How to find the slope of a linear function?

Understand the Problem

The question is asking how to determine the slope of a linear function, which is a fundamental concept in algebra. The slope represents the rate of change of the function and can be calculated using the formula (change in y) / (change in x) between two points on the line.

Answer

The slope $m$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Steps to Solve

Identify two points on the line

To calculate the slope, you need two points on the line, which can be expressed as $(x_1, y_1)$ and $(x_2, y_2)$. Make sure these points are clear and that you have their coordinates.

Calculate the difference in y-coordinates

Subtract the y-coordinate of the first point from the y-coordinate of the second point:

$$ \Delta y = y_2 - y_1 $$

Calculate the difference in x-coordinates

Subtract the x-coordinate of the first point from the x-coordinate of the second point:

$$ \Delta x = x_2 - x_1 $$

Calculate the slope

Now, use the differences found in steps 2 and 3 to calculate the slope $m$ using the formula:

$$ m = \frac{\Delta y}{\Delta x} $$

Interpret the slope

The calculated slope indicates how much $y$ changes for a unit change in $x$. A positive slope means the function is increasing, while a negative slope means the function is decreasing.

The slope $m$ is calculated using the formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

More Information

The slope represents the steepness of the line and the direction of the change. A slope of zero indicates a horizontal line, while undefined (i.e., a vertical line where $x_1 = x_2$) indicates infinite steepness.

Tips

Forgetting to subtract in the correct order: Remember that $\Delta y$ should always be $y_2 - y_1$ and not the other way around.
Mixing up the coordinates: Ensure $x_1$ is correctly paired with $y_1$ and $x_2$ with $y_2$.

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