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 4 right parenthesis, left parenthesis 0 comma 3 right parenthesis, left parenthesis 3 comma 2 right parenthesis, and left parenthesis 6 comma 1 right parenthesis.

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.

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

To determine the initial value and the rate of change of the linear function represented by the line in the coordinate plane, we can analyze the points provided.

1. Rate of Change (Slope): The rate of change, or slope (m), can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] We can take two points on the line, for example, \((0, 3)\) and \((3, 2)\):

   • Point 1: \( (0, 3) \)
   • Point 2: \( (3, 2) \)

   Plugging these points into the slope formula: \[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \]

   So, the rate of change is \(-0.333\) (rounded to three decimal places).

2. Initial Value (y-intercept): The initial value of the linear function, which is the y-intercept, can be found directly from the point where the line crosses the y-axis. From the points, we see it crosses the y-axis at \((0, 3)\).

Thus, the results are:

• The initial value is \(3\).
• The rate of change is \(-0.333\).

In conclusion:

• The initial value is 3, and the rate of change is -0.333.