In the past few lessons, students have reasoned symbolically and formally about quadratic expressions. This activity ties that work back to the quadratic functions that represent situations from earlier in the unit.

Students are given two expressions that represent a familiar quadratic function. They identify these expressions by their form and justify why the two are equivalent, using what they learned about expanding factored expressions. Students also study the graph that represents the function and interpret the features of the graph in context. Students are asked to interpret the meaning of the horizontal and vertical intercepts rather than the - and -intercepts because these variables are not used to define the function.

Some students may not recognize that the quadratic terms are the same if they identify the squared term as rather than . Some may ignore the subtraction sign in , or associate it with the rather than the . Show them a simpler problem expression such as and rewrite it as . Then, ask them to rewrite the expression defining in a similar way.

Make sure that:

• Students see that the expression is in standard form, even though it is written as . Clarify that -16 is (the coefficient of the squared term), 78 is (the coefficient of the linear term), and 10 is (the constant term).
• Students can explain the equivalence of and , using a strategy from earlier lessons (for example, drawing a diagram or applying the distributive property).

Ask students if they notice any connections between the two equations and the features of the graph. Some may notice that the 10 in the standard form of the equation tells us that the vertical intercept is . Others may predict that the subtraction of 5 has something to do with the horizontal intercepts or may not see any connections. Any observation is fine at this point, because students will look closely at equations and graphs, starting in the next activity.

The goal of this activity is to uncover the connections between the - and -intercepts on the graph and the parameters of quadratic expressions in standard and factored form. Earlier in the unit, students learned that the zeros of a quadratic function are -values that produce a -value of 0, and that the zeros of a function are the -coordinates of the -intercepts of the graph. This idea comes into focus in this activity.

Note that in the examples given here, the -coordinate of the -intercept is always equal to the product of the zeros, but this is not the case with all graphs representing quadratic functions. For example, the graph representing  intercepts the -axis at 2 and -2, but the -intercept is . If students make this observation, consider acknowledging that this seems to be true for these graphs and consider prompting students to revisit this observation in upcoming lessons (in which students will also study graphs) to see if it is always true.

At this point, students are just noticing that numbers in the factored expressions have something to do with intercepts on the graph. In a later lesson students will explore why they are related.

Invite students to share their observations on how the numbers in the quadratic expressions relate to the intercepts of the graphs. Then ask them to share their predictions for the - and -intercepts of the graph of function , defined by . Discuss with students:

• “How did you find the -intercept of the graph of function without graphing?" (By looking at the intercepts in other graphs with similar factors. In the examples, expressions of the form have and for the -intercepts.)
• “How did you find the -intercept?” (By writing the expression in the standard form and seeing what the constant term is. By evaluating the expression at .)

Demonstrate graphing using the technology available in your classroom. Point to the intercepts. If using Desmos, show students that you can click on the intercepts to reveal the coordinates of the points.

Remind students that earlier in the unit we learned that the -intercepts of a graph tell us the zeros of the function, or the input values that produce an output of 0. Highlight that, because an expression in factored form can tell us about the -intercept of the graph, this form is also handy for telling us about the zeros of the function that the expression represents.