1. **State the problem:** We are given a function $f(x) = x$ and a graph of another function $g(x)$ which is a transformation of $f(x)$. We need to find the equation for $g(x)$ from the given options. 2. **Understand the transformations:** The function $f(x) = x$ is a straight line through the origin with slope 1. 3. **Analyze the graph of $g(x)$:** The graph of $g(x)$ is parallel to $f(x)$ but shifted upward and horizontally. It starts near $(0,8)$ and goes to $(4,10)$. 4. **Check vertical shift:** Since $f(0) = 0$ and $g(0) \approx 8$, there is a vertical shift of about 9 units (from options, vertical shifts are +9). 5. **Check horizontal shift and scaling:** The slope of $g$ is the same as $f$ (both lines are parallel), so the coefficient multiplying $f$ must be 1 or a factor that keeps slope 1. 6. **Test each option:** - Option A: $g(x) = \frac{1}{3} f(x + 3) + 9 = \frac{1}{3} (x + 3) + 9 = \frac{x}{3} + 1 + 9 = \frac{x}{3} + 10$ slope $= \frac{1}{3}$ (not parallel to $f$) - Option B: $g(x) = 3 f(x - 3) + 9 = 3(x - 3) + 9 = 3x - 9 + 9 = 3x$ slope $= 3$ (not parallel) - Option C: $g(x) = \frac{1}{3} f(x - 3) + 9 = \frac{1}{3} (x - 3) + 9 = \frac{x}{3} - 1 + 9 = \frac{x}{3} + 8$ slope $= \frac{1}{3}$ (not parallel) - Option D: $g(x) = 3 f(x + 3) + 9 = 3(x + 3) + 9 = 3x + 9 + 9 = 3x + 18$ slope $= 3$ (not parallel) 7. **Re-examine slope:** The graph of $g$ is parallel to $f$, so slope must be 1. None of the options have slope 1. 8. **Check if options might be misinterpreted:** Since $f(x) = x$, $f(x + 3) = x + 3$, $f(x - 3) = x - 3$. 9. **Try to find $g(x)$ from points:** Using points $(0,8)$ and $(4,10)$ for $g$: Slope $m = \frac{10 - 8}{4 - 0} = \frac{2}{4} = \frac{1}{2}$ Equation: $y - 8 = \frac{1}{2}(x - 0) \Rightarrow y = \frac{1}{2} x + 8$ 10. **Compare with options:** None match $g(x) = \frac{1}{2} x + 8$ exactly. 11. **Check if options can be simplified to $\frac{1}{2} x + 8$:** Option C: $g(x) = \frac{1}{3} (x - 3) + 9 = \frac{x}{3} - 1 + 9 = \frac{x}{3} + 8$ slope $= \frac{1}{3}$ close but not $\frac{1}{2}$. Option A: $\frac{x}{3} + 10$ no. Option B and D have slope 3. 12. **Conclusion:** None of the options exactly match the observed slope $\frac{1}{2}$. However, the closest is option C with slope $\frac{1}{3}$ and vertical shift 8. Since the problem asks to choose one answer, and the graph is shifted right by 3 units and up by 9 units, option C is the best fit. **Final answer:** $$g(x) = \frac{1}{3} f(x - 3) + 9$$