Lesson 9

Comparing Graphs

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

9.1: Population Growth

This graph shows the populations of Baltimore and Cleveland in the 20th century. \(B(t)\) is the population of Baltimore in year \(t\). \(C(t)\) is the population of Cleveland in year \(t\).

Two functions. year and population.
  1. Estimate \(B(1930)\) and explain what it means in this situation.
  2. Here are pairs of statements about the two populations. In each pair, which statement is true? Be prepared to explain how you know.

    1. \(B(2000) > C(2000)\) or \(B(2000) < C(2000)\)
    2. \(B(1900) = C(1900)\) or \(B(1900) > C(1900)\)
  3. Were the two cities’ populations ever the same? If so, when?

9.2: Wired or Wireless?

\(H(t)\) is the percentage of homes in the United States that have a landline phone in year \(t\). \(C(t)\) is the percentage of homes with only a cell phone. Here are the graphs of \(H\) and \(C\).

Graph of 2 functions. Blue solid line represents percentage of homes in US with a landline phone in year t. Red dotted line represents the percentage of homes with only a cell phone.
  1. Estimate \(H(2006)\) and \(C(2006)\). Explain what each value tells us about the phones.
  2. What is the approximate solution to \(C(t)=20\)? Explain what the solution means in this situation.
  3. Determine if each equation is true. Be prepared to explain how you know.

    1. \(C(2011) = H(2011)\)
    2. \(C(2015) = H(2015)\)
  4. Between 2004 and 2015, did the percentage of homes with landlines decrease at the same rate at which the percentage of cell-phones-only homes increased? Explain or show your reasoning.


  1. Explain why the statement \(C(t) + H(t) \leq 100\) is true in this situation.
  2. What value does \(C(t) + H(t)\) appear to take between 2004 and 2017? How much does this value vary in that interval?

9.3: Audience of TV Shows

The number of people who watched a TV episode is a function of that show’s episode number. Here are three graphs of three functions—\(A, B\), and \(C\)—representing three different TV shows.

Show A

10 data points on coordinate plane.

Show B

A graph. 

Show C

10 data points on coordinate plane.
  1. Match each description with a graph that could represent the situation described. One of the descriptions has no corresponding graph.

    1. This show has a good core audience. They had a guest star in the fifth episode that brought in some new viewers, but most of them stopped watching after that.
    2. This show is one of the most popular shows, and its audience keeps increasing.
    3. This show has a small audience, but it’s improving, so more people are noticing.
    4. This show started out huge. Even though it feels like it crashed, it still has more viewers than another show.
  2. Which is greatest, \(A(7)\), \(B(7)\), or \(C(7)\)? Explain what the answer tells us about the shows.
  3. Sketch a graph of the viewership of the fourth TV show that did not have a matching graph.

    A blank graph, with origin O. The horizontal axis, episode number, scale from 0 to 10 by 2s. The vertical axis, audience in millions, scale is 6 units.

9.4: Functions $f$ and $g$

  1. Here are graphs that represent two functions, \(f\) and \(g\).

    Decide which function value is greater for each given input. Be prepared to explain your reasoning.

    1. \(f(2)\) or \(g(2)\)
    2. \(f(4)\) or \(g(4)\)
    3. \(f(6)\) or \(g(6)\)
    4. \(f(8)\) or \(g(8)\)
    A graph. 

    ​​​​​

  2. Is there a value of \(x\) at which the equation \(f(x)=g(x)\) is true? Explain your reasoning.
  3. Identify at least two values of \(x\) at which the inequality \(f(x) < g(x)\) is true.

Summary

Graphs are very useful for comparing two or more functions. Here are graphs of functions \(C\) and \(T\), which give the populations (in millions) of California and Texas in year \(x\).

Graph of the populations of two states over time on a coordinate plane. 
What can we tell about the
populations?
How can we tell? How can we convey this
with function notation?

In the early 1900s, California had a smaller population than Texas.

The graph of \(C\) is below the graph of \(T\) when \(x\) is 1900.

\(C(1900) < T(1900)\)

Around 1935, the two states had the same population of about 5 million people.

The graphs intersect at about \((1935, 5)\).

\(C(1935) = 5\) and \(T(1935)=5\), and \(C(1935)=T(1935)\)

After 1935, California has had more people than Texas.

When \(x\) is greater than 1935, the graph of \(C(x)\) is above that of \(T(x)\) .

\(C(x) > T(x)\) for \(x>1935\)

Both populations have increased over time, with no periods of decline.

Both graphs slant upward from left to right.

 

From 1900 to 2010, the population of California has risen faster than that of Texas. California had a greater average rate of change.

If we draw a line to connect the points for 1900 and 2010 on each graph, the line for \(C\) has a greater slope than that for \(T\).

Equation. 

 

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.