Question

which functions have a vertex with a \\(x\\)-value of 0? select three options.

\\(f(x) = |x|\\)

\\(f(x) = |x| + 3\\)

\\(f(x) = |x + 3|\\)

\\(f(x) = |x| - 6\\)

\\(f(x) = |x + 3| - 6\\)

Explanation:

Understand the vertex form

Using the Absolute Value Functions knowledge point

$$ f(x) = a|x - h| + k \implies \text{Vertex} = (h, k) $$

Analyze the horizontal translation

Using the Function Translations knowledge point

$$ h = 0 \iff \text{No horizontal translation} \iff f(x) = a|x| + k $$

Evaluate each function option

Using the Absolute Value Functions and Function Translations knowledge points

$$
LATEXBLOCK0
$$

Answer:

• (A) \(f(x) = |x|\) (Correct answer)
• (B) \(f(x) = |x| + 3\) (Correct answer)
• (C) \(f(x) = |x + 3|\)
• (D) \(f(x) = |x| - 6\) (Correct answer)
• (E) \(f(x) = |x + 3| - 6\)