In what a point on the graph means you saw that the graph of $y = kx$ passes through $(0, 0)$ and $(1, k)$, where $k$ is the unit rate. Now look at how steep that line is.
Why it works
In $y = kx$, every step of 1 to the right adds $k$ to $y$. With $k = 2$ the line climbs 2 for each step; with $k = 5$ it climbs 5. So a bigger unit rate makes a steeper line. The steepness of a line is called its slope, and for $y = kx$ the slope is exactly the unit rate $k$.
Key idea
The graph of a proportional relationship $y = kx$ is a line through the origin, and its slope is the unit rate $k = \frac{y}{x}$. To compare two proportional relationships, compare their unit rates: the steeper line has the greater rate.
Worked examples
Worked example: From a table
3 pounds of apples cost \$7.50 and 6 pounds cost \$15. What is the unit rate?
- $k = \frac{y}{x} = \frac{7.50}{3} = 2.5$.
- Check with the other pair: $\frac{15}{6} = 2.5$.
Answer: \$2.50 per pound; the graph of $y = 2.5x$ has slope 2.5
Worked example: Compare two forms
Pump A follows $y = 4x$ (gallons after $x$ minutes). Pump B’s graph passes through $(5, 18)$. Which pump is faster?
- Pump A’s unit rate is 4 gallons per minute.
- Pump B’s is $\frac{18}{5} = 3.6$ gallons per minute.
- 4 is greater, so A’s line is steeper.
Answer: Pump A
Worked example: From a graph
A proportional graph passes through $(4, 6)$. Write its equation.
- $k = \frac{6}{4} = \frac{3}{2}$.
Answer: $y = \frac{3}{2}x$
Common mistake
Dividing the wrong way, $\frac{x}{y}$ instead of $\frac{y}{x}$. The unit rate is the amount of $y$ for each 1 of $x$. Check: at $x = 1$, $y$ should equal $k$.