Comparing Graphs

5 min

Narrative

In this Warm-up, students compare functions by analyzing graphs and statements in function notation. The work here prepares students to make more sophisticated comparisons later in the lesson.

Student Task

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

<p>Two functions. year and population.</p>

  1. Estimate B(1930)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)B(2000) > C(2000) or B(2000)<C(2000)B(2000) < C(2000)
    2. B(1900)=C(1900)B(1900) = C(1900) or B(1900)>C(1900)B(1900) > C(1900)
  3. Were the two cities’ populations ever the same? If so, when?

Sample Response

  1. B(1930)B(1930) is about 800 thousand. It’s the population of Baltimore in 1930.
    1. B(2000)>C(2000)B(2000)>C(2000)
    2. B(1900)>C(1900)B(1900)>C(1900)
  3. Yes. The populations were the same in roughly 1910 and 1943.

Synthesis

Invite students to share their response to the first question. After students give a reasonable estimate of the population of Baltimore (about 800,000), display the statement B(1930)=800,000B(1930) = 800,000 for all to see. Make sure students can interpret it to mean “In 1930, the population of Baltimore was about 800,000 people.”

Next, ask students to explain how they knew which statement in each pair of inequalities is true and how they knew that there were two points in time when Baltimore and Cleveland had the same population.

Ask students how we could use function notation to express that the populations of Baltimore and Cleveland were equal in 1910. If no students mention B(1910)=C(1910)B(1910)=C(1910) or B(t)=C(t)B(t)=C(t) for t=1910t=1910, bring these up and display these statements for all to see.

Standards
Addressing
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • F-IF.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  • HSF-IF.B.4·For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. <span>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.</span>
Building Toward
  • A-REI.11·Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • A-REI.11·Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • A-REI.11·Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • A-REI.11·Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • A-REI.11·Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • HSA-REI.D.11·Explain why the <span class="math">\(x\)</span>-coordinates of the points where the graphs of the equations <span class="math">\(y = f(x)\)</span> and <span class="math">\(y = g(x)\)</span> intersect are the solutions of the equation <span class="math">\(f(x) = g(x)\)</span>; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where <span class="math">\(f(x)\)</span> and/or <span class="math">\(g(x)\)</span> are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

20 min

15 min

10 min