Lesson 7

Using Graphs to Find Average Rate of Change

Let’s measure how quickly the output of a function changes.

7.1: Temperature Drop

Here are the recorded temperatures at three different times on a winter evening. 

time 4 p.m. 6 p.m. 10 p.m.
temperature \(25^\circ F\) \(17^\circ F\) \(8^\circ F\)
  • Tyler says the temperature dropped faster between 4 p.m. and 6 p.m. 
  • Mai says the temperature dropped faster between 6 p.m. and 10 p.m.

Who do you agree with? Explain your reasoning.

7.2: Drop Some More

The table and graphs show a more complete picture of the temperature changes on the same winter day. The function \(T\) gives the temperature in degrees Fahrenheit, \(h\) hours since noon.

\(h\) \(T(h)\)
0 18
1 19
2 20
3 20
4 25
5 23
6 17
7 15
8 11
9 11
10 8
11 6
12 7
Graph of discrete points, \(x y\) plane, origin \(O\).
  1. Find the average rate of change for the following intervals. Explain or show your reasoning.

    1. between noon and 1 p.m.
    2. between noon and 4 p.m.
    3. between noon and midnight
  2. Remember Mai and Tyler’s disagreement? Use average rate of change to show which time period—4 p.m. to 6 p.m. or 6 p.m. to 10 p.m.—experienced a faster temperature drop.


  1. Over what interval did the temperature decrease the most rapidly?
  2. Over what interval did the temperature increase the most rapidly?

7.3: Populations of Two States

The graphs show the populations of California and Texas over time.

Graph of the populations of two states over time on a coordinate plane. 
    1. Estimate the average rate of change in the population in each state between 1970 and 2010. Show your reasoning.
    2. In this situation, what does each rate of change mean?
  1. Which state’s population grew more quickly between 1900 and 2000? Show your reasoning.

Summary

Here is a graph of one day’s temperature as a function of time. 

Graph. Horizontal axis, 9 o’clock to 8 o’clock, time. Vertical axis, 25 to 50 by 5’s, temperature in degrees Fahrenheit. Graph is linear piecewise. Starts at 0 comma 35.

The temperature was \(35 ^\circ F\) at 9 a.m. and \(45 ^\circ F\) at 2 p.m., an increase of \(10^\circ F\) over those 5 hours.

The increase wasn't constant, however. The temperature rose from 9 a.m. and 10 a.m., stayed steady for an hour, then rose again.

  • On average, how fast was the temperature rising between 9 a.m. and 2 p.m.?

    Let's calculate the average rate of change and measure the temperature change per hour. We do that by finding the difference in the temperature between 9 a.m. and 2 p.m. and dividing it by the number of hours in that interval.

    \(\text{average rate of change}=\dfrac{45-35}{5}=\dfrac{10}{5}=2\)

    On average, the temperature between 9 a.m. and 2 p.m. increased \(2^\circ F\) per hour.

  • How quickly was the temperature falling between 2 p.m. and 8 p.m.?

    \(\text{average rate of change}=\dfrac{30-45}{6}=\dfrac{\text-15}{6}=\text-2.5\)

    On average, the temperature between 2 p.m. and 8 p.m. dropped by \(2.5 ^\circ F\) per hour.

In general, we can calculate the average rate of change of a function \(f\), between input values \(a\) and \(b\), by dividing the difference in the outputs by the difference in the inputs.

\(\text{average rate of change}=\dfrac{f(b)-f(a)}{b-a}\)

If the two points on the graph of the function are \((a, f(a))\) and \((b, f(b))\), the average rate of change is the slope of the line that connects the two points.

Graph of curve. X axis, negative 2 to 4. Y axis, 0 to 3. Curve goes through points a comma f of a and b comma f of b. A dotted line connects the 2 points.

Glossary Entries

  • average rate of change

    The average rate of change of a function \(f\) between inputs \(a\) and \(b\) is the change in the outputs divided by the change in the inputs: \(\frac{f(b)-f(a)}{b-a}\). It is the slope of the line joining \((a,f(a))\) and \((b, f(b))\) on the graph.

  • decreasing (function)

    A function is decreasing if its outputs get smaller as the inputs get larger, resulting in a downward sloping graph as you move from left to right.

    A function can also be decreasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is decreasing for \(x \ge 0\) because the graph slopes downward to the right of the vertical axis.

  • horizontal intercept

    The horizontal intercept of a graph is the point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), the horizontal intercept is also called the \(x\)-intercept. The horizontal intercept of the graph of \(2x + 4y = 12\) is \((6,0)\).

    The term is sometimes used to refer only to the \(x\)-coordinate of the point where the graph crosses the horizontal axis.

  • increasing (function)

    A function is increasing if its outputs get larger as the inputs get larger, resulting in an upward sloping graph as you move from left to right.

    A function can also be increasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is increasing for \(x \le 0\) because the graph slopes upward to the left of the vertical axis.

  • maximum

    A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the graph of the function is the corresponding highest point on the graph.

  • minimum

    A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the graph of the function is the corresponding lowest point on the graph.

  • vertical intercept

    The vertical intercept of a graph is the point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), the vertical intercept is also called the \(y\)-intercept.

    Also, the term is sometimes used to mean just the \(y\)-coordinate of the point where the graph crosses the vertical axis. The vertical intercept of the graph of \(y = 3x - 5\) is \((0,\text-5)\), or just -5.