# Lesson 7

Using Graphs to Find Average Rate of Change

### Lesson Narrative

Previously, students have characterized how functions are changing qualitatively, by describing them as increasing, staying constant, or decreasing in value. In earlier units and prior to this course, students have also computed and compared the slopes of line graphs and interpreted them in terms of rates of change. In this lesson, students learn to characterize changes in functions quantitatively, by using average rates of change.

Students learn that average rate of change can be used to measure how fast a function changes over a given interval. This can be done when we know the input-output pairs that mark the interval of interest, or by estimating them from a graph.

Attention to units is important in computing or estimating average rates of change, because units give meaning to how much the output quantity changes relative to the input. In thinking carefully about appropriate units to use, students practice attending to precision (MP6).

Students also engage in aspects of mathematical modeling (MP4) when they use a data set or a graph to compute average rates of change and then use it to analyze a situation or make predictions.

### Learning Goals

Teacher Facing

• Given a graph of a function, estimate or calculate the average rate of change over a specified interval.
• Recognize that the slope of a line joining two points on a graph of a function is the average rate of change.
• Understand that the average rate of change describes how fast the output of a function changes for every unit of change in the input.

### Student Facing

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

### Student Facing

• I understand the meaning of the term “average rate of change.”
• When given a graph of a function, I can estimate or calculate the average rate of change between two points.

Building Towards

### 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.