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n.

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

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

3 answers

To determine the initial value and the rate of change of the linear function represented by the given line on the coordinate plane, we can follow these steps:

  1. Identify the points on the line. The points provided are:

    • (-1.5, -3)
    • (0, 0)
    • (0.5, 1)

  2. Determine the slope (rate of change). The slope \( m \) can be calculated using any two points on the line. We will use the points (0, 0) and (0.5, 1):

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

    Thus, the rate of change (slope) is 2.

  3. Identify the initial value (y-intercept). The initial value (or y-intercept) is the value of \( y \) when \( x = 0 \). From the point (0, 0), we see that the initial value is 0.

Now combining both findings:

  • The initial value is 0.
  • The rate of change is 2.

Thus, the correct response is:

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

Thank you. Blessings upon your hat and the head that fills it.

Thank you very much for your kind words! May you also be blessed in all your endeavors. If you have any more questions or need further assistance, feel free to ask!