1. **State the problem:** We are given the quadratic expression $2x^2 - 11x + 9$ and want to analyze its graph and find the minimum point $P$. 2. **Rewrite the quadratic in vertex form:** The vertex form of a quadratic is $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex (minimum or maximum point). 3. **Complete the square:** Start with $$y = 2x^2 - 11x + 9$$ Factor out the coefficient of $x^2$ from the first two terms: $$y = 2\left(x^2 - \frac{11}{2}x\right) + 9$$ Find the term to complete the square: $$\left(\frac{\frac{11}{2}}{2}\right)^2 = \left(\frac{11}{4}\right)^2 = \frac{121}{16}$$ Add and subtract inside the parentheses: $$y = 2\left(x^2 - \frac{11}{2}x + \frac{121}{16} - \frac{121}{16}\right) + 9$$ Rewrite as: $$y = 2\left(\left(x - \frac{11}{4}\right)^2 - \frac{121}{16}\right) + 9$$ Distribute 2: $$y = 2\left(x - \frac{11}{4}\right)^2 - 2 \times \frac{121}{16} + 9 = 2\left(x - \frac{11}{4}\right)^2 - \frac{242}{16} + 9$$ Simplify constants: $$9 = \frac{144}{16}$$ So: $$y = 2\left(x - \frac{11}{4}\right)^2 - \frac{242}{16} + \frac{144}{16} = 2\left(x - \frac{11}{4}\right)^2 - \frac{98}{16}$$ Simplify fraction: $$\frac{98}{16} = \frac{49}{8}$$ Final vertex form: $$y = 2\left(x - \frac{11}{4}\right)^2 - \frac{49}{8}$$ 4. **Interpretation:** The vertex is at $$\left(\frac{11}{4}, -\frac{49}{8}\right)$$ which is the minimum point $P$ because the coefficient of the squared term ($2$) is positive. 5. **Summary:** The quadratic $2x^2 - 11x + 9$ has a minimum point at $$P = \left(\frac{11}{4}, -\frac{49}{8}\right)$$ and its graph is a parabola opening upwards.