4.7: Rates of Change
What if when you bought a koi fish it measured 4 inches long? Three months later you measure it again and it is 6 inches long? At what rate is the fish growing? After completing this Concept, you'll be able to find the rate of change taking place in problems like this one.
Watch This
CK12 Foundation: 0407S Rate of Change (H264)
Guidance
The slope of a function that describes real, measurable quantities is often called a rate of change. In that case the slope refers to a change in one quantity \begin{align*}(y)\end{align*}
Example A
A candle has a starting length of 10 inches. 30 minutes after lighting it, the length is 7 inches. Determine the rate of change in length of the candle as it burns. Determine how long the candle takes to completely burn to nothing.
Solution
First we’ll graph the function to visualize what is happening. We have 2 points to start with: we know that at the moment the candle is lit (\begin{align*}time = 0\end{align*}
The rate of change of the candle’s length is simply the slope of the line. Since we have our 2 points \begin{align*}(x_1, y_1) = (0, 10)\end{align*}
\begin{align*}\text{Rate of change}& = \frac{y_2  y_1}{x_2 x_1}\\ &= \frac{(7 \ \text{inches})(10 \ \text{inches})}{(30 \ \text{minutes})(0 \ \text{minutes})}\\ &= \frac{3 \ \text{inches}}{30 \ \text{minutes}}\\ &= 0.1 \ \text{inches per minute}\end{align*}
Note that the slope is negative. A negative rate of change means that the quantity is decreasing with time—just as we would expect the length of a burning candle to do.
To find the point when the candle reaches zero length, we can simply read the \begin{align*}x\end{align*}
\begin{align*}\text{Length burned} & = \text{rate} \times \text{time}\\
10 & = 0.1 \times 100\end{align*}
Since the candle length was originally 10 inches, our equation confirms that 100 minutes is the time taken.
Example B
The population of fish in a certain lake increased from 370 to 420 over the months of March and April. At what rate is the population increasing?
Solution
Here we don’t have two points from which we can get \begin{align*}x\end{align*}
The change in \begin{align*}y\end{align*}
Interpret a Graph to Compare Rates of Change
Example C
The graph below represents a trip made by a large delivery truck on a particular day. During the day the truck made two deliveries, one taking an hour and the other taking two hours. Identify what is happening in the first 3 stages of the trip (stages A through C).
Solution:
The first 3 stages of the trip are:
A. The truck sets off and travels 80 miles in 2 hours.
B. The truck covers no distance for 2 hours.
C. The truck covers \begin{align*}(120  80) = 40 \ \text{miles}\end{align*}
A. \begin{align*}\text{Rate of change} = \frac{\Delta y}{\Delta x}=\frac{80 \ \text{miles}}{2 \ \text{hours}}=40\ \text{miles per hour}\end{align*}
Notice that the rate of change is a speed—or rather, a velocity. (The difference between the two is that velocity has a direction, and speed does not. In other words, velocity can be either positive or negative, with negative velocity representing travel in the opposite direction. You’ll see the difference more clearly in part E.)
Since velocity equals distance divided by time, the slope (or rate of change) of a distancetime graph is always a velocity.
So during the first part of the trip, the truck travels at a constant speed of 40 mph for 2 hours, covering a distance of 80 miles.
B. The slope here is 0, so the rate of change is 0 mph. The truck is stationary for one hour. This is the first delivery stop.
C. \begin{align*}\text{Rate of change} = \frac{\Delta y}{\Delta x}=\frac{(120  80) \ \text{miles}}{(43) \ \text{hours}} = 40 \ \text{miles per hour}.\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Rate of Change
Vocabulary

Slope is a measure of change in the vertical direction for each step in the horizontal direction. Slope is often represented as “\begin{align*}m\end{align*}
m ”.  Slope can be expressed as \begin{align*}\frac{\text{rise}}{\text{run}}\end{align*}
riserun , or \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}ΔyΔx .  The slope between two points \begin{align*}(x_1, y_1)\end{align*}
(x1,y1) and \begin{align*}(x_2, y_2)\end{align*}(x2,y2) is equal to \begin{align*}\frac{y_2  y_1}{x_2 x_1}\end{align*}y2−y1x2−x1 . 
Horizontal lines (where \begin{align*}y = a\end{align*}
y=a constant) all have a slope of 0. 
Vertical lines (where \begin{align*}x = a\end{align*}
x=a constant) all have an infinite (or undefined) slope.  The slope (or rate of change) of a distancetime graph is a velocity.
Guided Practice
Continue where we left of in Example C, by identifying what is happening in the last two stages of the trip (stages D and E).
Solution:
The last two stages of the trip are:
D. The truck covers no distance for 1 hour.
E. The truck covers 120 miles in 2 hours.
Let's find the slopes:
D. Here the slope is 0, so the rate of change is 0 mph. The truck is stationary for two hours. This is the second delivery stop. At this point the truck is 120 miles from the start position.
E.
\begin{align*}\text{Rate of change} &= \frac{\Delta y}{\Delta x}\\ &=\frac{(0120) \ \text{miles}}{(86) \ \text{hours}}\\ &=\frac{120 \ \text{miles}}{2 \ \text{hours}}\\ &=60 \ \text{miles per hour.}\end{align*}
The truck is traveling at negative 60 mph.
Wait – a negative speed? Does that mean that the truck is reversing? Well, probably not. It’s actually the velocity and not the speed that is negative, and a negative velocity simply means that the distance from the starting position is decreasing with time. The truck is driving in the opposite direction – back to where it started from. Since it no longer has 2 heavy loads, it travels faster (60 mph instead of 40 mph), covering the 120 mile return trip in 2 hours. Its speed is 60 mph, and its velocity is 60 mph, because it is traveling in the opposite direction from when it started out.
Practice
For 16, the graph below is a distancetime graph for Mark’s three and a half mile cycle ride to school. During this ride, he rode on cycle paths but the terrain was hilly. He rode slower up hills and faster down them. He stopped once at a traffic light and at one point he stopped to mend a punctured tire. The graph shows his distance from home at any given time. Identify each section of the graph accordingly.
 Section A.
 Section B.
 Section C.
 Section D.
 Section E.
 Section F.
For 712, approximate the slope of each part of Mark's ride.
 Section A.
 Section B.
 Section C.
 Section D.
 Section E.
 Section F.
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Horizontally
Horizontally means written across in rows.Slope
Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the over the change in the .” The symbol for slope isVertically
Vertically means written up and down in columns.Image Attributions
Here you'll learn how to find the rate of change of a function. You'll also compare rates of change on a graph to interpret what is happening.