Question

the graph of $f(x) = |x|$ is transformed to $g(x) = |x + 1| - 7$. on which interval is the function decreasing?
$\bigcirc$ $(-\infty, -7)$
$\bigcirc$ $(-\infty, -1)$
$\bigcirc$ $(-\infty, 1)$
$\bigcirc$ $(-\infty, 7)$

Explanation:

Step1: Recall absolute value function

The parent function \( f(x) = |x| \) has a vertex at \( (0,0) \) and is decreasing on \( (-\infty, 0) \) and increasing on \( (0, \infty) \).

Step2: Analyze transformation of \( g(x) \)

For \( g(x) = |x + 1| - 7 \), the transformation is a horizontal shift left by 1 unit (since \( x \to x + 1 \)) and a vertical shift down by 7 units. The vertex of \( g(x) \) is at \( (-1, -7) \).

Step3: Determine decreasing interval

The absolute value function \( g(x) = |x + 1| - 7 \) has the same shape as \( f(x) = |x| \) but shifted. For \( y = |x - h| + k \), the function is decreasing when \( x < h \). Here, \( h = -1 \), so the function is decreasing on \( (-\infty, -1) \).

Answer:

\( (-\infty, -1) \) (corresponding to the option with this interval, e.g., if the options are as given, the second option: \( (-\infty, -1) \))