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.

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. 

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

Function

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.)

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.

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.

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.