Unit 8, Lesson 7
Using Graphs to Find Average Rate of Change
Integrated Math 1
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The table and graph show a more complete picture of the temperature changes on the same winter day. The function gives the temperature in degrees Fahrenheit, hours since noon.
| 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,
plane, origin
. Horizontal axis scale 0 to 12 by 2’s, labeled “hours since noon”. Vertical axis scale 0 to 30 by 10’s, labeled “temperature in degrees F”. The points have coordinates (0 comma 18), (1 comma 19), (2 comma 20), (3 comma 20), (4 comma 25), (5 comma 23), (6 comma 17), (7 comma 15), (8 comma 11), (9 comma 11), (10 comma 8), (11 comma 6) and (12 comma 7).
Find the average rate of change for the following intervals. Explain or show your reasoning.
The graphs show the populations of California and Texas over time.
Graph of the populations of two states over time on a coordinate plane. Horizontal axis scale 1900 to 2010 by 10’s, labeled “year”. Vertical axis scale 0 to 40 by 10’s, labeled “population in millions”. California has a solid blue graph and Texas is a red dashed graph. California has the points (0 comma 2), (1940 comma 5), (1970 comma 20) and (2010 comma 37). Texas has the points (0 comma 4), (1940 comma 5), (1960 comma 10), (1970 comma 11) and (2010 comma 25).
Here is a graph of one day’s temperature as a function of time.
The temperature was at 9 a.m. and at 2 p.m., an increase of 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.
On average, the temperature between 9 a.m. and 2 p.m. increased per hour.
How quickly was the temperature falling between 2 p.m. and 8 p.m.?
On average, the temperature between 2 p.m. and 8 p.m. dropped by per hour.
In general, we can calculate the average rate of change of a function between input values and by dividing the difference in the outputs by the difference in the inputs.
If the two points on the graph of the function are and , the average rate of change is the slope of the line that connects the two points.
The average rate of change of a function is a ratio that describes how fast one quantity changes with respect to another.
The average rate of change for function between inputs and is the change in the outputs divided by the change in the inputs: . It is the slope of the line that connects and on the graph.