Invite students to share their observations and predictions.

To help students make sense of the shifts in -intercepts, relate the expressions in standard form to equivalent expressions in factored form. Remind students that the factored form allows us to see the zeros of a quadratic function and the -intercepts of the graph representing the function. Writing the expression as lets us see that the zeros are 0 and 20 (because and both give an output of 0), and the -intercepts are and . Likewise, can be written as , so the -intercepts of  and  make sense.

Discuss with students:

• “What if the squared term has a negative coefficient, as in ? How can we use the equivalent expression in factored form to make sense of the -intercepts?” (Factoring out in gives , so the -intercepts are at and , because and both give an output of 0.)
• “Can we still predict the -coordinate of the vertex of a graph if the coefficient of the squared term is negative? Why or why not?” (Yes. When there are two -intercepts, the vertex is still halfway between the two -intercepts.)
• “What if there isn’t a linear term but there is a constant term? How did you find the -intercepts without graphing? Were you able to write the expression in factored form?” (It was not as easy to tell what the -intercepts were or to write the expressions in factored form. For , there were no -intercepts, but for , there were two.)

Tell students they will continue to make sense of quadratic equations and their graphs in the rest of this unit and in an upcoming unit.

Time permitting, consider asking students to predict how the graph of would change if we add both a linear term and a constant term, for example, adding and then . “How would the the graph of differ from that of ?” Students are likely to say that the graph would have a -intercept of , but the -intercepts are now harder to determine. Tell students that they will learn how to find the zeros and the -intercepts of the graph representing such functions in a later unit.

Select students to share their strategies for writing equations. Highlight explanations such as:

• When the graph opens upward, the  term in standard form has a positive coefficient. When it opens downward, the  term has a negative coefficient.
• When two -intercepts are known, we can write the equations in factored form.
• When the -intercept is not at and the graph is centered on the -axis, the equation in standard form will have the form of (no linear term) and  is the -intercept. (A linear term would have shifted the graph horizontally as well as vertically so it would not be centered on the -axis.)
• When  is the only -intercept, the -intercept, and the vertex, the equation in standard form has only the squared term (). If there were a linear term, the graph would not be centered on the -axis. If there were a constant term, the graph would move up or down and either cross the -axis in two places or not cross it at all (that is, there would either be two -intercepts or none).

Consider discussing Graph F, as it is unlike most graphs that students have seen so far (it has only one -intercept at and one -intercept that is not at the origin). Here a couple of approaches:

• Help students see that the factored form is still useful here. Students may guess that is one of the factors (because multiplying any number by would give an output of 0) but may be unsure about the second factor. Point out that the second factor is also because there is no other -intercept. Expanding into standard form gives , which allows us to see that the -intercept is .
• If we shift this graph down 4 units, it will intercept the -axis at 0 and 4 and its equation would be or (per what we learned earlier). To arrive at the Graph F, we would have to move it back up 4, making the equation .