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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 find the initial value and the rate of change of the linear function represented by the downward-slanting line on the coordinate plane, we can analyze the given points:

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

1. Rate of Change: The rate of change (slope) can be calculated using two points on the line. Let's use the points \( (0, 1) \) and \( (2, 0) \).

   The formula for the slope \( m \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates: \[ m = \frac{0 - 1}{2 - 0} = \frac{-1}{2} = -0.5 \]

2. Initial Value: The initial value (y-intercept) is the y-coordinate where the line crosses the y-axis. From the point \( (0, 1) \), we see that the y-intercept is: \[ 1 \]

Therefore, we can summarize the results as follows:

• The initial value is \( 1 \)
• The rate of change is \( -0.5 \)

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