A skateboard ramp, a phone charging, a runner on a track: in each case one quantity changes at a steady pace compared with another. In this chapter you will learn to measure that pace with a single number, the unit rate, and to read it on a graph as the slope of a line.
1. Proportional relationships and unit rate
Two quantities are in a proportional relationship when one is always a fixed multiple of the other. If \(x\) and \(y\) are proportional, the quotient \(\dfrac{y}{x}\) is the same for every pair of values (with \(x \neq 0\)).
The unit rate is the amount of \(y\) for exactly 1 unit of \(x\). In a proportional relationship it is called the constant of proportionality \(k\), and the relationship is written \(y = kx\).
A hiker walks 7.5 miles in 3 hours. The unit rate is \(\dfrac{7.5}{3} = 2.5\) miles per hour, so the distance \(d\) after \(t\) hours is \(d = 2.5t\). After 4 hours she has walked \(2.5 \times 4 = 10\) miles.
The unit rate depends on which quantity goes on top. Miles per hour is \(\dfrac{\text{miles}}{\text{hours}}\); hours per mile is the reciprocal. Always read the question: “per” tells you what goes in the denominator.
2. Graphing proportional relationships
To graph a proportional relationship, make a table of values, plot the ordered pairs on the coordinate plane, and connect them. The result always has two features.
The graph of a proportional relationship is a straight line that passes through the origin \((0, 0)\). If the line is straight but misses the origin, or if it is curved, the relationship is not proportional.
On the graph above, the points \(P(2, 5)\) and \(Q(4, 10)\) both satisfy \(d = 2.5t\). Notice that when the time doubles from 2 to 4 hours, the distance doubles too, from 5 to 10 miles.
The line \(y = 2x + 1\) is straight and increasing, but it crosses the \(y\)-axis at 1, not at 0. When \(x = 1\), \(y = 3\) and the ratio is 3; when \(x = 2\), \(y = 5\) and the ratio is 2.5. The ratios differ, so this is not proportional.
3. Slope as rate of change
Between two points on a line, we compare how far we go up or down (the rise) with how far we go across (the run).
The slope of a line is \(m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{change in } y}{\text{change in } x}\). It is the rate of change of \(y\) with respect to \(x\).
On the line above, going from \(A(1, 2)\) to \(B(5, 5)\) means moving 4 units to the right (run) and 3 units up (rise), so the slope is \(\dfrac{3}{4}\). For a proportional relationship \(y = kx\), the slope equals the unit rate: \(m = k\). In the hiker example, the slope 2.5 means “2.5 miles for each hour.”
A line that goes up from left to right has a positive slope; one that goes down has a negative slope.
4. Slope from a graph
- Choose two points that sit exactly on grid intersections.
- Starting from the left point, count the run to the right.
- Count the rise: up is positive, down is negative.
- Write \(m = \dfrac{\text{rise}}{\text{run}}\) and simplify. Give the unit if the situation has one.
In the hiker graph, from \(P(2, 5)\) to \(Q(4, 10)\) the run is 2 and the rise is 5. The slope is \(\dfrac{5}{2} = 2.5\) miles per hour, matching the unit rate.
On my home planet we say: “run first, then rise.” Walk across the grid before you climb, and you will never mix up the numerator and the denominator!
5. Slope from two points
When you only have coordinates, there is no need to draw anything.
For points \((x_1, y_1)\) and \((x_2, y_2)\) with \(x_1 \neq x_2\):
\[ m = \dfrac{y_2 - y_1}{x_2 - x_1} \]
Subtract the coordinates in the same order on top and bottom. Swapping the two points changes both signs, so the quotient stays the same.
For \(A(-2, 7)\) and \(B(4, -5)\): \(m = \dfrac{-5 - 7}{4 - (-2)} = \dfrac{-12}{6} = -2\). The line falls 2 units for each 1 unit to the right.
Special cases. Two points with the same \(y\) give a horizontal line, rise 0, slope 0. Two points with the same \(x\) give a vertical line: the run is 0 and the slope is undefined, because we cannot divide by zero.
6. Comparing proportional relationships
To compare two relationships, find the unit rate of each, even when one is given as a table, one as a graph, and one as an equation. The larger unit rate is the faster rate, and its line is steeper.
Cyclist A’s graph passes through \((0, 0)\) and \((2, 24)\), so her speed is \(\dfrac{24}{2} = 12\) miles per hour. Cyclist B follows \(d = 10.5t\), so his speed is 10.5 miles per hour. Since \(12 > 10.5\), cyclist A is faster, and her line is steeper.
7. Similar triangles and slope
Why does a line have one slope, no matter which two points you pick? The reason is similar triangles. Draw a right triangle under the line with a horizontal leg (run) and a vertical leg (rise). A bigger triangle drawn the same way has the same angles, so the two triangles are similar, and their corresponding sides are in the same ratio.
All slope triangles drawn on the same line are similar, so \(\dfrac{\text{rise}}{\text{run}}\) has the same value for every pair of points on that line.
In the figure, the small triangle has run 3 and rise 2; the large one has run 9 and rise 6. Both ratios simplify to \(\dfrac{2}{3}\).
A zip line has a small slope triangle with run 5 feet and rise 2 feet. A larger triangle on the same cable has run 35 feet. The triangles are similar, so the rise is \(35 \times \dfrac{2}{5} = 14\) feet.
Key takeaways
- A relationship is proportional when \(\dfrac{y}{x}\) is constant; the constant is the unit rate \(k\), and \(y = kx\).
- Its graph is a straight line through the origin.
- Slope \(= \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}\); for \(y = kx\) the slope is \(k\).
- Up to the right means positive slope, down to the right means negative slope; horizontal is 0, vertical is undefined.
- To compare relationships, compare unit rates; the steeper line has the greater rate.
- Similar triangles explain why the slope is the same between any two points of a line.
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