1. **State the problem:** We need to graph the quadratic function $$y = -x^2 - 10x - 21$$ and plot 5 points including the roots and the vertex. 2. **Formula and rules:** The quadratic function is in the form $$y = ax^2 + bx + c$$ where $$a = -1$$, $$b = -10$$, and $$c = -21$$. 3. **Find the vertex:** The vertex $$x$$-coordinate is given by $$x = -\frac{b}{2a} = -\frac{-10}{2 \times -1} = -\frac{-10}{-2} = -5$$. 4. **Calculate the vertex $$y$$-coordinate:** $$y = -(-5)^2 - 10(-5) - 21 = -25 + 50 - 21 = 4$$. So the vertex is at $$(-5, 4)$$. 5. **Find the roots:** Solve $$-x^2 - 10x - 21 = 0$$. Multiply both sides by $$-1$$ to simplify: $$\cancel{-1} \times (-x^2 - 10x - 21) = \cancel{-1} \times 0$$ $$x^2 + 10x + 21 = 0$$. 6. **Factor the quadratic:** $$x^2 + 10x + 21 = (x + 3)(x + 7) = 0$$. 7. **Roots are:** $$x = -3$$ and $$x = -7$$. 8. **Calculate $$y$$ values for points near the vertex:** - For $$x = -4$$: $$y = -(-4)^2 - 10(-4) - 21 = -16 + 40 - 21 = 3$$. - For $$x = -6$$: $$y = -(-6)^2 - 10(-6) - 21 = -36 + 60 - 21 = 3$$. 9. **Points to plot:** - Roots: $$(-3, 0)$$ and $$(-7, 0)$$ - Vertex: $$(-5, 4)$$ - Additional points: $$(-4, 3)$$ and $$(-6, 3)$$ These 5 points show the shape of the parabola. **Final answer:** The parabola opens downward with vertex at $$(-5, 4)$$ and roots at $$(-3, 0)$$ and $$(-7, 0)$$. Additional points $$(-4, 3)$$ and $$(-6, 3)$$ help plot the curve.