Question

use transformations of the absolute value function, $f(x)=|x|$, to graph the function $h(x)=3|x + 3|$. what transformations are needed in order to obtain the graph of $h(x)$ from the graph of $f(x)$? select all that apply. a. horizontal shift 3 units to the left b. horizontal shift 3 units to the right c. reflection about the x - axis d. vertical stretch by a factor of 3 e. horizontal stretch by a factor of 3 f. vertical shift 3 units up

Explanation:

Step1: Analyze horizontal shift

For a function \( f(x) \), the transformation \( f(x + a) \) (where \( a>0 \)) is a horizontal shift \( a \) units to the left. Here, \( h(x)=3|x + 3|=3f(x + 3) \), so compared to \( f(x)=|x| \), there is a horizontal shift 3 units to the left. So option A is correct, B is incorrect.

Step2: Analyze reflection

The function \( h(x) \) has a positive coefficient (3) in front of the absolute value, so there is no reflection about the x - axis (reflection about x - axis would be \( -f(x) \) form). So option C is incorrect.

Step3: Analyze vertical stretch

For a function \( f(x) \), the transformation \( af(x) \) (where \( a > 1\)) is a vertical stretch by a factor of \( a \). Here, \( h(x)=3f(x + 3) \), with \( a = 3>1 \), so there is a vertical stretch by a factor of 3. So option D is correct.

Step4: Analyze vertical shift

The function \( h(x) \) is in the form \( 3|x + 3| \), there is no vertical shift (vertical shift would be of the form \( f(x)+k \) or \( f(x)-k \)). So option F is incorrect. And option E is incorrect as horizontal stretch is of the form \( f(bx) \) with \( 0 < b<1 \) for stretch, here we have \( x + 3 \) not \( bx \) with \( b=\frac{1}{3} \) or other, so no horizontal stretch by factor 3.

Answer:

A. Horizontal shift 3 units to the left, D. Vertical stretch by a factor of 3