Minimum Value 024A32

1. **State the problem:** We need to find the minimum value of the quadratic function $$f(x) = x^2 - 10x + 22.8$$ to the nearest hundredth. 2. **Recall the formula for the vertex of a quadratic function:** For a quadratic function in the form $$f(x) = ax^2 + bx + c$$, the vertex (which gives the minimum or maximum value) occurs at $$x = -\frac{b}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -10$$, and $$c = 22.8$$. 4. **Calculate the x-coordinate of the vertex:** $$x = -\frac{-10}{2 \times 1} = \frac{10}{2} = 5$$ 5. **Find the minimum value by evaluating $$f(x)$$ at $$x=5$$:** $$f(5) = 5^2 - 10 \times 5 + 22.8 = 25 - 50 + 22.8 = -25 + 22.8 = -2.2$$ 6. **Conclusion:** The minimum value of the function is $$-2.2$$, which to the nearest hundredth is $$-2.20$$. **Final answer:** The minimum value of $$f(x)$$ is $$\boxed{-2.20}$$.