Linear Functions

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A linear function is an expression that when graphed, results in a straight line. The name is slightly revealing—when something is “linear”, it means it progresses in a straight line. Here, you will learn how to recognize, and draw graphs of, linear functions.

Theory

Linear Function

A linear function can be written in the form

f (x) = a x + b

where $a$ is the slope, and $b$ is the $y$ -intercept, the place the graph intersects with the $y$ -axis (and where $x = 0$ ).

Finding the Slope and the Constant Term

You can find the slope of a line if you have the coordinates of two points on that line. Call the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ . You use the following formulas for the slope $a$ and the constant term $b$ :

Rule

The Slope of a Linear Function

The straight line that goes through the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ has the slope

a = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

and the constant term

b = y_{1} - a x_{1}

A straight line with slope a=-1 and y-intercept at (0,1)

Rule

Important Attributes of the Linear Function

The slope $a$ tells you how much $y$ increases or decreases by as $x$ increases by 1.
If $a > 0$ , the graph rises towards the right, meaning $y$ is increasing as $x$ increases. If $a < 0$ , the graph sinks towards the right, meaning $y$ is decreasing as $x$ increases.
The graph intersects the $y$ -axis at the point $b$ , which is why it is known as the $y$ -intercept.
The graph is a straight line that with coordinates $(x, y) = (x, f (x))$ .

Example 1

Find the slope of the straight line that passes through the points $(5, 2)$ and $(3, 6)$ , and find the $y$ -intercept.

You set $(x_{1}, y_{1})$ equal to $(3, 6)$ and $(x_{2}, y_{2})$ equal to $(5, 2)$ . (The calculations would still work even if you switched the points.) You get:

a = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{2 - 6}{5 - 3} = \frac{- 4}{2} = - 2

You now know that $y$ decreases by 2 when $x$ increases by 1. In other words, the graph slopes downward by 2 when it moves 1 to the right.

Let’s see what the $y$ -intercept is:

\begin{array}{l} b & = y_{1} - a x_{1} \\ = 6 - (- 2) \times 3 \\ = 6 + 6 \\ = 12 \end{array}

Thus, the point $y$ -intercept is $(0, 12)$ .

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Activities 2