Absolute Value Functions

Lesson

An equation of the form $|ax+b|=c$|`a``x`+`b`|=`c`, where $a,b$`a`,`b` and $c$`c` are constants, can have zero, one or two solutions depending on the constants values.

The graph of the linear absolute value function, depicted in green in the diagram, has the formula $y=|\frac{1}{2}x+1|$`y`=|12`x`+1|.

By letting the function value $y=c$`y`=`c` be fixed successively at $c=-1.5$`c`=−1.5, $c=0$`c`=0 and $c=1$`c`=1, we form three equations

$|\frac{1}{2}x+1|=-1.5$|12`x`+1|=−1.5

$|\frac{1}{2}x+1|=0$|12`x`+1|=0

$|\frac{1}{2}x+1|=1$|12`x`+1|=1

From the graph we see that the first of these has no solution, the second has exacly one solution ($x=-2$`x`=−2) and the third has two solutions ($x=-4$`x`=−4 and $x=0$`x`=0).

It must always be true that an equation $|ax+b|=c$|`a``x`+`b`|=`c` has no solutions if $c$`c` is negative, exactly one solution if $c=0$`c`=0 and two solutions if $c$`c` is a positive number.

When $c=0$`c`=0, the solution is $x=-\frac{b}{a}$`x`=−`b``a`. You should check that this is the value that makes the expression equal to zero.

Solve $\left|\frac{2}{3}x+\frac{1}{3}\right|=0$|23`x`+13|=0.

We need to solve either $\frac{2}{3}x+\frac{1}{3}=0$23`x`+13=0 or $-\left(\frac{2}{3}x+\frac{1}{3}\right)=0$−(23`x`+13)=0. (The two equations are the same.)

Thus, $x=-\frac{1}{2}$`x`=−12.

Note that this is indeed $-\frac{b}{a}=-\frac{\frac{1}{3}}{\frac{2}{3}}$−`b``a`=−1323.

Solve $\left|\frac{2}{3}x+\frac{1}{3}\right|=12$|23`x`+13|=12.

To obtain both solutions, we consider the two parts of the function definition and hence write down the two equations

$\frac{2}{3}x+\frac{1}{3}=12$23`x`+13=12

$-\left(\frac{2}{3}x+\frac{1}{3}\right)=12$−(23`x`+13)=12

The first of these gives $x=\frac{3}{2}\left(12-\frac{1}{3}\right)=17\frac{1}{2}$`x`=32(12−13)=1712.

The second equation gives $x=\frac{3}{2}\left(-12-\frac{1}{3}\right)=-18\frac{1}{2}$`x`=32(−12−13)=−1812.

Consider the function $f(x)=\left|4x-4\right|$`f`(`x`)=|4`x`−4|.

Plot the function $f(x)$

`f`(`x`).Loading Graph...Hence determine the number of solutions to the equation $\left|4x-4\right|=-2$|4

`x`−4|=−2.

Consider the function $f(x)=\left|2x+4\right|$`f`(`x`)=|2`x`+4|.

The graph of the function is given. On the same set of axes, plot the points at which $f(x)=2$

`f`(`x`)=2.Loading Graph...How many solutions does the equation $\left|2x+4\right|=2$|2

`x`+4|=2 have?Using the result of part (a) or otherwise, find the solutions to the equation $\left|2x+4\right|=2$|2

`x`+4|=2.State your solutions on the same line, separated by a comma.

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

Apply graphical methods in solving problems