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The initial value is the \(y\)-coordinate of the point where the line crosses the \(y\)-axis The rate of change is the slope of the line The slope of a line passing through two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) is given by \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

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You're correct in your definitions related to a linear equation and the characteristics of a line. Let's summarize and clarify these concepts:

  1. Initial Value (y-intercept):

    • The initial value refers to the \(y\)-coordinate of the point where the line crosses the \(y\)-axis. This point is represented as \((0, b)\), where \(b\) is the y-intercept. In the context of the linear equation of the form \(y = mx + b\), \(b\) is the \(y\)-intercept.

  2. Rate of Change (Slope):

    • The rate of change of a line is described by its slope, often denoted as \(m\). The slope indicates how much \(y\) changes for a unit change in \(x\).
    • The formula for calculating the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

  3. Equation of a Line:

    • A linear function can be represented in slope-intercept form as: \[ y = mx + b \]
    • In this equation, \(m\) represents the slope (rate of change) and \(b\) represents the \(y\)-intercept (initial value).

With these definitions in mind, you can analyze linear relationships, graph lines, and understand the behavior of functions in linear equations. If you have specific points or scenarios you'd like to work through, feel free to provide them!

Determine the initial value and the rate of change of the linear function given in the graph. (1 point) The initial value is , and the rate of change is .

To determine the initial value (y-intercept) and the rate of change (slope) of a linear function from a graph, follow these steps:

  1. Identifying the Initial Value:

    • The initial value is where the line crosses the \(y\)-axis. Look for the point where the graph intersects the \(y\)-axis. This point will give you the initial value \(b\).

  2. Determining the Rate of Change (Slope):

    • The rate of change, or slope \(m\), can be calculated using two points on the line. Choose two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) on the line. Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
    • This will give you the rate of change of the function.

Since you mentioned a graph but didn't provide a specific one to refer to, I can't give you exact values. However, using the steps above, you should be able to read the initial value and rate of change from the graph.

If you're looking at a specific graph, please describe it or provide the coordinates of points, and I can help you find those values!