Let’s use tables, equations, and graphs to answer questions about proportional relationships.
Decide whether each equation is true or false. Be prepared to explain your reasoning.
Explore the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.
| $x$ | $y$ | |
|---|---|---|
| row 1 |
Grab and drag the point until you see the equation $y=\frac 32 x$.
| $x$ | $y$ | $\frac{y}{x}$ | |
|---|---|---|---|
| row 1 | |||
| row 2 | |||
| row 3 |
Equation of the line: _______________________________
| $x$ | $y$ | $\frac{y}{x}$ | |
|---|---|---|---|
| row 1 | |||
| row 2 | |||
| row 3 |
The graph of an equation of the form $y = kx$, where $k$ is a positive number, is a line through $(0,0)$ and the point $(1,k)$.
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 first 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 each line. Use $t$ to represent time in minutes, and $h$ to represent number of hot dogs.
What does the constant of proportionality tell you in each case?
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.
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?
Imagine that a faucet is leaking at a constant rate and that every 2 minutes, 10 milliliters of water leaks from the faucet. There is a proportional relationship between the volume of water and elapsed time.
Let’s use $v$ to represent volume in milliliters and $t$ to represent time in minutes. Here are graphs and equations that represent both ways of thinking about this relationship:
Even though the relationship between time and volume is the same, we are making a different choice in each case about which variable to view as the independent variable. The graph on the left has $v$ as the independent variable, and the graph on the right has $t$ as the independent variable.
