A graph with (-3 , 4) , (0 , 3) , (3,2) , and (6 , 1)

Question

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.

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 given the points (-3, 4), (0, 3), (3, 2), and (6, 1), we will follow these steps:

1. Identify the Initial Value

The initial value of a linear function (often referred to as the y-intercept) is the output when the input (x) is 0. From the points provided, we observe:

• (0, 3) implies that when \(x = 0\), \(y = 3\).

Thus, the initial value is 3.

2. Calculate the Rate of Change

The rate of change (slope) of a linear function can be calculated using any two points on the line. The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

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

We can use the points (0, 3) and (6, 1) for our calculation:

• Let \((x_1, y_1) = (0, 3)\) and \((x_2, y_2) = (6, 1)\).

Substituting in the values:

\[ m = \frac{1 - 3}{6 - 0} = \frac{-2}{6} = -\frac{1}{3} \]

Summary

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

Thus, the final answers are:

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

Answer 2

you sure

Answer 3

Let’s double-check the calculations step by step to ensure everything is correct.

1. Identify the Initial Value

The initial value (or \(y\)-intercept) occurs when \(x = 0\). Looking at the provided points:

• From point (0, 3), we see that when \(x = 0\), \(y = 3\).

Thus, the initial value is indeed 3.

2. Calculate the Rate of Change

To find the rate of change (slope), we can use the formula for the slope \(m\):

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

Let's select two points and calculate the slope:

• Using points (0, 3) and (6, 1):
  • Here, \(x_1 = 0\), \(y_1 = 3\), \(x_2 = 6\), and \(y_2 = 1\).

Calculating the slope:

\[ m = \frac{1 - 3}{6 - 0} = \frac{-2}{6} = -\frac{1}{3} \]

Converting that to decimal form gives:

\[ m = -0.3333... \text{ (approximately)} \]

When rounded to three decimal places, this gives us -0.333.

Conclusion

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

So to confirm: The initial value is 3, and the rate of change is -0.333.

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