Question

1. explain what an independent variable is.
2. explain what a dependent variable is.
3. describe how first differences of number sequences relate to linear equations.
4. describe the graph of a linear equation.
5. telecommunications the costs associated with a particular cellular phone plan are shown below. find the first differences and write an equation to represent the data pattern. (example 1)

hours per month 0 1 2 3 4 5
total cost ($) 25 37 49 61 73 85

1. transportation a hovercraft ferry crossing the english channel averages 47 miles per hour. write an equation, make a table, and draw a graph to show the distance the hovercraft travels for times from 0 to 3 hours. (example 2)
2. contests the prize for a baking contest is $200. if a contestant spends $6 per batch on ingredients to practice a new recipe, write an equation to describe the profit, p, the contestant will make if he wins in terms of the number of practice batches, b, used to perfect the recipe. (example 3)

practice and apply
find the first differences for each data set, and write an equation to represent the data pattern.

1. 0 1 2 3 4 5 6

17 23 29 35 41 47 53

1. 0 1 2 3 4 5 6

25 31 37 43 49 55 61

1. 0 1 2 3 4 5 6

60 180 300 420 540 660 780

1. 0 1 2 3 4 5 6

0 3 6 9 12 15 18

Explanation:

Answer:

1. The independent variable is the variable that is manipulated or changed in an experiment or mathematical relationship. It is the input or cause, and its values are chosen freely.
2. The dependent variable is the variable that depends on the independent variable. It is the output or effect, and its value is determined by the value of the independent variable.
3. In a linear - number sequence, if the first differences are constant, the sequence can be modeled by a linear equation. The constant first - difference is the slope of the linear equation.
4. The graph of a linear equation is a straight line. It can be characterized by its slope (the rate of change) and y - intercept (the value of y when x = 0).
5. First differences: 12, 12, 12, 12, 12; Equation: \(y = 12x+25\)
6. Equation: \(d = 47t\) (where \(d\) is distance and \(t\) is time); Table:

\(t\) (hours)	\(d\) (miles)
1	47
2	94
3	141

1. Equation: \(p=200 - 6b\)
2. First differences: 6, 6, 6, 6, 6, 6; Equation: \(y = 6x+17\)
3. First differences: 6, 6, 6, 6, 6, 6; Equation: \(y = 6x+25\)
4. First differences: 120, 120, 120, 120, 120, 120; Equation: \(y = 120x+60\)
5. First differences: 3, 3, 3, 3, 3, 3; Equation: \(y = 3x\)