Question

write your answer in the form a|x - h| + k, where a, h, and k are integers or simplified fractions.
q(x) =

Explanation:

Step1: Identify the vertex

The vertex form of an absolute value function is \( g(x) = a|x - h| + k \), where \((h, k)\) is the vertex of the V - shaped graph. From the graph, the vertex is at \((0, - 2)\), so \( h = 0 \) and \( k=-2 \).

Step2: Determine the value of \( a \)

We can use another point on the graph to find \( a \). Let's use the point \((2,0)\). Substitute \( x = 2 \), \( y = 0 \), \( h = 0 \) and \( k=-2 \) into the equation \( g(x)=a|x - h|+k \):

$$ 0=a|2 - 0|+(-2) $$

$$ 0 = 2a-2 $$

Add 2 to both sides:

$$ 2a=2 $$

Divide both sides by 2:

$$ a = 1 $$

Step3: Write the function

Substitute \( a = 1 \), \( h = 0 \) and \( k=-2 \) into the vertex form \( g(x)=a|x - h|+k \):

$$ g(x)=1|x - 0|+(-2)=|x|-2 $$

Answer:

\( g(x)=|x|-2 \) (or \( g(x) = 1|x - 0| - 2 \))