Absolute Value Functions

Lesson

The essential ideas about how the inclusion of various constants and coefficients can affect an absolute value function are introduced in another chapter. Two other earlier chapters are relevant: A, B.

The following remarks apply to functions generally. We give some examples relating to the absolute function.

Suppose a function $f$`f` is defined on a subset of the real numbers. We can create a new function $f_1$`f`1 by adding a constant $A$`A` to $f$`f`. Thus, $f_1=f+A$`f`1=`f`+`A` and every function value of $f$`f` has had the constant $A$`A` added to it. This means that the graph of $f_1$`f`1 is the graph of $f$`f` transposed vertically by the amount $A$`A`.

Let $f(x)=|x|$`f`(`x`)=|`x`| be defined on the interval $[-3,3]$[−3,3]. The range of $f$`f` is the interval $[0,3]$[0,3].

Construct a new function $f_1=f+1$`f`1=`f`+1. That is, $f_1=|x|+1$`f`1=|`x`|+1. The domain of $f_1$`f`1 is the same as that of $f$`f` but the range of $f_1$`f`1 is the interval $[1,4]$[1,4].

As before, let a function $f$`f` be defined on the interval $[-3,3]$[−3,3]. We can create a new function $f_2$`f`2 by multiplying $f(x)$`f`(`x`) by a constant $B$`B`. Thus, $f_2(x)=Bf(x)$`f`2(`x`)=`B``f`(`x`) and every function value has been multiplied by $B$`B`. The graph of $f_2$`f`2 is the graph of $f$`f` stretched vertically by the factor $B$`B`. The stretch includes a reflection about the $x$`x`-axis if the sign of $B$`B` is negative.

Let $f(x)=|x|$`f`(`x`)=|`x`| be defined on the interval $[-3,3]$[−3,3]. A new function $f_2$`f`2 is defined by $f_2(x)=-2|x|$`f`2(`x`)=−2|`x`|.

Since the range of $f$`f` is $[0,3]$[0,3], the range of $f_2$`f`2 must be $[-6,0]$[−6,0] as every function value is multiplied by $-2$−2.

If the variable $x$`x` is multiplied by a constant $C$`C` before the function is applied, the result is essentially (after a possible adjustment to the domain) a re-scaling of the horizontal axis. If $f_3(x)=f(Cx)$`f`3(`x`)=`f`(`C``x`), the value of $f_3$`f`3 at $x$`x` is the value of $f$`f` at $C\times x$`C`×`x`. In the case of the absolute value function, the result is similar to that of the previous example except that reflection in the $x$`x`-axis does not occur.

If $f_4(x)=|2x|$`f`4(`x`)=|2`x`| we can write this as $f_4(x)=|2||x|=2|x|$`f`4(`x`)=|2||`x`|=2|`x`| . However, it is also true that if $f_4(x)=|-2x|$`f`4(`x`)=|−2`x`|, we have $f_4(x)=|-2||x|=2|x|$`f`4(`x`)=|−2||`x`|=2|`x`|.

In any function, if a constant $D$`D` is added to the variable $x$`x` before the function is applied, the effect is a shift of the graph to the left by the amount $D$`D`. So, if $f_5(x)=f(x+3)=|x+3|$`f`5(`x`)=`f`(`x`+3)=|`x`+3|, the graph of $f_5$`f`5 is the graph of $f$`f` shifted $3$3 units to the left.

If the graph of $y=\left|x\right|$`y`=|`x`| is translated $5$5 units down, what is the equation of the new graph?

If the graph of $y=\left|x\right|$`y`=|`x`| is stretched vertically by a factor of $4$4 and reflected across the $x$`x`-axis, what is the equation of the new graph?

Use the graph of $y=f\left(x\right)$`y`=`f`(`x`) to graph $y=3f\left(x\right)$`y`=3`f`(`x`).

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The following applet may also be useful to explore concepts relating to transformations of functions.

Start with the basic linear function $y=x$`y`=`x`, change the value of $b$`b` and note the change in the original function. What is the corresponding change in the absolute value function.

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