8.)The data in the table illustrate a linear function.
x |-3| 0| 3|6|
y |-5|-3|-1|1|
A.) -3/2
B.)-2/3
C.)2/3
D.)3/2
To determine whether the data in the table represents a linear function, we need to check if there is a constant rate of change between the values of x and y.
The rate of change is found by calculating the difference in y-values divided by the difference in x-values.
Let's calculate the rate of change for each set of points:
For the first set of points (-3, -5) and (0, -3):
Rate of change = (-3 - (-5)) / (0 - (-3)) = 2 / 3
For the second set of points (0, -3) and (3, -1):
Rate of change = (-1 - (-3)) / (3 - 0) = 2 / 3
For the third set of points (3, -1) and (6, 1):
Rate of change = (1 - (-1)) / (6 - 3) = 2 / 3
Since the rate of change is the same for all sets of points, we can conclude that the data represents a linear function.
Now, let's find the equation of the line using the formula for a linear function, y = mx + b, where m represents the slope and b represents the y-intercept.
We can choose any two points from the table to calculate the slope. Let's choose the first set of points (-3, -5) and (0, -3):
Slope (m) = (Change in y) / (Change in x) = (-3 - (-5)) / (0 - (-3)) = 2 / 3
Now, let's substitute one of the points into the equation to find the y-intercept. Let's use the point (0, -3):
-3 = (2 / 3) * 0 + b
-3 = 0 + b
b = -3
Therefore, the equation of the line is y = (2/3)x - 3.
Now, let's determine the answer choice that matches the slope (m) of the line:
From the given answer choices:
A.) -3/2
B.) -2/3
C.) 2/3
D.) 3/2
The correct answer is C) 2/3.