How is the graph of $y= | 3/4x +6 |$ related to the graph of $y = | x |$ ? Describe the order of the transformations too.

Question

The two possible answers I arrived at were depending on if you factor out the coefficient of 3/4 in front of the x.

If you do, the function becomes $y= | 3/4(x + 8) |$

• first transformation is a horizontal expansion by a factor of 4/3

• second transformation is a horizontal translation of 8 units left

If you leave function as is without factoring out the 3/4 coefficient in front of the x. The function remains as given $y = |3/4x+6|$

The other possible answer without factoring out the 3/4 coefficient of x is:

• first transformation is a horizontal translation of 6 units left
• second transformation is a horizontal expansion by a factor of 4/3

Answer 1

Both are correct.

To see why the first is correct, consider the functions $f(x)=|x|,$ $g(x)=\frac34 x,$ and $h_1(x)=x+8,$ and show that $y=(f\circ g\circ h_1)(x).$

To see why the second is correct, consider the functions $f(x)=|x|,$ $g(x)=\frac34 x,$ and $h_2(x)=x+6,$ and show that $y=(f\circ h_2\circ g)(x).$