The rate of change of a linear function represents how much the output of the function changes for a corresponding change in the input. In graphical terms, the rate of change is reflected by the slope of the line representing the linear function.
Key Concepts:
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Linear Function: A linear function can be represented in the form \( y = mx + b \), where:
- \( m \) is the slope (rate of change),
- \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \)).
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Slope of the Line: The slope (rate of change) is calculated as: \[ \text{slope} = \frac{\Delta y}{\Delta x} \] where \( \Delta y \) is the change in the y-values and \( \Delta x \) is the change in the x-values between two points on the line.
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Graphing: When you plot a linear function on a graph:
- A positive slope means the line rises as it moves from left to right.
- A negative slope means the line falls as it moves from left to right.
- A slope of zero means the line is horizontal, indicating no change in \( y \) as \( x \) changes.
Example:
Consider the linear function defined by the equation \( y = 2x + 1 \):
- Here, the slope \( m = 2 \), which indicates that for every increase of 1 in \( x \), \( y \) increases by 2.
- If you plot this function, you would see a straight line that rises steeply due to the positive slope.
Finding the Rate of Change from a Graph:
- Identify two points on the line, such as \( (x_1, y_1) \) and \( (x_2, y_2) \).
- Calculate the changes:
- \( \Delta y = y_2 - y_1 \)
- \( \Delta x = x_2 - x_1 \)
- Use the slope formula to find the rate of change: \[ \text{slope} = \frac{\Delta y}{\Delta x} \]
In practice, on a platform like IXL, you may be asked to analyze graphs, identify slopes, or even select the correct linear function based on given points or characteristics. If you have specific questions or examples from IXL you'd like assistance with, feel free to share!
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