Question

which functions have a vertex with a x - value of 0? select three options.\\(\square f(x)=|x|\\)\\(\square f(x)=|x| + 3\\)\\(\square f(x)=|x + 3|\\)\\(\square f(x)=|x| - 6\\)\\(\square f(x)=|x + 3| - 6\\)

Explanation:

Step1: Recall the vertex form of absolute value function

The general form of an absolute value function is \( f(x) = |x - h| + k \), where the vertex is at \( (h, k) \). We need to find the functions where the \( x \)-value of the vertex (i.e., \( h \)) is 0.

Step2: Analyze each function

• For \( f(x)=|x| \), we can rewrite it as \( f(x)=|x - 0|+0 \). Here, \( h = 0 \), so the vertex is at \( (0,0) \).
• For \( f(x)=|x| + 3 \), rewrite it as \( f(x)=|x - 0|+3 \). Here, \( h = 0 \), so the vertex is at \( (0,3) \).
• For \( f(x)=|x + 3| \), rewrite it as \( f(x)=|x - (- 3)|+0 \). Here, \( h=-3\), so the vertex is at \( (-3,0) \).
• For \( f(x)=|x|-6 \), rewrite it as \( f(x)=|x - 0|-6 \). Here, \( h = 0 \), so the vertex is at \( (0,-6) \).
• For \( f(x)=|x + 3|-6 \), rewrite it as \( f(x)=|x - (-3)|-6 \). Here, \( h = - 3\), so the vertex is at \( (-3,-6) \).

Answer:

A. \( f(x) = |x| \)
B. \( f(x) = |x| + 3 \)
D. \( f(x) = |x| - 6 \)