Lesson 17

Changing the Vertex

17.1: Graphs of Two Functions (5 minutes)

Warm-up

In this warm-up, students revisit quadratic expressions in different forms and what the expressions reveal about the graphs that represent them. They match equations and graphs representing two quadratic functions. The axes of the graphs are unlabeled, so students need to reason abstractly about the parameters in the expressions and the features of the graphs.

Student Facing

Two graphs representing functions in x y plane, origin O.

Here are graphs representing the functions \(f\) and \(g\), given by \(f(x) = x(x+6)\) and \(g(x) = x(x+6)+4\).

  1. Which graph represents each function? Explain how you know.
  2. Where does the graph of \(f\) meet the \(x\)-axis? Explain how you know.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Invite students to share their matches and explanations. If one or more explanations in the Student Response are not mentioned, bring them up so that students can see multiple ways of reasoning about the equations and graphs.

17.2: Shifting the Graph (15 minutes)

Activity

In this activity, students apply what they know about the vertex form to modify a given quadratic expression such that its graph is translated in particular ways to produce a different vertex. Students also consider (by analyzing an argument) whether modifying the squared term of a quadratic expression in standard form produces the same translation on the graph as does modifying the squared term of an expression in vertex form.

As they modify expressions to translate graphs, students make use of structure (MP7). As they analyze an argument about how changing a quadratic expression in standard form affects the graph, they practice explaining their thinking and critiquing the reasoning of others (MP3).

Launch

Remind students that they have studied the connections between quadratic expressions and their graphs over several lessons. They have matched expressions and graphs, and sketched graphs of given expressions. Now they will think about how to change one or more parts of a quadratic expression so that its graph has certain features.

Before beginning the activity, ask students to describe the graph representing \(y=x^2\). Make sure students recall that the graph has a vertex at the origin and opens up. If needed, ask them to name some points that the are on the graph, for example, \((\text-3, 9), (\text-2,4), (\text-1, 1), (0,0), (1,1), (2, 4), (3,9)\).

Give students access to graphing technology, but tell students to answer the first question without using technology.

Consider arranging them in groups of 2 and encouraging them to discuss the last question with a partner after some quiet think time. If students use a graph to illustrate that Kiran’s statement is incorrect, encourage them to also use the expression in their justification.

Student Facing

  1. How would you change the equation \(y=x^2\) so that the vertex of the graph of the new equation is located at the following coordinates and the graph opens as described?
    1. \((0,11)\), opens up
    2. \((7,11)\), opens up
    3. \((7,\text-3)\), opens down
  2. Use graphing technology to verify your predictions. Adjust your equations if necessary.
  3. Kiran graphed the equation \(y=x^2+1\) and noticed that the vertex is at \((0,1)\). He changed the equation to \(y=(x-3)^2+1\) and saw that the graph shifted 3 units to the right and the vertex is now at \((3,1)\).

    Next, he graphed the equation \(y= x^2 +2x +1\), observed that the vertex is at \((\text-1,0)\). Kiran thought, “If I change the squared term \(x^2\) to \((x-5)^2\), the graph will move 5 units to the right and the vertex will be at \((4,0)\).”

    Do you agree with Kiran? Explain or show your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select students to share their equations for the first question and their explanations for how they knew what modifications to make.

Then, focus the discussion on the third question and how students knew that, when a quadratic expression is in standard form, adding a constant term before squaring the input variable does not translate the graph the same way as when the expression is in vertex form. At this point, students are not expected to come up with a rigorous justification as to why the graph will not translate as Kiran described. They are only to make this observation and consider ways to explain it.

Solicit as many explanations as time permits. If no one mentioned that the expression \(x^2+2x+1\) is not in vertex form and that its parameters do not relate to the graph the same way, bring these points up.

If time permits, consider pointing out that in \(x^2+2x+1\) the input \(x\) shows up in the squared term and the linear term. If we subtract 5 from \(x\) before it is squared but do not subtract 5 from \(x\) before it is multiplied by 2, then the graph does not shift horizontally. (If we graph \((x-5)^2 + 2(x-5) + 1\), the graph does shift 5 units to the right.) In a later course, students will look more closely at the effects on the graph of replacing \(f(x)\) by, for instance, \(f(x) + k\) and \(f(x + k)\).

17.3: A Peanut Jumping over a Wall (15 minutes)

Activity

In this activity, students again apply what they learned about quadratic expressions and their graphs. This time, they solve a contextual problem. Students consider how to change the parameters of a quadratic expression that models the movement of a digitally animated object—a peanut—such that it travels in a parabolic path and meets certain restrictions (MP4).

Providing time to notice and wonder will elicit the idea that this equation shows the horizontal and vertical movement of the peanut, which will be useful when students adjust the equation in a later activity. While students may notice and wonder many things about the graphs and equation, connecting the vertex of the graph to the parameters in the equation are the important discussion points. This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

For the second question, there are many possible values of \(h\) and \(k\) that students could use. As students work, notice those who choose to:

  • Shift the starting position of the peanut closer to the wall (translating the graph horizontally to the right).
  • Stay in the same starting position but shift the vertex to a point closer to the wall.
  • Shift both the peanut and the vertex of the parabolic path.

If students choose to use graphing technology to help them experiment with expressions or verify their predictions, they practice choosing tools strategically (MP5).

Launch

You may wish to use the applet during the synthesis to demonstrate student equations. There is a copy of the applet with an option to show the path of the peanut.

Representing, Conversing: MLR7 Compare and Connect. As students share their values of \(h\) and \(k\) with the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the \(h\) and \(k\) represent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness
Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content. 
Supports accessibility for: Language; Conceptual processing

Student Facing

Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to show the path of the main character, an animated peanut, jumping over a wall.

Mai uses the equation \(y = \text-0.1(x-h)^2+k\) to represent the path of the jump. \(y\) represents the height of the peanut as a function of the horizontal distance it travels \(x\). On the screen, the base of the wall is located at \((22, 0)\), with the top of the wall at \((22,4.5)\).

The dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall.

Graph of the path of a peanut’s jump.
  1. What are the values of \(k\) and \(h\) in this equation?
  2. This applet is programmed to use Mai’s equation. Experiment with the applet to find an equation to get the peanut over the wall, and keep it on the screen. Be prepared to explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Launch

Ask students to read the first couple of paragraphs in the statement. Then, display this graph for all to see. Tell students that when Mai used the equation \(y = \text -0.1(x-16)^2+7\), this is the path of the peanut’s jump.

Graph of parabola 

Ask students what they notice and wonder about the equation and the graph. Make sure students see the connection between the parameters in the equation and the vertex of the graph before they proceed with the activity.

Provide continued access to graphing technology, in case students choose to use it.

Representing, Conversing: MLR7 Compare and Connect. As students share their values of \(h\) and \(k\) with the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the \(h\) and \(k\) represent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness
Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content. 
Supports accessibility for: Language; Conceptual processing

Student Facing

Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to model the path of the main character, an animated peanut, jumping over a wall.

two peanuts with wire arms and legs on purple background

Mai uses the equation \(y = \text-0.1(x-h)^2+k\) to represent the path of the jump. \(y\) represents the height of the peanut as a function of the horizontal distance it travels, \(x\).

On the screen, the base of the wall is located at \((22, 0)\), with the top of the wall at \((22,4.5)\). The dashed curve in the picture shows the graph of 1 equation Mai tried, where the peanut fails to make it over the wall.

Graph of the path of a peanut’s jump.
  1. What are the values of \(h\) and \(k\) in this equation?
  2. Starting with Mai’s equation, choose values for \(h\) and \(k\) that will guarantee the peanut stays on the screen but also makes it over the wall. Be prepared to explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Some students may randomly choose values for \(h\) and \(k\) until they find a combination whose graph meets the requirements. Encourage them to reason about the problem more systematically, by considering what they learned about the vertex form.

Activity Synthesis

Select students with contrasting strategies to share their equations and explanations for the second question. To verify that students’ proposed equations meet the animation criteria, consider graphing students’ proposed equations for all to see. Highlight the three strategies mentioned in the Activity Narrative.

Point out one distinction between the graphs in this activity and previous graphs that represent projectile motions. In earlier cases, the output represented the height of an object as a function of time, so the graph of the function and physical trajectory of the object may be very different (for example, the object may be going straight up and then fall straight down, but the graph shows a parabola). Here, they see a model where the height of an object is a function of horizontal distance, so the path of the object in a projectile motion does resemble the graph representing the function.

17.4: Smiley Face (10 minutes)

Optional activity

This activity is optional. Use it to allow students to practice writing quadratic expressions to produce several graphs with particular features. Here students will need to know how to restrict the domain of the graph produced using the technology available to them. If students are not already familiar with how to do so, demonstrate how to do so with a simple quadratic function, for example: \(y=x^2\), where the domain is \(\text{-}2 \leq x \leq 2\).

The activity is unlike the ones students have seen before and requires students to transfer and apply their understanding in new ways. It is a chance for students to make sense of a problem and persevere in solving it (MP1).

Launch

Provide access to devices that can run Desmos or other graphing technology.

Student Facing

Do you see 2 “eyes” and a smiling “mouth” on the graph? The 3 arcs on the graph all represent quadratic functions that were initially defined by \(y=x^2\), but whose equations were later modified.

  1. Write equations to represent each curve in the smiley face.
  2. What domain is used for each function to create this graph?
Left eye, Opens down, Vertex at -2 comma 50. Right eye, opens down, vertex = 2 comma 50. Mouth, opens up, vertex = 0 comma 10.

 

Student Response

For access, consult one of our IM Certified Partners.

Lesson Synthesis

Lesson Synthesis

Help students to synthesize what they learned by reflecting on questions such as:

  • “Suppose we add \(p\), a positive number, to the squared term in \(y=2x^2\) to give \(y=2x^2 + p\). How does this change affect the vertex of the graph?” (It moves the graph \(p\) units up. The \(x\)-coordinate of the vertex is the same but the \(y\)-coordinate increases by \(p\).)
  • “Suppose we replace the \(x^2\) in \(y=x^2\) with \((x+q)^2\), where \(q\) is a positive number. How does that change the vertex of the graph?” (It shifts the graph \(q\) units to the left. The vertex is now \((\text-q,0)\).)
  • “Suppose we replace \(x^2\)in \(y=x^2\) with \((x+q)^2\), where \(q\) is a negative number, and then add \(p\), also a negative number. How is the vertex of the graph representing \(y=(x+q)^2 + p\) different than that of \(y=x^2\)?” (Its vertex is shifted \(q\) units to the right and \(p\) units down.)
  • “What is the vertex of the graph representing \(y=\text{-}x^2 - 9\)? Does the graph open up or down?” (\((0, \text{-}9)\). It opens down, due to the negative coefficient of \(x^2\).)
  • “What equation will translate the graph so that its vertex is at \((\text-5, 2)\) but it still opens down?” (\(y=\text{-}(x+5)^2 +2\))

17.5: Cool-down - Nudging A Graph (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

The graphs of \(y = x^2\), \(y = x^2 + 12\) and \(y = (x+3)^2\) all have the same shape but their locations are different. The graph that represents \(y=x^2\) has its vertex at \((0,0)\).

Three parabolas in x y plane, origin O.

Notice that adding 12 to \(x^2\) raises the graph by 12 units, so the vertex of that graph is at \((0,12)\). Replacing \(x^2\) with \((x+3)^2\) shifts the graph 3 units to the left, so the vertex is now at \((\text-3,0)\).

We can also shift a graph both horizontally and vertically.

The graph that represents \(y = (x+3)^2 +12\) will look like that for \(y = x^2\) but it will be shifted 12 units up and 3 units to the left. Its vertex is at \((\text-3,12)\).

Two parabolas in x y plane, origin O.

The graph representing the equation \(y = \text{-}(x+3)^2 + 12\) has the same vertex at \((\text-3,12)\), but because the squared term \((x+3)^2\) is multiplied by a negative number, the graph is flipped over horizontally, so that it opens downward.

Two parabolas in x y plane, origin O.