Lesson 22

Rewriting Quadratic Expressions in Vertex Form

22.1: Three Expressions, One Function (5 minutes)

Warm-up

This warm-up reminds students about features of the graph that are visible in the different forms of expressions defining a quadratic function.

Student Facing

These expressions each define the same function.

\(x^2 + 6x +8 \qquad (x+2)(x+4) \qquad (x+3)^2-1\)

Without graphing or doing any calculations, determine where the following features would be on a graph that represents the function.

  1. the vertex
  2. the \(x\)-intercepts
  3. the \(y\)-intercept

Student Response

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

Activity Synthesis

Invite students to share how they would locate the specified features on a graph. Make sure students are reminded that:

  • The constant term in the standard form tells us the \(y\)-intercept.
  • The factored form shows us the \(x\)-intercepts.
  • The vertex form reveals the vertex.

Consider using graphing technology to demonstrate that the three expressions appear to produce the same graph. (We can verify algebraically that the three expressions define the same function, but we can’t be sure that the three expressions define the same function just by looking at the graph.) Label the vertex, \(x\)-intercepts, and \(y\)-intercept. 

22.2: Back and Forth (15 minutes)

Activity

The goal of this activity is for students to recognize rewriting a quadratic expression from standard form to vertex form essentially entails completing the square.

Students had quite a bit of experience completing the square, mostly in the context of solving equations, in which they know to add or subtract the same number from both sides of an equation to keep the equation true. Here students are dealing with expressions and need to be careful to keep each expression equivalent to the original. If they add a number to the expression, they need to remember to subtract the same number to keep the value of the expression unchanged.

Launch

Arrange students in groups of 2. Give students quiet time to think about the first two questions and then time to share their thinking with a partner.

Pause for a class discussion after the second question and invite one or more students to demonstrate their strategy. Highlight approaches that involve completing the square. If no students took that approach, display and discuss the following:

\(\displaystyle \begin {align} &x^2 - 6x + 2\\ &x^2 - 6x + 9 - 9 + 2 \\&(x - 3)^2 - 7 \end {align}\)

Then, ask students to proceed with the rest of the activity. Provide access to devices that can run Desmos or other graphing technology.

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their reasoning about how to convert standard form back into vertex form, scribe their thinking on a visible display. Use annotations and labels to show the structure of the steps using the form included in the launch. For example, labeling the first step as “standard form,” labeling +9 as “add nine to complete the square,” and drawing an arrow to the -9 and labeling it as “keeping it equivalent.” Continue to add in any key phrases students contributed that illustrate the steps.
Supports accessibility for: Visual-spatial processing; Conceptual processing

Student Facing

  1. Here are two expressions in vertex form. Rewrite each expression in standard form. Show your reasoning.

    1. \((x+5)^2+1\)
    2. \((x-3)^2-7\)
  2. Think about the steps you took, and about reversing them. Try converting one or both of the expressions in standard form back into vertex form. Explain how you go about converting the expressions.
  3. Test your strategy by rewriting \(x^2+10x+9\) in vertex form.
  4. Let’s check the expression you rewrote in vertex form.

    1. Use graphing technology to graph both \(x^2+10x+9\) and your new expression. Does it appear that they define the same function?
    2. If you convert your expression in vertex form back into standard form, do you get \(x^2+10x+9\)?

Student Response

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

Anticipated Misconceptions

Some students may wonder why they need to both add and subtract a number from the standard form expression in order to complete the square. In previous lessons, students completed the square while solving an equation, often adding the same number to each side of the equal sign to maintain equality. Here, there is no equal sign. Emphasize that each move must generate an equivalent expression. Show a few expressions such as 5, \(5+0\), \(5-2+2\), \(5+7-7\), \(5+8\), and \(5+8-3\). Ask them which ones are equal to 5. Next, ask them to write 2 more expressions that include the number 5 and also equal 5. Encourage students to notice that the sum of the numbers excluding 5 must be 0.

Activity Synthesis

Invite students to share their graph and response to the last question. Emphasize that while graphing is a quick way to check whether two expressions define the same function, it is not always reliable. If we make an algebraic error, the graph would almost certainly show it. But having two graphs that appear to be identical does not prove that two expressions are indeed equivalent. We can only be sure that two expressions define the same function by showing equivalence algebraically. For example, when students convert the expression in vertex form back to standard form, does it produce the original expression?

Make sure students understand that whatever operation is performed on an expression to complete the square, it should not change the value of the expression. Adding opposites (for example, 9 and -9, or -25 and 25), or adding and subtracting the same number, has the effect of adding 0, which keeps the original and the transformed expressions equivalent.

22.3: Inconvenient Coefficients (15 minutes)

Activity

In the first activity, students rewrote quadratic expressions whose squared term has a coefficient of 1 into vertex form. They did so by completing the square. In this activity, they transform expressions whose squared term has a coefficient other than 1 into vertex form.

Students learn that one way to deal with an inconvenient \(a\) in \(ax^2+bx+c\) is to rewrite the expression as a product of \(a\) and an expression (which now has a leading coefficient of 1), complete the square for the latter, and redistribute \(a\) afterward to obtain an equivalent expression in vertex form.

Launch

Keep students in groups of 2. Display these expressions for all to see and explain that these are some other expressions to rewrite in vertex form so that we could identify the vertex of their graph.

\(\displaystyle 3x^2 + 12x + 9 \\ \text-2x^2 -4x +6 \\ 4x^2 +24x + 20 \\ \text-x^2 +20x\)

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. 

Students may notice:

  • In all of the expressions, the coefficient of the squared term is not 1.
  • Each of the expressions contains terms that have a common factor.
  • The coefficients of some squared terms are negative.
  • The last expression has no constant term.

Students may wonder:

  • How to write these expressions in vertex form.
  • Whether it is still possible to write the last expression in vertex form given that it has no constant term.

Invite a few students to share what they noticed and wondered, and then to begin the activity.

Pause for a class discussion after the first question. Display the worked example. Ask students to share their explanation for each step and record their explanation for all to see. Make sure students understand the rationale for each step and how to check that their expression is equivalent to the original (by converting it back into standard form).

Students have learned in an earlier unit that a positive \(a\) in \(a(x-h)^2 +k\) means an upward-opening graph and may offer this explanation for the last part of the question. It is not necessary to focus on the direction of the opening of the parabola here, as students will explore it further in an upcoming lesson.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Allow students to discuss the steps and then provide students with printed slips of explanations from the student responses. Divide the slips between partners and encourage them to work together to identify which slip matches with each step. Allow them to keep the steps nearby and put them next to their steps as they check their work and use them as a guide.
Supports accessibility for: Organization; Attention

Student Facing

    1. Here is one way to rewrite \(3x^2 + 12x + 9\) in vertex form. Study the steps and write a brief explanation of what is happening at each step.

      \(\displaystyle \begin {align} 3x^2 + 12x + 9 &\qquad& \text{Original expression}\\\\ 3(x^2 + 4x + 3)\\\\ 3(x^2 + 4x + 3 + 1 - 1)\\\\ 3(x^2+4x+4 -1)\\\\ 3\left((x+2)^2 -1\right) \\\\3(x+2)^2 - 3 \end {align}\)

    2. What is the vertex of the graph that represents this expression?
    3. Does the graph open upward or downward? Explain how you know.
  1. Rewrite each expression in vertex form. Show your reasoning.

    1. \(\text-2x^2 -4x +6\)
    2. \(4x^2 +24x + 20\)
    3. \(\text-x^2 +20x\)

Student Response

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

Student Facing

Are you ready for more?

  1. Write \(f(x) = 2(x-3)(x-9)\) in vertex form without completing the square. (Hint: Think about finding the zeros of the function.) Explain your reasoning.
  2. Write \(g(x)=2(x-3)(x-9)+21\) in vertex form without completing the square. Explain your reasoning.

Student Response

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

Activity Synthesis

Select students to display their responses for all to see. Discuss questions such as:

  • “How did you know what factor to use to rewrite the original expression?” (Find a factor that all terms in an expression have in common. Use the coefficient of the squared term so that, after the expression is rewritten, the squared term has a coefficient of 1.)
  • “In \(4x^2 +24x + 20\), both 2 and 4 are common factors. Is one number better than the other for our purposes here?” (4 is a better factor to use because it allows the squared term to have a coefficient of 1, which makes it easier to complete the square. If we use 2, the squared term still has an inconvenient coefficient that is neither 1 nor a perfect square.)
  • “How can we check if the expression in vertex form is equivalent to the original expression?” (One way is to convert it back into standard form and see if it is the same expression as the original.)

22.4: Info Gap: Features of Functions (20 minutes)

Optional activity

This optional Info Gap activity gives students an opportunity to determine and request the information needed to write expressions that define quadratic functions with certain graphical features. To do so, students need to consider what they learned about the structure of quadratic expressions in various forms (MP7).

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Because students are expected to make use of structure and construct logical arguments about how the structure helps them write expressions, technology is not an appropriate tool.

4 info gap cards; features of functions.
 

Here is the text of the cards for reference and planning. Note that the questions on card 1 have many possible correct answers, but no possible expressions for \(f\) and \(g\) can define the same function.

Launch

Tell students they will continue to write expressions in different forms that define quadratic functions. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Since this activity was designed to be completed without technology, ask students to put away any devices.

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving features of quadratic functions. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation 
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

  1. Silently read the information on your card.
  2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
  3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
  4. Read the problem card, and solve the problem independently.
  5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to answer the question.
  2. Ask your partner for the specific information that you need.
  3. Explain to your partner how you are using the information to solve the problem.
  4. When you have enough information, share the problem card with your partner, and solve the problem independently.
  5. Read the data card, and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Student Response

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

Activity Synthesis

After students have completed their work, discuss the correct answers to the questions and any difficulties that came up.

Highlight for students that different forms of quadratic expressions are useful in different ways, so it helps to be able to move flexibly across forms. For example, a quadratic expression in factored form makes it straightforward to tell the zeros of the function that the expression defines and the \(x\)-intercepts of its graph. The vertex form makes it easy to identify the coordinates of the vertex of a graph of function.

To know whether two expressions define the same function, we can rewrite the expression in an equivalent form. There are many tools at our disposal. For instance, we can rewrite an expression into factored form, apply the distributive property to expand a factored expression, rearrange parts of an expression, combine like terms, or complete the square.

Lesson Synthesis

Lesson Synthesis

Display the equation \(f(x)=\text-3x^2+24x-36\) for all to see. Ask students to think about features of a graph of function\(f\), as many as they can, without creating a graph. To scaffold this work, consider displaying the list of features of a graph.

  • \(y\)-intercept
  • \(x\)-intercepts
  • coordinates of vertex

Solicit responses from students on how they found each feature of a graph of \(f\).

Highlight that the \(x\)-intercepts and the coordinates of the vertex aren’t easy to spot in the standard form of the expression as given, but we have techniques to rewrite the expression in other forms.

  • To rewrite in factored form, we can use the distributive property twice:

    \(\text-3x^2+24x-36\\ \text-3(x^2-8x+12)\\ \text-3(x-2)(x-6)\)

  • Once the expression is in factored form, we can tell that the \(x\)-intercepts of a graph of \(f\) are at \((2,0)\) and \((6,0)\).
  • To rewrite in vertex from, we can complete the square:

    \(\text-3(x^2-8x+12)\\ \text-3(x^2-8x+12+4-4)\\ \text-3(x^2-8x+16-4)\\ \text-3((x-4)^2-4)\\ \text-3(x-4)^2+12\)

  • Once the expression is in vertex form, we can tell that the vertex of a graph of \(f\) is at \((4,12).\)

Time permitting, demonstrate that with all of this information, we can now sketch a reasonable graph of \(f\) by hand.

22.5: Cool-down - Rewrite This Expression (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Remember that a quadratic function can be defined by equivalent expressions in different forms, which enable us to see different features of its graph. For example, these expressions define the same function:

\(\begin {align} (x-3)(x-7) \qquad &\text {factored form}\\ x^2-10x+21 \qquad &\text {standard form}\\ (x-5)^2-4 \qquad &\text {vertex form}\end {align}\)

  • From factored form, we can tell that the \(x\)-intercepts are \((3,0)\) and \((7,0)\).
  • From standard form, we can tell that the \(y\)-intercept is \((0,21)\).
  • From vertex form, we can tell that the vertex is \((5,\text-4)\).
Graph of a quadratic function on a grid.

Recall that a function expressed in vertex form is written as: \(a(x-h)^2 + k\). The values of \(h\) and \(k\) reveal the vertex of the graph: \((h, k)\) are the coordinates of the vertex. In this example, \(a\) is 1, \(h\) is 5, and \(k\) is -4.

  • If we have an expression in vertex form, we can rewrite it in standard form by using the distributive property and combining like terms.

    Let’s say we want to rewrite \((x-1)^2-4\) in standard form.

    \(\displaystyle \begin {align} &(x-1)^2-4 \\ &(x-1)(x-1)-4 \\& x^2-2x+1-4 \\&x^2-2x-3 \end {align}\)

  • If we have an expression in standard form, we can rewrite it in vertex form by completing the square.

    Let’s rewrite \(x^2 + 10x +24\) in vertex form.

    A perfect square would be \(x^2 + 10x +25\), so we need to add 1. Adding 1, however, would change the expression. To keep the new expression equivalent to the original one, we will need to both add 1 and subtract 1.

    \(\displaystyle \begin {align} &x^2 + 10x +24 \\& x^2 + 10x +24 + 1 - 1 \\ &x^2 + 10x + 25 - 1 \\ &(x+5)^2-1 \end {align}\)

  • Let’s rewrite another expression in vertex form: \(\text-2x^2 + 12x - 30\).

    To make it easier to complete the square, we can use the distributive property to rewrite the expression with -2 as a factor, which gives \(\text-2 (x^2 -6x + 15)\).

    For the expression in the parentheses to be a perfect square, we need \(x^2 -6x + 9\). We have 15 in the expression, so we can subtract 6 from it to get 9, and then add 6 again to keep the value of the expression unchanged. Then, we can rewrite \(x^2-6x+9\) in factored form.

    \(\displaystyle \begin {align} &\text-2x^2 + 12x -30 \\&\text-2(x^2 -6x +15) \\& \text-2(x^2 -6x +15 -6 +6) \\ &\text-2(x^2 -6x +9 +6) \\ &\text-2\left((x-3)^2 + 6\right) \end {align}\)

    This expression is not yet in vertex form, however. To finish up, we need to apply the distributive property again so that the expression is of the form \(a(x-h)^2 + k\):

    \(\displaystyle \begin {align} &\text-2\left((x-3)^2 + 6\right) \\&\text-2(x-3)^2 -12 \end {align}\)

    When written in this form, we can see that the vertex of the graph representing \(\text-2(x-3)^2 -12\) is \((3, \text-12)\).