In this lesson, students begin to analyze graphs of functions and use them to answer questions about a context. Students also look at what happens over intervals of input values and learn that graphs can be viewed as dynamic objects that tell stories.
In the temperature activity, students connect specific features of the graph, such as the highest point, with specific features of the contextual situation, i.e., the highest temperature of the day and when it was attained. In the activity about garbage production, students investigate what happens over ranges of input values. The graph tells us how much garbage was produced at certain times and we can also determine if the amount of garbage was increasing or decreasing over time.
As students learn to interpret graphs in terms of a context and use them to answer questions, they learn an important skill in mathematical modeling (MP4).
- Describe (orally and in writing) a graph of a function as “increasing” or “decreasing” over an interval, and explain (orally) the reasoning.
- Interpret (orally and in writing) a graph of temperature as a function of time, using language such as “input” and “output”.
Let’s interpret graphs of functions.
- I can explain the story told by the graph of a function.
A dependent variable represents the output of a function.
For example, suppose we need to buy 20 pieces of fruit and decide to buy apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of bananas is the dependent variable because it depends on the number of apples.
An independent variable represents the input of a function.
For example, suppose we need to buy 20 pieces of fruit and decide to buy some apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of apples is the independent variable because we can choose any number for it.
A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.
For example, \(r\) is the radius of this circle with center \(O\).