Linear Functions

A linear function is an expression that gives you a straight line. The name is slightly revealing. Here, you will learn how to draw and recognize linear functions.

Theory

Linear Function

A linear function can be written on the form

f (x) = a x + b,

where $a$ is the slope, and $b$ is the intersection with the $y$ -axis.

Finding the Slope and the Constant Term

You can find the slope of a line if you have the coordinates of two points on the 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 intersecting the y-axis in (0,1)

Rule

Important Attributes of the Linear Function

The slope $a$ tells you how much the graph is increasing/decreasing when $x$ increases by 1.
If $a > 0$ , the graph rises towards the right, and if $a < 0$ , the graph is sinking towards the right.
The graph intersects the $y$ -axis in the point $b$ .
The graph is a straight line with coordinates $(x, y) = (x, f (x))$ .

Example 1

Find the slope of the straight line that goes through the points $(5, 2)$ and $(3, 6)$ , and find where it intersects with the $y$ -axis

You choose $(x_{1}, y_{1})$ to be $(3, 6)$ and $(x_{2}, y_{2}) = (5, 2)$ . The calculations would 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 the line decreases by 2 when you move one place to the right. Let’s see what the point of intersection with the $y$ -axis is: $\begin{array}{l} b & = y_{1} - a x_{1} \\ = 6 - (- 2) \times 3 \\ = 6 + 6 \\ = 12 . \end{array}$

Thus, the point of intersection with the $y$ -axis is $(0, 12)$ .

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Activities 1

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Activities 2