Question

determine whether the relationship each table or graph is linear. if so, explain your reasoning. (example 1)
1
cost of electricity to run personal computer
time (h) | cost (¢)
5 | 15
8 | 24
12 | 36
24 | 72

Explanation:

Step1: Calculate the rate of change (slope) between consecutive points.

For the first two points \((5, 15)\) and \((8, 24)\):
The slope \(m = \frac{y_2 - y_1}{x_2 - x_1}=\frac{24 - 15}{8 - 5}=\frac{9}{3} = 3\).

Step2: Check the slope between the next two points \((8, 24)\) and \((12, 36)\):

\(m=\frac{36 - 24}{12 - 8}=\frac{12}{4}=3\).

Step3: Check the slope between \((12, 36)\) and \((24, 72)\):

\(m=\frac{72 - 36}{24 - 12}=\frac{36}{12}=3\).

Since the rate of change (slope) is constant (\(m = 3\)) for all consecutive pairs of points, the relationship is linear. The equation of the line can be found using the point - slope form \(y - y_1=m(x - x_1)\). Using the point \((5,15)\) and \(m = 3\), we get \(y-15 = 3(x - 5)\), which simplifies to \(y=3x\) (where \(x\) is time in hours and \(y\) is cost in cents).

Answer:

The relationship is linear because the rate of change (slope) between any two points in the table is constant (equal to 3). The equation representing the relationship is \(y = 3x\) (where \(x\) is the time in hours and \(y\) is the cost in cents).