graph the equation shown below by transforming the given graph of the parent function. \n$y = |3x|$

Explanation:

Step1: Identify the parent function

The parent function of ( y = |3x| ) is ( y = |x| ). The graph of ( y = |x| ) is a V - shaped graph with the vertex at the origin ((0,0)), and the slope of the right - hand side (where ( x\geq0 )) is ( 1 ) and the slope of the left - hand side (where ( x < 0 )) is ( - 1 ).

Step2: Analyze the transformation

For a function of the form ( y=|ax|) (where ( a>0 )), the transformation from the parent function ( y = |x| ) is a horizontal compression. The general rule for horizontal compressions/stretches: If we have a function ( y = f(kx) ), when ( k>1 ), the graph of ( y = f(x) ) is horizontally compressed by a factor of ( \frac{1}{k} ).

In the function ( y = |3x| ), we can rewrite it as ( y=\left|3\left(x\right)\right| ). Here, ( k = 3>1 ), so the graph of ( y = |x| ) is horizontally compressed by a factor of ( \frac{1}{3} ).

To understand the effect on the points: Let's take a point ((x,y)) on the parent function ( y = |x| ). For the function ( y=|3x| ), we can find the corresponding ( x' ) such that when ( y = |3x'|=|x| ), we have ( 3x'=x ) (for ( x\geq0 )) or ( 3x'=-x ) (for ( x < 0 )), so ( x'=\frac{x}{3} ).

For example, on the parent function ( y = |x| ), when ( x = 3 ), ( y = 3 ). For the function ( y=|3x| ), when ( y = 3 ), we solve ( |3x|=3 ), which gives ( 3x = 3) or ( 3x=-3 ), so ( x = 1 ) or ( x=-1 ). So the point ((3,3)) on ( y = |x| ) is transformed to ((1,3)) on ( y = |3x| ), and the point (( - 3,3)) on ( y = |x| ) is transformed to ((-1,3)) on ( y = |3x| ).

To graph ( y = |3x| ):

• The vertex remains at ((0,0)) because when ( x = 0 ), ( y=|3\times0| = 0).
• For the right - hand side (( x\geq0 )), the equation is ( y = 3x ). The slope is ( 3 ) (steeper than the slope of ( y = |x| ) which is ( 1 )). We can plot points: when ( x = 1 ), ( y=3\times1 = 3); when ( x = 2 ), ( y = 3\times2=6); when ( x=\frac{1}{3} ), ( y = 3\times\frac{1}{3}=1).
• For the left - hand side (( x < 0 )), the equation is ( y=- 3x ) (since ( |3x|=-3x ) when ( x < 0 )). When ( x=-1 ), ( y=-3\times(-1) = 3); when ( x = - 2 ), ( y=-3\times(-2)=6); when ( x=-\frac{1}{3} ), ( y=-3\times(-\frac{1}{3}) = 1).

The graph of ( y = |3x| ) will have a steeper slope (slope of ( 3 ) on the right and ( - 3 ) on the left) compared to the parent function ( y = |x| ), and it is a horizontal compression of the graph of ( y = |x| ) by a factor of ( \frac{1}{3} ).

(Note: Since the problem is about graphing by transformation, the key steps are identifying the parent function, the type of transformation (horizontal compression), and then determining the effect on the points and the shape of the graph.)

Answer:

To graph ( y = |3x| ), we start with the parent function ( y=|x| ) (a V - shaped graph with vertex at ((0,0)), slope ( 1 ) for ( x\geq0 ) and slope ( - 1 ) for ( x < 0)). The function ( y = |3x| ) represents a horizontal compression of ( y = |x| ) by a factor of ( \frac{1}{3} ). The graph of ( y = |3x| ) has its vertex at ((0,0)), a slope of ( 3 ) for ( x\geq0 ) (equation ( y = 3x )) and a slope of ( - 3 ) for ( x < 0) (equation ( y=-3x )). We can plot points such as ((0,0)), ((1,3)), ((-1,3)), ((\frac{1}{3},1)), ((-\frac{1}{3},1)) etc. and draw the V - shaped graph with a steeper slope than ( y = |x| ) due to the horizontal compression.