Lesson 16

This warm-up highlights the three forms of quadratic expressions students have seen so far. It reinforces how each form gives different insights into the graph of a quadratic function.

Arrange students in groups of 2. Give students a minute of quiet think time, and then ask them to share their thinking with a partner.

Invite students to share their responses and explanations. Encourage students to name specific features of the graph of \(f\). If time permits, consider asking students to use all of the info they can gather to sketch a graph.

If not already mentioned by students, point out that while the standard form allows us to find the \(y\)-intercept more readily, the \(y\)-intercept can always be found by evaluating the function at \(x=0\).

In this activity, students graph functions in vertex form and deepen their understanding of the connections between a quadratic expression in this form and the corresponding graph.

Students have previously observed that the graph of a function expressed in the form of \(y=a(x-h)^2 + k\) opens up when \(a\) is positive and opens down when \(a\) is negative. This means that they could sketch a rough graph that goes through a particular vertex and determine the direction of the opening. The work here reinforces and builds on that earlier observation. Students identify two additional points that are on the graph and use them to help sketch a curve. They also reason about whether the vertex represents the minimum or maximum of the function by analyzing the relative values of the coordinate pairs on either side of the vertex. 

There are multiple opportunities for students to look for and make use structure (MP7) here.

• Given the coordinates of the vertex and of one other point, they may reason that the graph opens up or down by comparing the \(y\)-values of the two points. For example, if the vertex is \((4,10)\) and one point is \((0,18)\), they may conclude that the vertex must be the minimum of the graph simply because the other known point has a greater \(y\)-value. 
• After identifying the coordinates of one point on the graph, students may use the symmetry of the graph of a quadratic function to quickly find the coordinates of a second point.
• When making sense of Priya’s reasoning in the last question, they might notice that in the expression defining \(p\), the \(\text-(x-4)^2\) portion has a value of 0 at the vertex (when \(x\) is 4), because \(\text-(4-4)^2\) is \(0^2\) or 0. For other values of \(x\), \((x-4)^2\) will always be positive because squaring a number always gives a positive result. But because of the negative coefficient, \(\text-(x-4)^2\) will always be negative or less than 0. This means that for all values of \(x\) other than 4, the value of \(p\) would be less than \(p(4)\). 

As the focus here is on graphing a function by identifying points on the graph and by using structure, technology is not an appropriate tool.

Display the two equations defining \(p\) and \(q\) for all to see. Give students a moment to think about how the equations are alike and how they are different. Invite students to share their comments.

If not mentioned by students, ask: “Where are the vertices of their graphs located?” (All of the equations have an \((x-4)^2\) in them and a 10 constant term. This means that the vertex for each graph is at the same location, at \((4,10)\).)

Remind students that, earlier in the unit, we learned that the vertex of a graph represents the maximum or the minimum value of a function. Ask students: 

• "How can we tell if the vertex of the graph of \(p\) represents the maximum or the minimum?" (If the graph opens upward, the vertex is the minimum. If it opens downward, it is the maximum.)
• "How can we tell if the graph of \(p\) opens upward or downward?" (In an earlier lesson, we saw that when a quadratic function is expressed in the form of \(y=a(x-h)^2+k\) and the coefficient \(a\) is negative, the graph opens downward. This means the vertex represents the maximum of the function. Or we could plot some coordinate pairs to help us visualize the graph.)

If no students mentioned finding additional points on each graph to determine whether the graph opens up or down, ask them about it.

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

Students may be unsure about what input value to choose to find additional points on each graph. Without telling students a specific value to use, encourage them to choose an \(x\)-value that is simple to evaluate and would help them sketch the graph. 

Invite students to share their graphs and how they went about finding the coordinates of one other point on the graph of \(p\) and two other points on the graph of \(q\). Highlight explanations that make use of the symmetry of the graph to identify an additional point on the graph once one point (aside from the vertex) is known.

Then, select students to explain their analysis of Priya’s reasoning. Make sure students see that the vertex of the graph of \(p\) cannot be the minimum value of the function (and thus cannot have an upward-opening graph) because there are other values of \(p\) that are less than 10. If the vertex was the minimum, no other values of \(p\) would be less than at \(p(4)\).  

Likewise, once we see that \((0,\text-6)\) is on the graph of \(p\) and its \(y\)-value is less than that of the vertex, we can reason that the vertex \((4,10)\) represents the maximum of the function and the graph would open downward. 

Consider showing a table such as these to clarify the input-output relationship in each function.

Function \(p\)

Function \(q\)

If time permits, discuss with students how we can also determine whether the vertex is the maximum or minimum by studying the structure of squared term in the vertex form. Let’s take function \(q\) as an example. Ask students:

• “In the expression defining \(q\), what is the value of \((x-4)^2\) when \(x\) is 4?” (0)
• “What is the value of \((x-4)^2\) when \(x\) is less than 4, say, when it is 1? Is it greater or less than 0?” (At \(x=1\), \((x-4)^2\)is 9, which is greater than 0.)
• “What about when \(x\) is greater than 4, say, when it is 5?” (At \(x=5\), \((x-4)^2\) is 1, which is also greater than 0.)
• “If the squared term \((x-4)^2\) is 0 when \(x\) is 4, and greater than 0 when \(x\) is any other number, does \(x=4\) give the minimum or the maximum point on the graph?” (the minimum)
• “Does multiplying \((x-4)^2\) by \(\frac12\) change the fact that the minimum is at \(x=4\)? Why or why not?” (No. Multiplying \((x-4)^2\) by \(\frac12\) halves all the values but they will still be positive or greater than 0. 0 times \(\frac12\) is still 0.)
• “What about adding 10 to \((x-4)^2\)?” (No. Adding 10 increases the value of \((x-4)^2\) by 10, regardless of the input.)

In this activity, students are given a set of equations and graphs of quadratic functions and are asked to match them. To do so, students apply their understanding about the connections between the two representations. To make the matches, they are to rely on the structure of a quadratic expression and how it relates to the graph, so technology is not an appropriate tool here.

As students work, encourage them to refine their descriptions of the equations and graphs using more precise language and mathematical terms (MP6). They also practice explaining their reasoning and critiquing the reasoning of others (MP3) as they take turns matching and supporting their matches.

As students discuss, listen for strategies they use to match the representations. Highlight productive and reliable strategies as well as use of precise language in the discussion.

Here are the equations and graphs for reference and planning:

Arrange students in groups of 2. Give each group a set of cut-up cards. Ask students to take turns matching a card containing an equation with a graph that represents it. The first partner identifies a match and explains why they think it is a match, while the other listens and works to understand. Then, they switch roles. If they disagree, they should discuss their thinking and work to reach an agreement. Ask students to record each match (write the equation and sketch the graph) and their reasoning. If time is limited, students may just record the label of the graph instead of sketching.

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

Some students may be challenged to identify the coordinates of the vertex on the graphs because the \(x\)- and \(y\)-axes use a different scale. Encourage students to write down coordinates of the vertex and the intercepts to help them match the graphs to the equations. Some students may have trouble identifying that one graph is steeper than the others. Provide them with a piece of patty-paper or a transparency and have them trace one graph onto it. They can overlay the traced graph on top of the other graphs to figure out which one is steeper. Students used similar tools to explore transformations in middle school.

Invite students to share aspects of the equations and graphs they found helpful for the matching. If not mentioned in students’ explanations, highlight the following features:

• The number being added or subtracted from \(x\) in the squared term, or the \(h\) in \((x-h)^2\):

  • If \(h\) is being subtracted from \(x\), \(h\) is a positive number and the \(x\)-coordinate of the vertex is positive.
  • If \(h\) is being added to \(x\) in the expression, then \(h\) is a negative number and the \(x\)-coordinate of the vertex is negative.

• The constant term in the quadratic expression in vertex form, or the \(k\) in \(a(x-h)^2+k\):

  • If \(k\) is positive, the \(y\)-coordinate of the vertex is also positive.
  • If \(k\) is negative, the \(y\)-coordinate of the vertex is also negative.

• The coefficient of the squared term, or the \(a\) in \(a(x-h)^2+k\):  We saw earlier that when the squared term has a negative coefficient, it makes all positive values negative. This means that:

  • If \(a\) is negative, the vertex represents the maximum (the graph of the quadratic function opens downward).
  • If \(a\) is positive, the vertex represents the minimum (the graph of the quadratic function opens upward).

• The magnitude of \(a\):

  • The larger \(|a|\) is, the narrower the opening of the graph is (or the “steeper” the graph is).

Students may also reason the other way around: by looking at the graphs first and then relate the features of the graphs to the equations.