Lesson 7

The purpose of this activity is to revisit connections made in the previous lesson about which parts of an expression do, and which do not, affect the intercepts of a graph of the function while preparing students for the activities that follow, in which they will consider appropriate graphing windows for specific polynomial functions.

Select students to share their ideas for how the two functions are alike and different. When discussing the last question, it is important students understand that there are many windows that work to see all the intercepts. Since the extra 3 in the second equation changes the vertical position of the points of the graph, however, only the vertical window size needs adjusting. Sometimes when adjusting windows, students change more than they need to and end up with a window that is either too zoomed-in or too zoomed-out on the vertical or horizontal scale (or both) to see important features of the graph.

The goal of this activity is for students to reason about what horizontal intercepts a graph of a polynomial must have based on the structure of the factored form of the polynomial (MP7). Students begin by considering a polynomial equation and graph and reasoning about why the graph window is insufficient since it only shows two out of three intercepts. Students then consider the horizontal intercepts of a different polynomial through the lens of selecting an appropriate viewing window before checking with graphing technology.

Provide access to devices that can run Desmos or other graphing technology. 

If students have trouble deciding what the constant term of \(f(x)\) is, remind them that it can be found quickly by multiplying the constant terms of each linear factor. Students can use the distributive property to understand why this is true.

Select students to share how they selected a viewing window for \(f(x)=(x+1)(x-1)(x-2)(x-28)\). In particular, ask students to state what structure in the factored form they focused on in order to select their window size.

If students question whether we can really be sure we’re seeing all the horizontal intercepts, tell them that in future lessons, they will learn more about what functions are doing to the far left and right of a graph and how we can be sure there are no other horizontal intercepts outside our viewing window. Later in this unit, they will see that all zeros of a function correspond to factors of the function.

This activity asks students to write possible equations for polynomials with specific horizontal intercepts, using their understanding that a factor of \((x-a)\) means that \(a\) is a zero of the function and therefore \((a,0)\) is a horizontal intercept. This is the first time students are asked to write equations this way, and the use of graphing technology to quickly check that the graphs of their equations have the desired features is encouraged.

Monitor for students including constant multipliers or using factors of the form \(ax-b\) where \(\frac{b}{a}\) is the horizontal intercept to share their equations during the discussion.

Provide access to devices that can run Desmos or other graphing technology.

Some students may think that, for example, a zero at \(x=2\) means that \((x+2)\) is a possible factor. Encourage these students to use graphing technology to check their work.

Display for all to see the five sets of horizontal intercepts from the activity. Throughout the discussion, add on equations that meet the criteria to help make clear that there are many equations that work. Select previously identified students to share their equations and reasoning about why their equations have the required horizontal intercepts. If not brought up by a student, ask about different factors that lead to a horizontal intercept of \(\frac12\). Make sure students understand that while both \((x-\frac12)\) and \((2x-1)\) are equal to zero when \(x=\frac12\), they are not identical and lead to different equations.