The purpose of this activity is for students to make connections between key features of graphs and equations in different forms. Students reason abstractly and quantitatively when making these connections (MP2). This activity focuses on linear equations, while the associated Algebra 1 lesson makes similar connections with quadratic equations. The form and parts of an equation can give us information about the graph of the equation: about the slope, the intercepts, or the position relative to the axes. Some forms allow us to predict features of a graph more easily.

• Slope-intercept form helps us name the slope and -intercept of the graph right away.
• The form can help us name the - and -intercepts. When either variable has a value of 0, we can find the other value fairly quickly.
• When the constant term is 0 (as in the form ), the graph goes through the origin.
• When the equation is a number, the graph is a horizontal line. When the graph is a number, the graph is a vertical line.

Monitor for students who use these different strategies:

• Make a quick sketch by plotting a few points.
• Reason about the - and -intercepts.
• Rewrite the equation to be in a form to easily see key features like the slope and intercepts.

The goal of this discussion is to highlight connections between equations and graphs.
Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “How do the slope and intercepts show up in each method?”
• “Are there any benefits or drawbacks to one representation compared to another?”

In this practice activity, students analyze whether a few different equations could be represented by a graph, propose a new equation that could be represented by other graphs, and then interpret the meaning of the intercepts of a linear equation representing a context. When working on these problems, students look for and make use of the structure of equations and graphs (MP7).

The purpose of this discussion is to make connections between key features of graphs and equations. Here are some questions for discussion:

• “How can you use an equation in the form , like or to find the - and -intercepts of the graph?” (When you substitute 0 into either variable, you can find the other value fairly quickly.)
• “How can you tell whether an equation has a graph that goes through the origin, like A or C?” (When the constant term is 0 (as in the form ), the graph goes through the origin.)
• “When an equation involves only one of the variables, what do you know about its graph?” (When the equation is a number, the graph is a horizontal line. When the graph is a number, the graph is a vertical line.)
• “What must be true about the coordinates of a -intercept?” (Its -coordinate is 0.)
• “What must be true about the coordinates of an -intercept?” (Its -coordinate is 0.)