Use the image to answer the question.

Question

<img alt="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. " height="800" src=>

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 .

Answer 1

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

From the problem, the points on the line are:

• (-3, 4)
• (0, 3)
• (3, 2)
• (6, 1)

Initial Value

The initial value of a linear function, also referred to as the y-intercept, is the y-coordinate when x = 0. From the given points, at \( x = 0 \), the y-coordinate is \( 3 \). Therefore, the initial value is \( 3 \).

Rate of Change

The rate of change of a linear function is equivalent to the slope of the line. The slope \( m \) can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

We can use any two points on the line. Let's use the points (0, 3) and (3, 2):

\[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} = -0.333 \]

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

Conclusion

The final answers are: The initial value is 3, and the rate of change is -0.333.

Answer 2

are you sure

Answer 3

Let's re-evaluate the calculations step-by-step to ensure we have the correct values for the initial value and the rate of change.

Initial Value

The initial value (or y-intercept) of the function is found where \( x = 0 \). Referring to the points given:

• At \( x = 0 \), the point is \( (0, 3) \).

Thus, the initial value is indeed 3.

Rate of Change (Slope)

To find the rate of change (slope) of the line, we can select any two points. Let's again use the points (0, 3) and (3, 2):

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \( (0, 3) \) and \( (3, 2) \):

• \( (x_1, y_1) = (0, 3) \)
• \( (x_2, y_2) = (3, 2) \)

Now, substituting into the slope formula:

\[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \]

This simplifies to:

\[ m = -\frac{1}{3} \approx -0.333 \]

Conclusion

After reviewing the calculations:

• The initial value is 3.
• The rate of change (slope) is approximately -0.333 when rounded to three decimal places.

These values are correct. Thank you for prompting a second look!