1. **State the problem:** We are given the function $f(x) = (x - 1)^{\frac{5}{2}}$ and want to understand its behavior and graph. 2. **Analyze the function:** The function is defined for $x \geq 1$ because the base $(x-1)$ must be non-negative to take a real-valued fractional power with denominator 2 (square root). 3. **Evaluate at key points:** - At $x=1$, $f(1) = (1-1)^{\frac{5}{2}} = 0$. - For $x > 1$, the function increases because raising a positive number to a positive power greater than 1 results in growth. 4. **Behavior near $x=1$:** Since the exponent is $\frac{5}{2} = 2.5$, the function grows like $(x-1)^{2.5}$ near 1, which is very flat near 1 but increases faster as $x$ moves away. 5. **Derivative for slope and extrema:** $$f'(x) = \frac{5}{2}(x-1)^{\frac{3}{2}}$$ Since $f'(x) > 0$ for $x > 1$, the function is strictly increasing on its domain. 6. **Summary:** The graph starts at $(1,0)$ and increases smoothly and steeply for $x > 1$ with no maxima or minima. **Final answer:** The function $f(x) = (x-1)^{\frac{5}{2}}$ is defined for $x \geq 1$, starts at zero at $x=1$, and increases monotonically with increasing slope as $x$ grows.