Features of Graphs

5 min

Narrative

This Warm-up is an opportunity to practice interpreting statements in function notation. It also draws attention to statements that correspond to the intercepts of a graph of a function (for instance, d(0)d(0) and d(m)=0d(m) = 0), preparing students to reason about them in the lesson (particularly in the second activity).

Student Task

Diego is walking home from school at a constant rate. This graph represents function dd, which gives his distance from home, in kilometers, mm minutes since leaving the school. 

<p>Graph of line. Horizontal axis, 0 to 24 by 4’s, minutes since leaving school. Vertical axis, 0 to 3 by point 5’s, kilometers from home. Line starts at about 0 comma 2 point 2 5, and ends at 20 comma 0.</p>

Use the graph to find or estimate:

  1. d(0)d(0)
  2. d(12)d(12)
  3. the solution to d(m)=1d(m)=1
  4. the solution to d(m)=0d(m)=0

Sample Response

  1. d(0)d(0) is approximately 2.25.
  2. d(12)d(12) is approximately 0.9.
  3. mm is approximately 11.
  4. m=20m=20.

Synthesis

Ask students to interpret each statement in function notation before soliciting their response to each question. Make sure students understand, for instance, that d(0)d(0) represents Diego’s distance from school at the time of leaving (or at 0 minutes) and d(m)=0d(m)=0 represents his distance from home being 0 km, mm minutes after leaving school.

If time permits, discuss with students:

  • “Is the relationship between time and Diego’s distance from school a linear function? How can we tell?” (Yes. The graph is a line, which means the function’s value changes at a constant rate.)
  • “Can we tell from the graph how far away Diego’s house is from school? How?” (Yes. From the graph, we can see that at the time he leaves school, he is 2.25 km from home.)
  • “Can we tell from the graph how long it took Diego to get home? How?” (Yes. From the graph, we can see the distance reaching 0 km when the time is 20 minutes.)
  • “Why does the graph slant downward (or have a negative slope)?” (As the input, mm, increases, the output, d(m)d(m), decreases.)
Standards
Building On
  • F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • HSF-IF.A.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

15 min

15 min