# Taxi!

Alignments to Content Standards:
A-REI.D.10
F-LE.B.5

## Task

Lauren keeps records of the distances she travels in a taxi and what she pays:

Distance, $d$, in miles | Fare, $F$, in dollars |
---|---|

3 | 8.25 |

5 | 12.75 |

11 | 26.25 |

- If you graph the ordered pairs $(d, F)$ from the table, they lie on a line. How can you tell this without graphing them?
- Show that the linear function in part (a) has equation $F=2.25d+1.5$.
- What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides?

## IM Commentary

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

## Solution

- The slope of the line segment connecting (3, 8.25) and (5, 12.75) is $\displaystyle \frac {12.75 - 8.25}{5 - 3} = 2.25$. The slope of the line segment connecting (5, 12.75) and (11, 26.25) is $\displaystyle \frac {26.25 - 12.75}{11 - 5} = 2.25$. Because the two line segments are connected and have the same slope, the three points lie on the same line.
- There is only one possible line in part (a), since two points determine a line. The graph of $F - 2.25 d + 1.5$ is a line, so if we show that each ordered pair satisfies it then we will know that it is the same line as in part (a). \begin{align} (3, 8.25)&: 2.25(3)+1.5 = 8.25\\ (5, 12.75)&: 2.25(5)+1.5 = 12.75\\ (11, 26.25)&: 2.25(11)+1.5 = 26.25\\ \end{align}
- The 2.25 represents the cost per mile for the ride. The 1.5 represents a fixed cost for every ride; it does not depend on the distance traveled.

## Taxi!

Lauren keeps records of the distances she travels in a taxi and what she pays:

Distance, $d$, in miles | Fare, $F$, in dollars |
---|---|

3 | 8.25 |

5 | 12.75 |

11 | 26.25 |

- If you graph the ordered pairs $(d, F)$ from the table, they lie on a line. How can you tell this without graphing them?
- Show that the linear function in part (a) has equation $F=2.25d+1.5$.
- What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides?