You're correct in your definitions related to a linear equation and the characteristics of a line. Let's summarize and clarify these concepts:
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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.
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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: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
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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!