Question

match the vertex to each absolute value function.

1. $f(x) = |x - 3|$ 2) $f(x) = |x + 3| + 2$ 3) $f(x) = |x| - 3$ 4) $f(x) = |x + 1|$

(there are boxes with numbers 1,2,3,4 connected to dashed boxes, and below are vertex options: (0, -1), (-1,0), (3,0), (0, -3), (-3,0), (3,0), (0,3), (-3,2), (3,2), (2, -3))

Explanation:

Step1: Recall the vertex form of absolute value functions.

The general form of an absolute value function is \( f(x) = |x - h| + k \), where the vertex is \((h, k)\).

Step2: Find the vertex for \( f(x) = |x - 3| \).

For \( f(x) = |x - 3| \), we can rewrite it as \( f(x) = |x - 3| + 0 \). So, \( h = 3 \) and \( k = 0 \). Thus, the vertex is \((3, 0)\).

Step3: Find the vertex for \( f(x) = |x + 3| + 2 \).

Rewrite \( |x + 3| \) as \( |x - (-3)| \). So, \( f(x) = |x - (-3)| + 2 \), where \( h = -3 \) and \( k = 2 \). Thus, the vertex is \((-3, 2)\).

Step4: Find the vertex for \( f(x) = |x| - 3 \).

Rewrite \( |x| \) as \( |x - 0| \), so \( f(x) = |x - 0| - 3 \). Here, \( h = 0 \) and \( k = -3 \). Thus, the vertex is \((0, -3)\).

Step5: Find the vertex for \( f(x) = |x + 1| \).

Rewrite \( |x + 1| \) as \( |x - (-1)| \), so \( f(x) = |x - (-1)| + 0 \). Here, \( h = -1 \) and \( k = 0 \). Thus, the vertex is \((-1, 0)\).

Answer:

1. \( f(x) = |x - 3| \) matches \((3, 0)\)
2. \( f(x) = |x + 3| + 2 \) matches \((-3, 2)\)
3. \( f(x) = |x| - 3 \) matches \((0, -3)\)
4. \( f(x) = |x + 1| \) matches \((-1, 0)\)