Question

which equation is represented by the graph below?
answer
\\( y = -2|x + 3| + 6 \\)
\\( y = -2|x + 3| - 6 \\)
\\( y = -2|x - 3| - 6 \\)
\\( y = -2|x - 3| + 6 \\)

Explanation:

Step1: Recall vertex form of absolute value

The vertex form is $y = a|x - h| + k$, where $(h,k)$ is the vertex.

Step2: Identify vertex from graph

The vertex is at $(-3,6)$, so $h=-3$, $k=6$.

Step3: Determine slope $a$

Take a point on the graph, e.g., $(-1,2)$. Substitute into the form: $2 = a|-1 - (-3)| + 6$. Simplify: $2 = a|2| + 6 \Rightarrow 2 = 2a + 6 \Rightarrow 2a = -4 \Rightarrow a=-2$.

Step4: Write the equation

Substitute $a=-2$, $h=-3$, $k=6$: $y = -2|x + 3| + 6$.

Answer:

y = -2|x + 3| + 6