape called a parabola. People often say that a parabola “opens upward” when the lowest point on the graph is the vertex (where the graph changes direction), and “opens downward” when the highest point on the graph is the vertex. Each coefficient in a quadratic expression written in standard form \(ax^2 + bx+ c\) tells us something important about the graph that represents it.

The graph of \(y=x^2\) is a parabola opening upward with vertex at \((0,0)\). Adding a constant term 5 gives \(y = x^2 + 5\) and raises the graph by 5 units. Subtracting 4 from \(x^2\) gives \(y=x^2-4\) and moves the graph 4 units down.

Coordinate plane, 3 graphs quadratic functions.  First, y = x squared + 5, minimum at 0 comma 5. Next y = x squared, minimum at the origin. Third, y = x squared minus 4, minimum at 0 comma negative 4.
\(x\) -3 -2 -1 0 1 2 3
\(x^2\) 9 4 1 0 1 4 9
\(x^2+5\) 14 9 6 5 6 9 14
\(x^2-4\) 5 0 -3 -4 -3 0 5

A table of values can help us see that adding 5 to \(x^2\) increases all the output values of \(y = x^2\) by 5, which explains why the graph moves up 5 units. Subtracting 4 from \(x^2\) decreases all the output values of \(y = x^2\) by 4, which explains why the graph shifts down by 4 units.

In general, the constant term of a quadratic expression in standard form influences the vertical position of the graph. An expression with no constant term (such as \(x^2\) or \(x^2 +9x\)) means that the constant term is 0, so the \(y\)-intercept of the graph is on the \(x\)-axis. It’s not shifted up or down relative to the \(x\)-axis.

The coefficient of the squared term in a quadratic function also tells us something about its graph. The coefficient of the squared term in \(y = x^2\) is 1. Its graph is a parabola that opens upward.

  • Multiplying \(x^2\) by a number greater than 1 makes the graph steeper, so the parabola is narrower than that representing \(x^2\).
  • Multiplying \(x^2\) by a number less than 1 but greater than 0 makes the graph less steep, so the parabola is wider than that representing \(x^2\).
  • Multiplying \(x^2\) by a number less than 0 makes the parabola open downward.
Coordinate plane, 4 quadratic functions.
\(x\) -3 -2 -1 0 1 2 3
\(x^2\) 9 4 1 0 1 4 9
\(2x^2\) 18 8 2 0 2 8 18
\(\text-2x^2\) -18 -8 -2 0 -2 -8 -18

If we compare the output values of \(2x^2\) and \(\text-2x^2\), we see that they are opposites, which suggests that one graph would be a reflection of the other across the \(x\)-axis.