Question: 27. f(x): V-shape opening up with vertex at (0, 0); key points appear at $(-2, 2)$, $(2, 2)$, $(-4, 4)$, and $(4, 4)$. g(x): V-shape opening down with vertex at $(0, -3)$; key points appear at $(-2, -5)$, $(2, -5)$, $(-4, -7)$, and $(4, -7)$.

1. **Problem Statement:** We are given two functions: - $f(x)$ is a V-shaped graph opening upwards with vertex at $(0,0)$ and points $(-2,2)$, $(2,2)$, $(-4,4)$, and $(4,4)$. - $g(x)$ is a V-shaped graph opening downwards with vertex at $(0,-3)$ and points $(-2,-5)$, $(2,-5)$, $(-4,-7)$, and $(4,-7)$. 2. **Identify the function forms:** - A V-shaped graph opening up or down is an absolute value function of the form: $$f(x) = a|x - h| + k$$ where $(h,k)$ is the vertex. 3. **Find $f(x)$:** - Vertex at $(0,0)$ means $h=0$, $k=0$. - Use point $(2,2)$ to find $a$: $$2 = a|2 - 0| + 0 = 2a \implies a = 1$$ - So, $$f(x) = |x|$$ 4. **Find $g(x)$:** - Vertex at $(0,-3)$ means $h=0$, $k=-3$. - The graph opens downwards, so $a$ is negative. - Use point $(2,-5)$: $$-5 = a|2 - 0| - 3 = 2a - 3$$ $$2a = -5 + 3 = -2 \implies a = -1$$ - So, $$g(x) = -|x| - 3$$ 5. **Summary:** - $$f(x) = |x|$$ - $$g(x) = -|x| - 3$$ These match the given points and vertex information.