1. **Stating the problem:** We are given two functions related to sound intensity and decibels: $$d(Y) = 10 \log_{10}(Y \cdot 10^{12})$$ $$g(Y) = 10 \log_{10}(Y)$$ We want to understand the relationship between these two functions and how the graph of $d(Y)$ relates to $g(Y)$. 2. **Formula and rules:** The decibel level for intensity $Y$ is given by $10 \log_{10}(Y)$, which is a logarithmic measure of intensity. 3. **Rewrite $d(Y)$:** $$d(Y) = 10 \log_{10}(Y \cdot 10^{12}) = 10 \left(\log_{10}(Y) + \log_{10}(10^{12})\right)$$ Using the logarithm property $\log_{10}(a \cdot b) = \log_{10}(a) + \log_{10}(b)$. 4. **Simplify $d(Y)$:** $$d(Y) = 10 \log_{10}(Y) + 10 \times 12 = 10 \log_{10}(Y) + 120$$ 5. **Interpretation:** This shows that $d(Y)$ is just $g(Y)$ shifted vertically upward by 120 decibels. 6. **Graph relationship:** - The graph of $g(Y)$ is $10 \log_{10}(Y)$. - The graph of $d(Y)$ is $g(Y) + 120$. This vertical translation explains why $d(Y)$ starts near 120 decibels when $g(Y)$ is near 0. **Final answer:** $$d(Y) = g(Y) + 120$$ This means $d(Y)$ is the function $g(Y)$ shifted up by 120 decibels.