This Warm-up refreshes the work in an earlier lesson. It prepares students to deepen their understanding of the factored form and the intercepts of a graph that represents a quadratic function.

As students work, notice how they find the -coordinate of the -intercept. Identify students who do so by evaluating . Ask them to share during a whole-class discussion.

Invite students to share their responses and reasoning. If not mentioned in students’ explanations, emphasize that:

• and must be 0 because the -coordinate of the -intercepts of the graph of any function is 0.
• must be 0 because the -coordinate of the -intercept of the graph of any function is 0.
• and correspond to the 1.6 and 2 in the factored expression. Students may reason that the graph shows that the positive -intercept is farther away from 0 than the negative -intercept, suggesting that is 2 and is -1.6. They may also use their observations from an earlier lesson: that when a factor in the expression shows a negative sign or subtraction, a corresponding intercept takes a positive value, and when a factor shows a positive sign or addition, a corresponding intercept takes a negative value. Either explanation is reasonable at this point.
• is the -intercept, so the value of  can be found by evaluating , which is -3.2, because .

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.

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.