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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. 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 −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 0, and the rate of change is 12
.
The initial value is 0, and the rate of change is Start Fraction 1 over 2 End Fraction .

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 2.
The initial value is 0, and the rate of change is 2.
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Answer 1

To determine the initial value (y-intercept) and the rate of change (slope) of the linear function represented by the line in the graph:

1. Initial Value (y-intercept): This is the value of y when x = 0. From the points given, we see that the line passes through the point (0, 0). Thus, the initial value is 0.

2. Rate of Change (slope): We can calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] We have at least two points: (0, 0) and (0.5, 1). Using these points:

   • \( (x_1, y_1) = (0, 0) \)
   • \( (x_2, y_2) = (0.5, 1) \)

   Substituting the values: \[ m = \frac{1 - 0}{0.5 - 0} = \frac{1}{0.5} = 2 \]

Thus, 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.