Question

Use the image to answer the question.

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from -5 to 5 in increments of 1. A downward slanting line is plotted on the coordinate plane. The downward slanting line with arrows on both ends passes through the following closed points: (-3, 4), (0, 3), (3, 2), and (6, 1).

Determine the initial value and the rate of change of the linear function as given in the graph. Round the answer to three decimal places as needed.

Answer 1

To determine the initial value and rate of change of the linear function, we can use the formula for the equation of a line: y = mx + b, where m represents the rate of change and b represents the initial value (or y-intercept).

Looking at the given points (-3, 4), (0, 3), (3, 2), and (6, 1), we can calculate the rate of change (m) by finding the difference in y-coordinates and dividing by the difference in x-coordinates:

m = (change in y) / (change in x)

For the points (-3, 4) and (0, 3):
change in y = 3 - 4 = -1
change in x = 0 - (-3) = 3

m = -1/3

Now we can substitute the values of one of the points into the equation y = mx + b to solve for b:

Using the point (0, 3):
3 = (-1/3)(0) + b
3 = b

So the initial value (b) is 3.

Therefore, the linear function can be written as y = -1/3x + 3.

The initial value is 3 and the rate of change is -1/3.