Identify Characteristics of Absolute Value Functions I

An expression of the form $ax+b$ax+b defines a function if we let $a$a and $b$b be constants and we let $x$x be a real number variable.

If we write $y=ax+b$y=ax+b, where $y$y is the value of the function at any given $x$x, we can draw a graph showing how $y$y varies with $x$x. The graph, in this case, is always a line and the function is called a linear function.

Except in the special case where $a=0$a=0, the function given by $y=ax+b$y=ax+b always has both positive and negative values. For example, if $y=5x-15$y=5x−15, $y$y will be negative whenever $x$x is less than $3$3 and positive when $x$x is greater than $3$3.

The effect of inserting the absolute value signs $|ax+b|$|ax+b| is to make all the values of the function positive. This is done by redefining the function as two functions: one for values of $x$x that would make the original function positive and another for values of $x$x that make the original function negative. Look at the following example to see how this works.

The graph of a linear absolute value function, therefore, has two lines: one for each part of the function definition.

In the diagram below, the lines corresponding to the two parts of a function definition are shown lightly with the 'composite' absolute value function drawn more heavily.

 

Other functions of the form $|ax+b|$|ax+b| have a shape similar to the one illustrated. The 'V' shape is typical with the vertex located on the horizontal axis at the point $x$x that makes $ax+b=0$ax+b=0.

When the coefficient $a$a is large, the 'V' shape is narrow. This is because $a$a controls the gradients. One side has a gradient of $a$a while the other has a gradient of $-a$−a.

 

If a negative sign is placed in front of the absolute value symbol, the effect is to invert the 'V' shape of the graph. All the function values are negative when $y=-|ax+b|$y=−|ax+b|.

 

Example 1

Where is the vertex of the graph of $y=\left|\frac{x}{2}+7\right|$y=|x2​+7|? And where does the graph cut the vertical axis?

For the vertex, we set $\frac{x}{2}+7=0$x2​+7=0 and solve for $x$x. This gives $x=-\frac{7}{2}=-3\frac{1}{2}$x=−72​=−312​. This $x$x-value is to the left of the vertical axis. So, we need to look at the part of the graph with positive slope to find the intercept on the vertical axis. (You could draw a rough diagram to verify this.)

Thus, we put $x=0$x=0 in the equation $y=\frac{x}{2}+7$y=x2​+7 and this gives $y=7$y=7.

 

Example 2

The two arms of the graph of a linear absolute value function have gradients $-1$−1 and $1$1 respectively. The vertex of the graph is at the point $x=3$x=3. Find the function definition.

The function has the form $|ax+b|$|ax+b| so that its two parts are given by $-(ax+b)$−(ax+b) and $ax+b$ax+b. Both of these have the value $0$0 when $x=3$x=3.

Thus, we can take $3a+b=0$3a+b=0. Now, $a$a is either $1$1 or $-1$−1. So, we could have $3+b=0$3+b=0 or $-3+b=0$−3+b=0. In the first case $b=-3$b=−3 and in the second case $b=3$b=3. Hence, we have two possibilities:
$y=x-3$y=x−3 and
$y=-x+3$y=−x+3

These are negatives of each other. So, the function definition we seek could be written $y=|-x+3|$y=|−x+3| or $y=|x-3|$y=|x−3|. In fact, these are the same. You should check that $y=0$y=0 when $x=3$x=3 and the two gradients are $-1$−1 and $1$1 .

 

Example 3

Example 4