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a graph shows 4 points.

(-3, 4)
(0, 3)
(3, 2)
(6, 1)

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)

3 answers

To determine the initial value (y-intercept) and the rate of change (slope) of the linear function represented by the given points, we will first find the slope using two of the points and then determine the y-intercept.

  1. Calculate the slope (rate of change): The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's use the points \((0, 3)\) and \((3, 2)\):

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

Calculating the slope: \[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \approx -0.333 \]

  1. Determine the initial value (y-intercept): The y-intercept is the value of \( y \) when \( x = 0 \). From the point \((0, 3)\), we can see that the initial value is: \[ b = 3 \]

  2. Summary of Results:

  • Initial value (y-intercept): \( 3 \)
  • Rate of change (slope): \( -0.333 \)

Thus, the initial value is \( 3 \) and the rate of change is approximately \( -0.333 \).

why 0.333? could you explain?

Certainly! The slope of a line, which is represented as the rate of change, is calculated by taking the difference in the \( y \)-values (vertical change) divided by the difference in the \( x \)-values (horizontal change) between two points.

Calculation Breakdown

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

  • Point 1 (x₁, y₁): \((0, 3)\)
  • Point 2 (x₂, y₂): \((3, 2)\)

Step-by-step calculation of the slope \( m \):

  1. Calculate the vertical change (difference in \( y \)-values): \[ y_2 - y_1 = 2 - 3 = -1 \]

  2. Calculate the horizontal change (difference in \( x \)-values): \[ x_2 - x_1 = 3 - 0 = 3 \]

  3. Now calculate the slope \( m \) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1}{3} \]

Result

So the calculated slope is: \[ m = -\frac{1}{3} \approx -0.33333\ldots \] When rounded to three decimal places, it is \(-0.333\).

Explanation of Slope Value

This implies that for every increase of 1 unit in \( x \), \( y \) decreases by approximately \( 0.333 \). Since this is a negative slope, it indicates that the line is sloping downwards as you move from left to right across the graph.

If you have further questions about the slope or its implications, feel free to ask!