This Warm-up highlights the three forms of quadratic expressions that students have seen so far. It reinforces how each form gives different insights into the graph of a quadratic function.
Launch
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.
Activity
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Student Task Statement
Expressions in different forms can be used to define the same function. Here are three ways to define a function, .
standard form
factored form
vertex form
Which form would you use if you want to find the following features of the graph of ? Be prepared to explain your reasoning.
the -intercepts
the vertex
the -intercept
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share their responses and explanations. Encourage students to name specific features of the graph of . If time permits, consider asking students to use all of the information 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 -intercept more readily, the -intercept can always be found by evaluating the function at .
In this activity, students graph functions in vertex form and deepen their understanding of the connections between a quadratic expression in this form and its corresponding graph.
Students have seen the effect of the sign of when graphing a function of the form . Here, students identify the vertex and two additional points on the graph to help them sketch a curve and determine whether the vertex is a minimum or maximum.
There are multiple opportunities for students to look for and make use of structure (MP7) here such as comparing the position of points to determine the type of extremum present, using symmetry to find additional points, and using the idea that squared terms are always non-negative to reason about a graph.
Because the focus here is on graphing a function by identifying points on the graph and by using structure, technology is not an appropriate tool.
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 the strategies that they use to match the representations. Highlight productive and reliable strategies, as well as use of precise language in the discussion.
Lesson Synthesis
To help students consolidate and articulate the ideas from this lesson, ask questions such as:
“Suppose we graph the equation . What is the vertex of the graph?”
“What are some ways to tell if the vertex is the maximum or the minimum point on the graph?” (Some explanations:
The squared term has a negative coefficient, -2. We saw earlier that when that coefficient is negative the graph opens downward, so the vertex is the maximum point.
When , the value is 5. When is greater than 7, is less than 5. When is less than 7, is also less than 5. This means that the points on either side of the vertex have smaller -values, so the vertex is the maximum.
When is 7, is 0. Multiplying it by -2 gives 0. When is other than 7, is a positive number. Multiplying it by -2 gives a negative number, so it is less than 0.)
“How can we verify that 5 is the -coordinate of the vertex?” (Evaluating the function at gives .)
“Does knowing the location of the vertex and whether the graph opens upward or downward enough for sketching the graph to represent ? If not, what else is needed?” (We still don’t know how “wide” or “narrow” the opening of the graph will be, so it helps to know another point on the graph. To find another point, we can choose an -value and find the corresponding -value. We can use that point to quickly find one more point with the same -coordinate because the graph is symmetric across the vertex.)
Student Lesson Summary
Not surprisingly, vertex form is especially helpful for finding the vertex of a graph of a quadratic function. For example, we can tell that the function, , given by has a vertex at .
We also noticed that, when the squared expression has a positive coefficient, the graph opens upward. This means that the vertex, , represents the minimum function value, .
Parabola in x y plane, origin O. X axis 0 to 6, by 1’s. Y axis 0 to 10, by 2s. Parabola opens upward with vertex at 3 comma 1. Left side of parabola crosses the y axis at 10.
But why does function take on its minimum value when is 3?
Here is one way to explain it: When , the squared term equals 0, because . When is any other value besides 3, the squared term is a positive number greater than 0. (Squaring any number results in a positive number.) This means that the output when will always be greater than the output when , so function has a minimum value at .
This table shows some values of the function for some values of . Notice that the output is the least when , and it increases both as increases and as it decreases.
0
1
2
3
4
5
6
10
5
2
1
2
5
10
The squared term sometimes has a negative coefficient, for instance in . The value that makes equal 0 is -4, because . Any other value makes greater than 0. But when is multiplied by a negative number like -2, the resulting expression, , ends up being negative. This means that the output when will always be less than the output when , so function has its maximum value when .
Remember that we can find the -intercept of the graph representing any function that we have seen. The -coordinate of the -intercept is the value of the function when . If is defined by , then the -intercept is because . Its vertex is at . Another point on the graph with the same -coordinate is located the same horizontal distance from the vertex but on the other side.
Parabola in x y plane, origin O. X axis negative 4 to 1, by 1’s. Y axis negative 8 to 2, by 2s. Parabola opens upward with vertex at negative 1 comma negative 5. Other points on the parabola given are negative 2 comma negative 4 and 0 comma negative 4.
to access Student Response.
Launch
Display the two equations defining and . 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 in them and a 10 constant term. This means that the vertex for each graph is at the same location, at .)
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 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 opens upward or downward?" (In an earlier lesson, we saw that when a quadratic function is expressed in the form and the coefficient 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 upward or downward, ask them about it.
Because this activity is designed to be completed without technology, ask students to put away any devices.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first-draft explanation of how Priya might have reasoned about whether the vertex is the minimum or maximum. Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Representation: Internalize Comprehension. Invite students to identify which details are needed to know which way the parabola opens. Display the sentence frame, “The next time I need to know which way a parabola opens, I will look for/pay attention to . . . .“ Supports accessibility for: Language
Activity
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Student Task Statement
Here are two equations that define quadratic functions.
The graph of passes through and , as shown on the coordinate plane.
Find the coordinates of another point on the graph of . Explain or show your reasoning. Then use the points to sketch and label the graph.
On the same coordinate plane, identify the vertex and two other points that are on the graph of . Explain or show your reasoning. Sketch and label the graph of .
Priya says, "Once I know that the vertex is , I can find out, without graphing, whether the vertex is the maximum or the minimum of function . I can just compare the coordinates of the vertex with the coordinates of a point on either side of it."
Complete the table, and then explain how Priya might have reasoned about whether the vertex is the minimum or maximum.
3
4
5
10
Student Response
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Building on Student Thinking
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 -value that is simple to evaluate and would help them sketch the graph.
Activity Synthesis
Invite students to share their graphs and how they went about finding the coordinates of one other point on the graph of and two other points on the graph of . Highlight explanations that make use of the symmetry of the graph to identify an additional point on the graph after one point (aside from the vertex) is known.
Next, select students to explain their analysis of Priya’s reasoning. Make sure students see that the vertex of the graph of cannot be the minimum value of the function (and thus cannot have an upward-opening graph) because there are other values of that are less than 10. If the vertex was the minimum, then no other values of would be less than the value at .
Likewise, once we see that is on the graph of and its -value is less than that of the vertex, we can reason that the vertex represents the maximum of the function and that the graph opens downward.
Consider showing tables, such as these, to clarify the input-output relationship in each function.
Function
0
1
2
3
4
5
6
-6
1
6
9
10
9
6
Function
0
1
2
3
4
5
6
18
12
10
12
If time permits, discuss with students how we can also determine whether the vertex is the maximum or minimum by studying the structure of the squared term in the vertex form. Let’s take function as an example. Ask students:
“In the expression defining , what is the value of when is 4?” (0)
“What is the value of when is less than 4, say, when it is 1? Is it greater or less than 0?” (At , is 9, which is greater than 0.)
“What about when is greater than 4, say, when it is 5?” (At , is 1, which is also greater than 0.)
“If the squared term is 0 when is 4, and greater than 0 when is any other number, does give the minimum or the maximum point on the graph?” (the minimum)
“Does multiplying by change the fact that the minimum is at ? Why or why not?” (No. Multiplying by halves all the values but they will still be positive or greater than 0. 0 times is still 0.)
“What about adding 10 to ?” (No. Adding 10 increases the value of by 10, regardless of the input.)
Launch
Arrange students in groups of 2, and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:
Give students 1 minute to sort the cards into categories of their choosing.
Pause the class after students have sorted the cards.
Select groups to share their categories and how they sorted their cards.
Discuss as many different types of categories as time allows.
Attend to the language that students use to describe their categories, equations, and graphs, giving them opportunities to describe their equations and graphs more precisely. Highlight the use of terms like “opens upward or downward,” “vertex,” and “intercept.” After a brief discussion, invite students to complete the remaining questions.
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed _____ , so I matched . . . .” Encourage students to respectfully challenge each other when they disagree. Advances: Conversing
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with, and introduce the remaining cards after students have completed their initial set of matches. Supports accessibility for: Conceptual Processing, Organization, Memory
Activity
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Student Task Statement
Your teacher will give you a set of cards containing an equation or a graph that represents a quadratic function. Take turns matching each equation to a graph that represents the same function. Record your matches, and be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Some students may be challenged to identify the coordinates of the vertex on the graphs because the - and -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.
Activity Synthesis
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 in the squared term, or the in :
If is being subtracted from , is a positive number, and the -coordinate of the vertex is positive.
If is being added to in the expression, then is a negative number, and the -coordinate of the vertex is negative.
The constant term in the quadratic expression in vertex form, or the in :
If is positive, the -coordinate of the vertex is also positive.
If is negative, the -coordinate of the vertex is also negative.
The coefficient of the squared term, or the in : We saw earlier that when the squared term has a negative coefficient, it makes all positive values negative. This means that:
If is negative, the vertex represents the maximum (the graph of the quadratic function opens downward).
If is positive, the vertex represents the minimum (the graph of the quadratic function opens upward).
The magnitude of :
The larger 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 relating the features of the graphs to the equations.
Standards Alignment
Building On
Addressing
HSF-IF.C.7.a
Graph linear and quadratic functions and show intercepts, maxima, and minima.
