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 $f_1$f1 by adding a constant $A$A to $f$f. Thus, $f_1=f+A$f1=f+A and every function value of $f$f has had the constant $A$A added to it. This means that the graph of $f_1$f1 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 $f_1=f+1$f1=f+1. That is, $f_1=|x|+1$f1=|x|+1. The domain of $f_1$f1 is the same as that of $f$f but the range of $f_1$f1 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 $f_2$f2 by multiplying $f(x)$f(x) by a constant $B$B. Thus, $f_2(x)=Bf(x)$f2(x)=Bf(x) and every function value has been multiplied by $B$B. The graph of $f_2$f2 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 $f_2$f2 is defined by $f_2(x)=-2|x|$f2(x)=−2|x|.
Since the range of $f$f is $[0,3]$[0,3], the range of $f_2$f2 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 $f_3(x)=f(Cx)$f3(x)=f(Cx), the value of $f_3$f3 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 $f_4(x)=|2x|$f4(x)=|2x| we can write this as $f_4(x)=|2||x|=2|x|$f4(x)=|2||x|=2|x| . However, it is also true that if $f_4(x)=|-2x|$f4(x)=|−2x|, we have $f_4(x)=|-2||x|=2|x|$f4(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 $f_5(x)=f(x+3)=|x+3|$f5(x)=f(x+3)=|x+3|, the graph of $f_5$f5 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=3f(x).
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