### Student Page 2.6.4 Finding Equations from Graphs and Contexts, Interpreting Slope and y-Intercept

#### 1.

Compare the graphs in Figure 2.6.4.1. Which ones are related to each other? How are they related? (Note: The green line is Graph A, blue is Graph B, black is Graph C, and red is Graph D)

#### 2.

##### (a)

Complete a table for each graph. Use Table 2.6.4.2, Table 2.6.4.3, Table 2.6.4.4, and Table 2.6.4.5.

\(x\) | \(y\) |
---|---|

0 | |

1 | |

2 | |

3 | |

4 | |

5 | |

6 |

\(x\) | \(y\) |
---|---|

0 | |

1 | |

2 | |

3 | |

4 | |

5 | |

6 |

\(x\) | \(y\) |
---|---|

0 | |

1 | |

2 | |

3 | |

4 | |

5 | |

6 |

\(x\) | \(y\) |
---|---|

0 | |

1 | |

2 | |

3 | |

4 | |

5 | |

6 |

##### (b)

Determine a recursive pattern for each table.

##### (c)

Use the recursive rule to find an explicit equation for each graph. Write the recursive pattern and the equation under each table.

#### 3.

##### (a)

What is slope? (See Task 2.4.2.4.e if you don't recall.)

##### (b)

How can you find the slope of a graph from a table?

##### (c)

How can you find the slope of a graph from an equation?

##### (d)

How can you find the slope directly from a graph? Show how the slope arises from the graph in each case.

#### 4.

##### (a)

What is the \(y\)-intercept?

##### (b)

How can you find the \(y\)-intercept from a table?

##### (c)

How can you find the \(y\)-intercept from an equation?

##### (d)

How can you find the \(y\)-intercept from a graph?

#### 5.

Consider the contexts that follow.

- Context A
Raffle tickets are $0.50 each. How much will you pay for \(T\) tickets?

- Context B
You have $6 to buy ride tickets at a fair. Each ticket costs $1. How much money do you have after you purchase \(T\) tickets?

- Context C
It costs $2 to get into a dance. At the dance you can buy glasses of punch for $0.50 each. How much will you spend for the dance and punch after buying \(G\) glasses of punch?

- Context D
The Rowing Team is selling cookies for $1 each. A friend is selling the cookies and allows you to pay later for cookies you buy beyond the $3 you have in your pocket. How much will you pay your friend if you buy \(C\) cookies? (Negative dollars are dollars you will pay your friend after the cookie sale.)

##### (a)

Create a table for each context in Student Page Exercise 2.6.4.5.

##### (b)

Graph the data in each table.

##### (c)

Determine a recursive rule then an explicit equation that fits each context in Student Page Exercise 2.6.4.5.

#### 6.

##### (a)

How can you find slope from a context?

##### (b)

How can you find the \(y\)-intercept from a context?

##### (c)

What does the slope mean in the contexts in Student Page Exercise 2.6.4.5?

##### (d)

What does the \(y\)-intercept mean in the contexts in Student Page Exercise 2.6.4.5?

#### 7.

Match the contexts in Student Page Exercise 2.6.4.6 to the graphs in Student Page Exercise 2.6.4.1. How do you know you are right?

#### 8.

Some students say that you can find the slope from a table by dividing the value of \(y\) by the value of \(x\) for a single data point.

##### (a)

Does this method ever work? If so, for which of the examples above will this method work? Why does the method work?

##### (b)

Try this method for finding slope on Table 2.6.4.3. Does the method work in this case? Why or why not?

##### (c)

For which of the examples above will this method for finding slope fail? Why does it fail?