Want to understand linear functions without having to read a textbook? This post describes these functions in a way that is illustrative and to the point.
Linear functions represent lines. Normally, they are presented with equations such as $$ y = mx + b $$ If you recall from school, \(m\) is the slope and \(b\) is the y-intercept.
Let’s plot the linear function \( y = 0.5 x \, – 1 \) below.
There are two things to look out for. The first is the y-intercept. Recall that the y-axis is the vertical line with all the numbers as shown above. The y-intercept is the intersection of the graph of \( y = 0.5 x \, – 1 \) with the y-axis, and that point is \( (0, -1) \). This is shown below. Note that the x-coordinate of \( (0, -1) \) is \( 0 \) and the y-coordinate of \( (0, -1) \) is \( -1 \).
If we let \( x = 0 \) and \( y = -1 \), then \( 0.5 x \, -1\) equals \(y \) because \( 0.5 \cdot 0 \, – 1 = -1 \). That is why the point \( (0, -1) \) is contained in the graph of \( y = 0.5 x \, – 1 \).
The second thing to look out for is the slope of the line. The slope tells you how far up (or down) a point would move if a point on that line moved one unit to the right. It measures the steepness of the line.
For instance, consider the y-intercept \( (0, -1) \), and imagine moving it one point to the right. This is what it would look like.
Notice how the point slightly moves upward. Why is that? The x-coordinate of the point is \( 1 \) and the y-coordinate of this point is \( -0.5 \). If we let \( x = 1 \) and \( y = -0.5 \), we find that \( 0.5 x \, -1\) equals \(y \) because \( 0.5 \cdot 1 \, – 1 = -0.5 \).
Look the points \( (0, -1) \) and \( (1, -0.5) \). Notice that when we shifted the above y-intercept to the right by one unit, the y-coordinate also increased by \( 0.5 \). This is because \( 0.5 \) is the slope of the line.
