FORMULABANK

Linear Functions 

A linear function describes a straight line on a graph. The general form of a linear function is:

$\mathrm{f(x)}=ax+b$

Where:

• a is the slope  representing how steep the line is.
• b is the y-intercept, which is the point where the line crosses the y-axis.

The Four Representations of Linear Functions

Linear functions can be represented in four different ways:

1. Algebraic : The equation of the line, written as $y=ax+b$. This shows how the value of y changes in relation to x.
2. Graphical: The visual representation of the function as a straight line on a coordinate plane.
3. Tabular: A table of values for x and corresponding y values, which can help plot the graph of the function.
4. Verbal Description: A written explanation of how the variables are related, describing the slope and intercept of the line.

Formula for Finding the Equation of a Line from Two Points

To calculate the equation of a linear function from two points, you can use the following formula:

$a=\frac{\mathrm{y\_2}-\mathrm{y\_1}}{\mathrm{x\_2}-\mathrm{x\_1}}$

The slope, $a$, represents how much the line rises or falls between two points $\mathrm{(x\_1,\; y\_1)}$ and $\mathrm{(x\_2,\; y\_2)}$. The equation of the line can then be written as:

$y=ax+b$

Formula for Finding the Equation of a Line from One Point and a Slope

If you know the slope and a point on the line, you can use the point-slope formula to find the equation of the line:

$y-\mathrm{y\_1}=a(x-\mathrm{x\_1})$

Where $\mathrm{(x\_1,\; y\_1)}$ is a known point on the line and $a$ is the slope.