# Lesson 8

Interpreting and Creating Graphs

### Lesson Narrative

By now, students have had multiple opportunities to interpret graphs of functions and to create them (primarily by plotting known input-output pairs of a function or by using descriptions of the situation). Students have also acquired essential vocabulary to communicate about graphs of functions, and used average rate of change as a way to measure how a function changes.

In this lesson, students apply these insights and skills to interpret or create graphs of functions that are less well defined and that model real-life situations that are more complex. The lesson includes two main activities about flag-raising and two optional activities that use other contexts.

Information about the functions is presented in the form of verbal descriptions, video clips, and images. More ambiguity is involved here than in cases students have previously encountered, so they will need to persevere in sense making and problem solving (MP1). At times, the information given may be inadequate, so students will need to make assumptions and decisions in order to produce graphs that show the desired behaviors or meet certain requirements. Along the way, students engage in important aspects of mathematical modeling (MP4).

### Learning Goals

Teacher Facing

• Given a verbal or visual representation of a situation, sketch a graph and show key features.
• Interpret the average rate of change in a situation.
• Practice interpreting key features of graphs and explaining (orally and in writing) their meaning in terms of a situation.

### Student Facing

Let’s sketch graphs to represent situations.

### Required Preparation

Devices are required for the digital version of the extension in the first main activity, Flag Raising (Part 1).

Prepare access to the video clips needed in the second main activity, Flag Raising (Part 2), and in the second optional activity, The Bouncing Ball.

### Student Facing

• I can explain the average rate of change of a function in terms of a situation.
• I can make sense of important features of a graph and explain what they mean in a situation.
• When given a description or a visual representation of a situation, I can sketch a graph that shows important features of the situation.

### 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.