Lesson 7

Accurate Representations

  • Let’s represent situations precisely.

7.1: Possible Rectangles

A rectangle has a perimeter of 40 centimeters.

Here are some lengths in centimeters. Determine if each one could be a length of this rectangle. Be prepared to explain your reasoning.

  1. 10
  2. 4.5
  3. 20
  4. 32
  5. -10
  6. 0
  7. \(\frac{21}{4}\)

7.2: Which Graph Represents It the Best?

For each situation, several graphs are given. Which graph represents the situation the best? Be prepared to explain your reasoning.

  1. The fine for an overdue book at the library is $0.25 per day, up to a maximum of $6. \(x\) represents time in days, \(y\) represents the fine in dollars.

    A

    Graph on a coordinate plane. Horizontal axis by twos. Vertical axis by ones. Graph starts at the origin and ends at 32 comma 8.

    B

    Graph of a line on the coordinate plane. Horizontal axis by twos. Vertical axis by ones. Line starts at origin and ends at 24 comma 6.

    C

    Graph of plotted points on coordinate plane.

    D

    Graph of plotted points on coordinate plane. 

  2. A tank that starts with 25 gallons of water drains at a rate of 2 gallons per minute. \(x\) represents time in minutes, and \(y\) represents volume of water in tank in gallons.

    A

    Graph of line on coordinate plane. 

    B

    Graph of line on coordinate plane. 

    C

    Graph of line on coordinate plane.

    D

    Graph of plotted points in a straight line on coordinate plane.

  3. Someone folds a paper in half, then in half again repeatedly. After each fold, the thickness of the folded paper increases. \(x\) represents number of folds, and \(y\) represents thickness in layers.

    A

    Graph of a function, origin O.

    B

    Graph on coordinate plane. Horizontal axis by twos. Vertical axis by fives. Graph starts a little above origin and increases in a curve in the first quadrant.

    C

    Plotted points graphed on coordinate plane.

    D

    Plotted points in a graph on coordinate plane. 

  4. A t-shirt company offers deals on bulk purchases. Shirts cost $5 each if you purchase less than 10, and they cost $4 each if you purchase 10 or more. \(x\) represents number of shirts purchased, and \(y\) represents cost of shirts in dollars.

    A

    Graphs y = 5x and y = 4x on coordinate plane. Horizontal axis by twos. Vertical axis by tens. Both graphs go through origin. 

    B

    Graphs of two lines on coordinate plane.

    C

    Graphs of two lines on coordinate plane. Horizontal axis by twos. Vertical axis by tens. Black line starts at origin and increases to end at 10 comma 50. Blue line starts at 10 comma 40 and increases to end at 22 comma 88.

    D

    Graphs of two sets of plotted points in straight lines on coordinate plane.

7.3: Refining the Representation

  1. At the concession stand, popcorn costs \$2 and bananas cost \$1. Clare spent \$16 on popcorn and bananas for her family. 

    Graph of a line on coordinate plane. 
    1. Explain why each of these points on the graph do not make sense in the situation:
      1. \((\text-2, 20)\)
      2. \((1.5, 13)\)
      3. \((10, \text-4)\)

    2. Sketch a graph that better represents the situation. Explain your reasoning.

      Blank coordinate plane. Horizontal axis, orders of popcorn, by twos. Vertical axis, number of bananas, by fours.

      ​​​​​​

  2. The height in feet of a baseball over time, in seconds, is modeled by the function \(h\) given by the equation \(h(t)=3+60t-16t^2\). A graph of the function is shown. 
    Graph on coordinate plane. Horizontal axis by ones. Vertical axis by twenties. Graph goes through the origin and through the point 3 point 8 comma 0.
    1. Choose two points that are on the graph but do not make sense with the context. Explain your reasoning.

    2. Sketch a graph that better represents the situation.

      Blank coordinate plane. Horizontal axis by ones from negative 1 to 4. Vertical axis by twenties from negative 180 to 60.
    3. What point represents the baseball landing on the ground?

Summary