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. An upward slanting line is plotted on the coordinate plane. The upward slanting line with arrows at both ends passes through the following closed points: left parenthesis negative 1.5 comma negative 3 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 0.5 comma 1 right parenthesis.

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

(1 point)
Responses

The initial value is 0, and the rate of change is 2.
The initial value is 0, and the rate of change is 2.

The initial value is −6
, and the rate of change is 2.
The initial value is negative 6 , and the rate of change is 2.

The initial value is 2, and the rate of change is 2.
The initial value is 2, and the rate of change is 2.

The initial value is 0, and the rate of change is 12
.

Answer 1

To determine the initial value and the rate of change of the linear function from the provided points, we can proceed as follows:

1. Identify the points: The points given are (−1.5, −3), (0, 0), and (0.5, 1).

2. Find the rate of change (slope):

   • We notice that the line passes through the points (0, 0) and (0.5, 1).
   • The formula for slope (rate of change) is: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
   • Using the points (0, 0) and (0.5, 1): \[ \text{slope} = \frac{1 - 0}{0.5 - 0} = \frac{1}{0.5} = 2 \]

3. Find the initial value (y-intercept):

   • The initial value is the y-coordinate where the line intersects the y-axis, which is indeed represented by the point (0, 0).
   • Therefore, the initial value is 0.

Putting this all together, the initial value is 0, and the rate of change is 2.

The correct response is: The initial value is 0, and the rate of change is 2.