Lesson 15

Vertex Form

15.1: Notice and Wonder: Two Sets of Equations (5 minutes)

Warm-up

This warm-up prompts students to analyze two sets of equations that they will study more closely in a later activity. In each set, the three equations define the same function but are written in different forms—factored form, standard form, and vertex form. Noticing and wondering about the features of the equations prepares students to reason later that the expressions defining each output are equivalent.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language used to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Launch

Display the two sets of equations for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions. Follow with a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

Set 1:

\(f(x)= x^2 +4x\)

\(g(x)=x(x+4)\)

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

Set 2:

\(p(x)=\text-x^2+6x-5\)

\(\\q(x)=(5-x)(x-1)\)

\(r(x)=\text-1(x-3)^2+4\)

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the reasoning on or near the relevant equations. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information, etc. 

15.2: A Whole New Form (10 minutes)

Activity

This activity introduces students to the vertex form. Students examine the parameters in expressions of this form and the graphs of functions defined by such expressions. They look for structure in the given representations and notice that there are connections between the numbers in each expression and the vertex of the corresponding graph (MP7).

Students also see that we can rewrite the expression in vertex form into another form and show that the expressions are equivalent. Writing equivalent expressions allows students to practice applying the distributive property to expand expressions containing two sums or two differences.

Launch

Give students a minute to read the task statement. Before students answer the questions, briefly discuss the worked example to make sure students can follow the reasoning that illustrates the equivalence of the expressions defining \(h\) and \(f\).

Student Facing

Here are two sets of equations for quadratic functions you saw earlier. In each set, the expressions that define the output are equivalent. 

Set 1:

\(f(x)= x^2 +4x\)

\(g(x)=x(x+4)\)

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

Set 2:

\(p(x)=\text-x^2+6x-5\)

\(\\q(x)=(5-x)(x-1)\)

\(r(x)=\text-1(x-3)^2+4\)

The expression that defines \(h\) is written in vertex form. We can show that it is equivalent to the expression defining \(f\) by expanding the expression:

\(\displaystyle \begin {align} (x+2)^2-4 &=(x+2)(x+2)-4\\ &=x^2+2x+2x+4-4\\ &=x^2+4x\\ \end{align}\)

  1. Show that the expressions defining \(r\) and \(p\) are equivalent.
  2. Here are graphs representing the quadratic functions. Why do you think expressions such as those defining \(h\) and \(r\) are said to be written in vertex form?

    Graph of $h$

    Parabola. Opens up. Vertex = -2 comma -4.

    Graph of $r$

    Parabola. Opens down. Vertex = 3 comma 4.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may incorrectly think that \((x-3)^2\) is \((x^2-3^2)\). Remind them that \((x-3)^2\) means \((x-3)(x-3)\). Consider pointing out the example in the task statement to help students show that the expressions for \(r\) and \(p\) are equivalent.

Activity Synthesis

Invite students to show that the expressions defining functions \(r\) and \(p\) are equivalent. Consider pointing out that at the moment it is easier to show equivalence by going from vertex form to standard form than from standard form to vertex form. In a future unit, we will look at how to do the latter.

Then, discuss questions such as:

  • “What information does a quadratic expression in the vertex form reveal? How does it show that information?” (It reveals the coordinates of the vertex of the parabola. The number in the parentheses seems to be related to the \(x\)-coordinate of the vertex. The number outside seems to be the \(y\)-coordinate of the vertex.)
  • “What doesn’t it tell us?” (It does not allow us to easily see the \(x\)- or \(y\)-intercepts.)
  • “Why do you think this form is used?” (Sometimes we want to know the maximum or the minimum of a function. It is helpful to be able to see it in the expression or equation defining the function.)
  • “Can you give an example of a situation when it might be useful to have the relationship modeled using an expression in vertex form?” (Examples: when we are interested in the maximum height of an object in projectile motion, or when we want to know the maximum revenue in a business model.)
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of vertex form. 
Supports accessibility for: Conceptual processing; Language

15.3: Playing with Parameters (20 minutes)

Activity

Earlier, students noticed that the numbers in a quadratic expression in vertex form are related to the coordinates of the vertex of the graph. Here, they investigate those connections closely. Just as they have done with expressions in standard and factored forms, students use technology to experiment with each parameter of expressions in vertex form and study the effects on the graph. In this process, they practice looking for regularity in 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).

The work also encourages students to begin looking at the structure of the form—a squared expression with a coefficient \(a\) and a constant term (MP7). In an upcoming lesson, they will further make sense of how this structure relates to the graph.

Launch

Provide access to graphing technology. Consider arranging students in groups of 2. For the first two questions involving graphing, ask one partner to operate the graphing technology and the other to record the group’s observations, and then to switch roles halfway.

Consider pausing for a class discussion after the second question so students can share their observations. When discussing the effects of adding a constant term to \((x-h)^2\), ask students how the vertical movement of the vertex affects the \(x\)-intercepts of the graph. If not mentioned by students, point out that in some cases it produces graphs that are away from the \(x\)-axis and thus have no \(x\)-intercepts.

Encourage students to put their graphing technology out of reach while they work on the third question.

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software. 
Supports accessibility for: Organization; Conceptual processing; Attention

Student Facing

  1. Using graphing technology, graph \(y=x^2\). Then, add different numbers to \(x\) before it is squared (for example, \(y=(x+4)^2\), \(y=(x-3)^2\)) and observe how the graph changes. Record your observations.
  2. Graph \(y=(x-1)^2\). Then, experiment with each of the following changes to the function and see how they affect the graph and the vertex:

    1. Adding different constant terms to \((x-1)^2\) (for example: \((x-1)^2+5\), \((x-1)^2-9\)).
    2. Multiplying \((x-1)^2\) by different coefficients (for example: \(y=3(x-1)^2\), \(y=\text-2(x-1)^2\)).
  3. Without graphing, predict the coordinates of the vertex of the graphs of these quadratic functions, and predict whether the graph opens up or opens down. Ignore the last row until the next question. 
    equations coordinates of vertex graph opens up or down?
    \(y=(x+10)^2\)    
    \(y = (x-4)^2 + 8\)    
    \(y = \text-(x-4)^2 +8\)    
    \(y=x^2 - 7\)    
    \(y= \frac12(x + 3)^2 -5\)    
    \(y= \text-(x+100)^2 + 50\)    
    \(y = a(x+m)^2 + n\)    
  4. Use graphing technology to check your predictions. If they are incorrect, revise them. Then, complete the last row of the table.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. What is the vertex of this graph?
  2. Find a quadratic equation whose graph has the same vertex and adjust it, if needed, so that it has the graph provided.
Parabola. Opens up. Vertex = 3 comma -4. Points on parabola include 0 comma 14, 2 comma -2, and 5 comma 4.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Consider displaying the incomplete table and inviting students to write in one of their correct responses. Then, ask students what they notice about the structure or composition of the expressions. Draw students’ attention to the following:

  • All expressions in the table have a squared term.
  • Most of the squared terms contain a sum or a difference of \(x\) and a number. For the one example where the squared term is not a sum or a difference, we can think of it as having a sum of \(x\) and 0.
  • The squared term may have a coefficient, which can be positive or negative.
  • Most expressions have a constant term. For the one example without a constant term, we can think of the constant term as 0.

To help students consolidate their observations, display the following sentence starters and ask students to complete them based on their work:

  • When a quadratic equation is in vertex form of \(y = a(x-h)^2 + k\), the coordinates of the vertex are \((\quad , \quad)\).
  • When the equation is graphed, the graph opens upward if . . .
  • The graph opens downward if . . .

If not mentioned by students, point out that when a quadratic equation is in vertex form of \(y = a(x-h)^2 + k\), the coordinates of the vertex are \((h,k)\). Also point out that when the equation is graphed, the graph of the equation opens upward if \(a\) is positive and opens downward if \(a\) is negative.

Tell students that, in a future lesson, we will take a closer look at how the parts of a quadratic equation in vertex form, \(y = a(x-h)^2 + k\), produce the behaviors they observed on the graph. 

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

To help students see the connections between the three different forms of quadratic expressions and deepen their understanding of each, consider asking questions such as:

  • “The standard form has a constant term: the \(c\) in \(ax^2+bx+c\). The vertex form also has a constant term: the \(k\) in \(a(x-h)^2 +k\). Are the two constant terms visible on the graph in the same way?” (No. The \(k\) in vertex form gives us the \(y\)-coordinate of the vertex. The \(c\) in standard form gives us the \(y\)-coordinate of the \(y\)-intercept. That said, changing these parameters has the same effect of moving the graph up or down.)
  • “In both the vertex and standard form, the squared term \(x^2\) has a coefficient (which could be 1). Does this coefficient affect the graph similar ways?” (Yes, they both influence the direction of the opening of function’s graph and how wide or narrow the opening is.)

15.4: Cool-down - Visualizing A Graph (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Sometimes the expressions that define quadratic functions are written in vertex form. For example, if the function \(f\) is defined by \((x-3)^2 + 4\), which is in vertex form, we can write \(f(x)=(x-3)^2 + 4\) and draw this graph to represent \(f\).

Parabola. Opens up. Vertex = 3 comma 4.

The vertex form can tell us about the coordinates of the vertex of the graph of a quadratic function. The expression \((x-3)^2\) reveals that the vertex has \(x\)-coordinate 3, and the constant term of 4 reveals its \(y\)-coordinate. Here the vertex represents the minimum value of the function \(f\), and its graph opens upward.

In general, a quadratic function expressed in vertex form is written as: \(\displaystyle y = a(x-h)^2 + k\) The vertex of its graph is at \((h,k)\). The graph of the quadratic function opens upward when the coefficient \(a\) is positive and opens downward when \(a\) is negative.

In future lessons, we will explore further how \(a\), \(h\), and \(k\) affect the graph of a quadratic function.