Linear equations resp. Linear Functions accompany you in most of the math lessons in school. Therefore it is important to understand and be able to use this function type. In this article you will learn exactly that with the help of explanations and illustrations.
Linear equations – What is it?
Linear equations can also be called first degree equations, because the variable only occurs in the first power. If this variable is x, it means that there are no parts with x², x³ and so on, but only x.
When you draw a linear function, you get a straight line.
Linear functions – formula explained in 2 steps
The general formula for linear equations is f ( x ) = m x + b.
- The b describes the y-axis intercept. So this is the point where the linear function intersects the y-axis.
- The slope is in m. This explains how flat or steep a function is. If the m is positive, the function increases and if the m is negative, it falls.
Linear functions – Determine slope easily
Now you already know what the slope is and where to find it in the formula. But maybe you ask yourself how to determine the slope from a drawing. For this you need a so called gradient triangle.
So you pick 2 points on the straight line and draw the triangle to those points. Then you look how long the two sides of the triangle are that you just drew to the function. Then divide the length of the vertical line by the length of the horizontal line.
Example: 3 / 1 = 3
So the slope is 3.
Linear functions – calculating the slope
With the two points you have selected on the line, you can also calculate the slope. Point 1 is ( 0 / 1 ), so \displaystyle x_1=0 and \displaystyle y_1=1 . The second point we had chosen is ( 1 / 4 ). Therefore \displaystyle x_2=1 and \displaystyle y_2=4 .
Now you calculate \displaystyle \frac .
If you now insert the values you get \displaystyle \frac=3 .
m = \displaystyle \frac
Draw linear functions
Given is the function f ( x ) = 2 x – 1.
The y-axis intercept is -1. So you mark the point ( – 1 / 0 ) on the y-axis. From there you go 1 to the right and 2 up, because the slope is positive. If the slope was negative, you would go down 2.
NoteWhenever there is no fraction in front of the x, you can think of the number as a fraction, because \displaystyle \frac = 2. The numerator is the length of the vertical line of the slope triangle and the denominator is the length of the horizontal line.
Linear functions – Determine zeros in the twinkling of an eye
Zeros are the intersections of a function with the x-axis. Linear functions usually have exactly one zero. Exceptions are linear equations with m = 0. These equations are horizontal and have no zeros or infinity if b = 0. This is because the function f ( x ) = 0 x + 0 lies on the x-axis.