Use the image to answer the question.

Question

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 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: left parenthesis negative 3 comma 2.5 right parenthesis, left parenthesis 0 comma 1 right parenthesis, and left parenthesis 2 comma 0 right parenthesis.

Determine the initial value and the rate of change of the linear function as given in the graph.

(1 point)
The initial value is
, and the rate of change is

Answer 1

To determine the initial value (the y-intercept) and the rate of change (slope) of the linear function represented by the line in the given illustration, we first analyze the closed points given:

• Points:
  • \((-3, 2.5)\)
  • \((0, 1)\)
  • \((2, 0)\)

1. Initial Value: The initial value or y-intercept is the value of \(y\) when \(x = 0\). From the points provided, when \(x = 0\), \(y = 1\). Therefore, the initial value is \(1\).

2. Rate of Change: The rate of change or slope (\(m\)) of the line can be calculated using any two points from the line. We can use the points \((0, 1)\) and \((2, 0)\):

   \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{2 - 0} = \frac{-1}{2} = -0.5 \]

Thus, the initial value and rate of change can be summarized as follows:

The initial value is 1, and the rate of change is -0.5.