Lesson 9

Comparing Graphs

Lesson Narrative

In this lesson, students deepen their understanding of functions by comparing representations of several functions relating the same pair of quantities. They analyze two or more graphs simultaneously, interpreting their relative features and their average rates of change in context.

Students also study comparative statements in function notation, such as $$A(x) = B(x)$$ or $$B(10) > A(10)$$, and explain them in terms of changes in population, changes in the trends of phone ownership, and the popularity of different television shows.

Students pay close attention to the intersection of two graphs in this lesson. Earlier in their study, they learned that a solution to a system of linear equations in two variables is a point where the graphs of the equations in the system intersect. Here, students recognize equations of the form $$f(x)=g(x)$$ to mean functions $$f$$ and $$g$$ having the same output value at the same input value. They see that a solution to such an equation is the $$x$$-coordinate of a point where the graphs of $$f$$ and $$g$$ intersect.

Making comparisons involves looking beyond individual pieces of information. To accurately relate the information from multiple representations requires careful and precise use of mathematical language and notation (MP6). Students continue reasoning abstractly and quantitatively (MP2) as they use their analyses of representations of functions to draw conclusions about the quantities in situations.

Learning Goals

Teacher Facing

• Compare key features of graphs of functions and interpret them in context.
• Interpret equations of the form $f(x)=g(x)$ in context and recognize that the solutions to such an equation are the $x$-coordinates of the points where the graphs of $f$ and $g$ intersect.
• Interpret statements about two or more functions written in function notation.

Student Facing

Let’s compare graphs of functions to learn about the situations they represent.

Student Facing

• I can compare the features of graphs of functions and explain what they mean in the situations represented.
• I can make sense of an equation of the form $f(x)=g(x)$ in terms of a situation and a graph, and know how to find the solutions.
• I can make sense of statements about two or more functions when they are written in function notation.

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.