1. **State the problem:** We need to graph the quadratic function $$y = x^2 + 8x + 7$$ and plot 5 points including the roots and the vertex. 2. **Formula and rules:** The quadratic function is in standard form $$y = ax^2 + bx + c$$ where $$a=1$$, $$b=8$$, and $$c=7$$. 3. **Find the roots:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. Calculate the discriminant: $$\Delta = b^2 - 4ac = 8^2 - 4 \times 1 \times 7 = 64 - 28 = 36$$ Calculate the roots: $$x = \frac{-8 \pm \sqrt{36}}{2 \times 1} = \frac{-8 \pm 6}{2}$$ Root 1: $$x = \frac{-8 + 6}{2} = \frac{-2}{2} = -1$$ Root 2: $$x = \frac{-8 - 6}{2} = \frac{-14}{2} = -7$$ 4. **Find the vertex:** The vertex $$x$$-coordinate is given by $$x = -\frac{b}{2a} = -\frac{8}{2 \times 1} = -4$$. Calculate the vertex $$y$$-coordinate: $$y = (-4)^2 + 8(-4) + 7 = 16 - 32 + 7 = -9$$ So the vertex is $$(-4, -9)$$. 5. **Choose two more points:** Pick $$x = -5$$ and $$x = -3$$ to find corresponding $$y$$ values. For $$x = -5$$: $$y = (-5)^2 + 8(-5) + 7 = 25 - 40 + 7 = -8$$ For $$x = -3$$: $$y = (-3)^2 + 8(-3) + 7 = 9 - 24 + 7 = -8$$ 6. **Summary of points to plot:** - Roots: $$(-7, 0)$$ and $$(-1, 0)$$ - Vertex: $$(-4, -9)$$ - Additional points: $$(-5, -8)$$ and $$(-3, -8)$$ These points define the parabola shape. 7. **Final answer:** The graph of $$y = x^2 + 8x + 7$$ has roots at $$x = -7$$ and $$x = -1$$, vertex at $$(-4, -9)$$, and passes through points $$(-5, -8)$$ and $$(-3, -8)$$.