Quadratic Function 059126

1. **State the problem:** We are given the function $$h(t) = -t^2 - 3t + 15$$ and we want to understand its properties. 2. **Identify the type of function:** This is a quadratic function of the form $$h(t) = at^2 + bt + c$$ where $$a = -1$$, $$b = -3$$, and $$c = 15$$. 3. **Important rules:** - The graph of a quadratic function is a parabola. - Since $$a = -1 < 0$$, the parabola opens downward. - The vertex formula for the parabola is $$t = -\frac{b}{2a}$$. 4. **Find the vertex:** $$t = -\frac{-3}{2 \times -1} = -\frac{-3}{-2} = -\frac{3}{2} = -1.5$$ 5. **Calculate the height at the vertex:** $$h(-1.5) = -(-1.5)^2 - 3(-1.5) + 15 = -2.25 + 4.5 + 15 = 17.25$$ 6. **Interpretation:** The vertex is at $$(-1.5, 17.25)$$, which is the maximum point of the parabola. 7. **Find the roots (where $$h(t) = 0$$):** Solve $$-t^2 - 3t + 15 = 0$$ or equivalently $$t^2 + 3t - 15 = 0$$. 8. **Use quadratic formula:** $$t = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-15)}}{2 \times 1} = \frac{-3 \pm \sqrt{9 + 60}}{2} = \frac{-3 \pm \sqrt{69}}{2}$$ 9. **Approximate roots:** $$\sqrt{69} \approx 8.3066$$ $$t_1 = \frac{-3 + 8.3066}{2} = 2.6533$$ $$t_2 = \frac{-3 - 8.3066}{2} = -5.6533$$ 10. **Summary:** - The parabola opens downward. - Maximum height is $$17.25$$ at $$t = -1.5$$. - Roots are approximately $$t = 2.65$$ and $$t = -5.65$$. This completes the analysis of the function $$h(t) = -t^2 - 3t + 15$$.