Question

use transformations of the absolute value function, $f(x)=|x|$, to graph the function $h(x)=-|x - 7|$.
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 stretch by a factor of 7
b. horizontal shift 7 units to the right
c. horizontal shift 7 units to the left
d. vertical shrink by a factor of $\frac{1}{7}$
e. reflection about the x - axis
f. vertical shift 7 units up

Explanation:

Step1: Analyze horizontal shift

For a function \( f(x) \), the transformation \( f(x - a) \) shifts the graph horizontally \( a \) units to the right. Here, \( h(x) = -|x - 7| \) can be seen as starting from \( f(x)=|x| \), then \( |x - 7| \) is \( f(x - 7) \), so this is a horizontal shift 7 units to the right.

Step2: Analyze reflection

The negative sign in front of the absolute value, \( -|x - 7| \), is a reflection about the x - axis of the function \( |x - 7| \). A reflection about the x - axis of a function \( y = g(x) \) is given by \( y=-g(x) \). So here, we have a reflection about the x - axis.

Step3: Eliminate other options

• Option A: There is no horizontal stretch by a factor of 7. The form \( |x - 7| \) is a shift, not a stretch. The horizontal stretch would be of the form \( |kx| \) where \( k

eq1 \), not \( |x - a| \).

• Option D: There is no vertical shrink. The vertical shrink would be of the form \( a|x| \) where \( 0 < a<1 \), here we have a reflection and a shift, not a shrink.
• Option C: A horizontal shift to the left would be \( f(x + a) \), but we have \( f(x - 7) \), so it's a shift to the right, not left.
• Option F: There is no vertical shift. The vertical shift would be of the form \( |x|+b \) or \( |x|-b \), here we have no such term.

Answer:

B. Horizontal shift 7 units to the right, E. Reflection about the x - axis