IM 7.2.13 Lesson: Two Graphs for Each Relationship

Decide whether the equation is true or false. Be prepared to explain your reasoning.

[math]\frac{3}{2}\cdot16=3\cdot8[/math]

[math]\frac{3}{4}\div\frac{1}{2}=\frac{6}{4}\div\frac{1}{4}[/math]

[math]\left(2.8\right)\cdot\left(13\right)=\left(0.7\right)\cdot\left(52\right)[/math]

Explore the graph. Start by dragging the gray bar on the left across the screen until you can see both the table and the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.

What stays the same and what changes in the table?

What stays the same and what changes in the equation?

What stays the same and what changes in the graph?

Choose one row from the table above and write it here.

To what does this row correspond to on the graph?

Do not move the point.

Choose three rows from the table, other than the origin. Record x and y, and compute y/x.

What do you notice? What does this have to do with the equation of the line?

Do not move the point. Check the box to view the coordinates . What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?

Drag the point to a different location.

Record the equation of the line, the coordinates of three points, and the value of [math]\frac{y}{x}[/math].

Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.

The graph of an equation of the form [math]y=kx[/math], where [math]k[/math] is a positive number, is a line through [math]\left(0,0\right)[/math] and the point [math]\left(1,k\right)[/math]. Name at least one line through [math]\left(0,0\right)[/math] that cannot be represented by an equation like this.

If you could draw the graphs of [i]all [/i]of the equations of this form in the same coordinate plane, what would it look like?

Andre and Jada were in a hot dog eating contest. Andre ate 10 hot dogs in 3 minutes. Jada ate 12 hot dogs in 5 minutes.

The points shown on the [i]first [/i]set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.

Draw two lines: one through the origin and Andre’s point, and one through the origin and Jada’s point. Write an equation for Andre's line. Use [math]t[/math] to represent time in minutes, and [math]h[/math] to represent number of hot dogs.

For each equation, what does the constant of proportionality tell you?[br]

The points shown on the second set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.[br]

Draw lines from the origin through each of the two points. Write an equation for each line. What does the constant of proportionality tell you in each case?[br]