Lesson 22

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

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. 

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.

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.

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.

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.

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.

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.

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.)

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.

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.

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.

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.