Question: Write the equation for $g(x)$ in terms of $f(x)$ in at least two different ways. g(x) = f(x) g(x) = g(x) = Graph: two sinusoidal curves on the same coordinate plane. The blue graph labeled $g(x)$ is the same shape as the red graph labeled $f(x)$, shifted upward by 8 units; equivalently, $f(x)$ is shifted downward by 8 units. position_hint = center User: how to do this

1. **State the problem:** We need to express $g(x)$ in terms of $f(x)$ given that the graph of $g(x)$ is the same as $f(x)$ but shifted upward by 8 units. 2. **Recall the vertical shift rule:** If a function $f(x)$ is shifted vertically by $k$ units, the new function is $f(x) + k$. 3. **Apply the rule:** Since $g(x)$ is $f(x)$ shifted upward by 8 units, we have: $$g(x) = f(x) + 8$$ 4. **Express $f(x)$ in terms of $g(x)$:** Since $f(x)$ is $g(x)$ shifted downward by 8 units, we can write: $$f(x) = g(x) - 8$$ 5. **Rewrite $g(x)$ in terms of $f(x)$ using the second expression:** From $f(x) = g(x) - 8$, solve for $g(x)$: $$g(x) = f(x) + 8$$ 6. **Alternative expression:** Another way to write $g(x)$ in terms of $f(x)$ is to isolate $g(x)$ from the first equation or use the inverse operation: $$g(x) - 8 = f(x)$$ or equivalently: $$g(x) = f(x) + 8$$ **Final answers:** $$g(x) = f(x) + 8$$ and equivalently, $$f(x) = g(x) - 8$$ These two equations show the relationship between $g(x)$ and $f(x)$ with the vertical shift of 8 units.