Question

A function$f$is given, and the indicated transformations are applied to its graph (in the given order). Write an equation for the final transformed graph.

$f\left(x\right)=\left|x\right|$; reflect in thelocalid="1645440736853" $x$-axis, shift$4$ units to the right and shift upward$3$ units

Short answer

The transformed equation is$f\left(x\right)=-|x-4|+3$

Step-by-step solution

Step 1. Concept of transformation in reflection in x or y axis and left or right and upward or downward direction.

To graph $y=-f\left(x\right)$, reflect the graph of $y=f\left(x\right)$in the $x$-axis

To graph $y=f(-x)$, reflect the graph of $y=f\left(x\right)$in the $y$-axis

To graph $y=f(x-c)$, shifts the graph to $y=f\left(x\right)$the right $c$units

And, to graph$y=f(x+c)$ shifts the graph$y=f\left(x\right)$ to the left$c$units

Adding a constant to a function shifts the graph upward or downward if the constant is positive or negative respectively.

Step 2. Given Problem.

Given that,

$f\left(x\right)=\left|x\right|$; reflect in the $x$-axis, shift $4$units to the right, and shift upward$3$ units

Step 3. Determining the transformed equation.

Here, the function reflected to $x$-axis then transformed equation is

$f\left(x\right)=-\left|x\right|$

And then function shifted $4$unit right, then transformed equation is

$f\left(x\right)=-|x-4|$

And then function shifted $3$unit upward, then transformed equation is

$f\left(x\right)=-|x-4|+3$