Some students may think that the numerical values in the equation correspond directly to the -intercepts in the graph and incorrectly state that and . Remind them that a graph shows all pairs of and values that make the equation true. Consider asking these students to try substituting -2 for and evaluating the expression to verify that .

Students will have opportunities to attend to the signs or the operations in quadratic expressions in factored form, so it is not essential that this misconception is corrected at this moment.

In this activity, students continue to make sense of their earlier observations about the connection between the factored form of a quadratic expression and the -intercepts of its graph. They study two very similar expressions:  and . They evaluate each at different values. The expressions are chosen so that students can shift their focus from the numbers to the operations, attending to precision (MP6) along the way.

Students notice that for , the values that produce a 0 output are 0 and -4, and for , those values are 0 and 4. This helps to explain why the signs of the -intercepts are the opposite of the signs in the factors. In an upcoming unit, they will further develop this understanding algebraically, in terms of the zero product property. For now, it suffices that they verify the -intercepts by evaluating the quadratic expression at the predicted values and by checking that the outputs are 0.

Students also observe, by looking for regularity in repeated reasoning, that the horizontal location of the vertex of the graph can be identified once the -intercepts are known: it is exactly halfway between the intercepts (MP8). This suggests that a quadratic expression in factored form can help us see both the -coordinates of the -intercepts and the -coordinate of the vertex of the graph. Some students may notice the horizontal location of the vertex could be determined using the halfway point between any two points with the same -coordinate because the graphs have reflection symmetry across a vertical line through the vertex.

As students work, look for those who use the table to determine the -intercepts, rather than by plotting the points first. Invite those students to share their strategy during the Activity Synthesis.

In this activity, students are generalizing the relationship between the zeros on the graph of a function and the factored form of the function, so technology is not an appropriate tool.

Discuss how students use the completed tables to help them find the -intercepts and the vertex of each graph. If not already mentioned in students’ explanations, highlight that:

• The values that produce 0 for the output are the zeros of the function. They tell us where the graph intersects the -axis. For , those values are 0 and -4, so the -intercepts are and . For , those values are 0 and 4, so the -intercepts are and .
• The vertex tells us the point at which the function reaches its maximum or minimum value. We can see the minimum value in each table. In the first table, is the lowest value. Because the values on one side of  mirror those on the other side, we can reason that -4 is the lowest value of function . In the second table, is the lowest value.
• The vertex is on the line that is halfway between each pair of -intercepts.

If time permits, consider asking students how they think the graphs of the functions given by and will compare.

• “Where will their -intercepts be?” ( and for the graph of function , and and for the graph of function )
• “Where will the -coordinate of the vertex of each graph be?” (2 for the graph of and -2 for that of )
• “How do we find the -coordinate of the vertex?” (Evaluate and .)
• “The -intercept of both graphs is . What is another point on each graph with the same -coordinate?” (The point 2 units to the right of the vertex of will have the same -coordinate, so . The point is on the graph of .)

Clarify that a table of values won’t always show the maximum or minimum values of a function. It also won’t always help us identify the -intercepts of a graph or the zeros of a quadratic function, especially if the zeros are not integers or the given expressions are more complex. Students will learn other ways to find the zeros of any function in another unit later in the course.

In this activity, students apply what they learned in earlier lessons to identify the -intercepts and the vertex of the graphs representing several quadratic functions. For example, they see that the -coordinate of the vertex is always halfway between those of the -intercepts, and the coordinate pairs on one side of the vertex mirror those on the other side. These observations help students locate the vertex of a graph once the -intercepts are known, and ultimately to sketch a graph of a quadratic function using at least three identifiable points. 

The given expressions here are in factored form, but some are unlike what students have previously seen, so students will need to transfer and generalize the reasoning strategies from earlier work (MP8). They also use graphing technology to graph the functions and to check their predictions.

In the digital version of the activity, students use graphing technology to graph the given functions. The graphing technology allows students to find specific points more easily. This activity works best when each student has access to graphing technology because it allows students to focus on the meaning of the points rather than on performing calculations. If students don’t have individual access, projecting the graph would be helpful during the Activity Synthesis.

Focus the discussion on how students determined the -intercepts and the -coordinate of the vertex of a graph, and how the coordinates of these points could help them sketch the graph. Ask questions such as:

• “For , how did you find the -intercepts without graphing?” (The suggests that one -intercept is , and evaluating does give an output of 0. The factor suggests that the second -intercept is , because is 0, and multiplying 0 by any number gives 0.)
• “How did you find the -value of the vertex?” (By finding the halfway point between 0 and 3, which is 1.5.)
• “How would you find the -coordinate of the vertex?” (By evaluating , which gives -4.5.)
• “The expression that defines function has the factor , where the constant term appears first and is subtracted from it. Did this affect how you determined the -intercepts? How so?” (It did not really affect how I found the -intercepts. I still wanted to know when that term is zero, which happens when .)
• “How did you sketch the graph representing ?” (By finding the -intercepts and the vertex. The intercepts are and , so the -coordinate of the vertex is -2, and the -coordinate is  which is -81. Those points are enough to sketch a graph.)