Lesson 3: Slope

New Concept

This lesson introduces slope, the single number that describes a line's steepness. You'll learn how to calculate this rate of change, $m = \frac{\Delta y}{\Delta x}$, and interpret it in real-world contexts like growth rates or costs.

What’s next

Next, you'll tackle interactive examples and practice cards to master calculating and interpreting slope in different scenarios.

Property

The variables $y$ and $x$ are proportional if

Examples

• The cost $C$ for gallons $g$ of gas is proportional. If 5 gallons cost 20 dollars, the constant is $k = \frac{20}{5} = 4$. The equation is $C = 4g$.
• The number of words $w$ you type is proportional to the minutes $m$ you spend typing. If you type 240 words in 4 minutes, the constant is $k = \frac{240}{4} = 60$. The equation is $w = 60m$.
• The length in centimeters $c$ is proportional to the length in inches $i$. Since 1 inch is 2.54 cm, the constant of proportionality is $k = 2.54$. The equation is $c = 2.54i$.

Explanation

Proportional variables have a constant ratio. This means one variable is always a fixed multiple of the other. Think of it like a recipe: doubling the ingredients doubles the serving size. Their graph is a straight line through the origin (0,0).

Property

The unit cost in each case is a ratio that measures the steepness of the graph. It tells us how much to increase the output variable per unit of increase in the input variable, that is, each time the input variable increases by 1. For proportional variables, this rate is the constant of proportionality, $k$. A larger constant of proportionality results in a steeper graph.

Examples

• A car traveling at 60 mph ($d=60t$) has a steeper distance-time graph than a car traveling at 40 mph ($d=40t$) because its rate of change is greater.
• A job that pays 20 dollars per hour ($P=20h$) will have a steeper earnings graph than a job that pays 15 dollars per hour ($P=15h$).
• If faucet A fills a tub at 4 gallons per minute ($V=4t$) and faucet B fills it at 2 gallons per minute ($V=2t$), the graph for faucet A is steeper.

Explanation

The constant of proportionality, or unit rate, determines the graph's steepness. A larger constant means the output value grows faster for each unit of input, resulting in a steeper line. It's a visual measure of the rate of change.

Property

The slope, $m$, of a line is the ratio

Examples

• A taxi ride costs 9 dollars for 2 miles and 15 dollars for 4 miles. The slope is the cost per mile: $m = \frac{15-9}{4-2} = \frac{6}{2} = 3$ dollars per mile.
• A plant is 10 cm tall in week 2 and 18 cm tall in week 6. The slope is its growth rate: $m = \frac{18-10}{6-2} = \frac{8}{4} = 2$ cm per week.
• A pool has 500 gallons of water at 10 am and 300 gallons at 2 pm. The slope is the rate of water loss: $m = \frac{300-500}{4-0} = \frac{-200}{4} = -50$ gallons per hour.

Explanation

Slope measures a line's steepness and direction. It's the 'rise over run' or the change in the vertical value ($\Delta y$) for every one unit of change in the horizontal value ($\Delta x$). It represents the constant rate of change for any straight line.

Property

When a variable decreases in value, we say that its change is negative. So, if the output-values of points on a line decrease as we move from left to right, we say that the slope of the line is negative. The change in the output variable, $\Delta y$, will be a negative number, resulting in a negative slope $m$.

Examples

• A phone's battery is at 90% and drops to 50% after 2 hours. The slope is the rate of battery drain: $m = \frac{50-90}{2-0} = \frac{-40}{2} = -20$ percent per hour.
• A submarine at a depth of 20 meters descends to 80 meters in 3 minutes. The slope is its rate of descent: $m = \frac{80-20}{3-0} = \frac{60}{3} = 20$ meters per minute is incorrect. The slope would be $m = \frac{80-20}{3-0} = \frac{60}{3} = 20$ but depth is often represented as negative, so let's rephrase. It starts at -20m and goes to -80m. Slope: $m = \frac{-80 - (-20)}{3-0} = \frac{-60}{3} = -20$ meters per minute.
• A block of ice weighs 12 pounds. After 4 hours in the sun, it weighs 4 pounds. The slope is its melting rate: $m = \frac{4-12}{4-0} = \frac{-8}{4} = -2$ pounds per hour.

Explanation

A decreasing graph has a negative slope. This means as you move from left to right along the x-axis, the line goes downwards. The output value decreases as the input value increases, representing a rate of decrease.