1. **State the problem:** Convert the quadratic function $y = -\frac{1}{2}(x + 6)^2 + 2$ into vertex form, zero form, and standard form. Find the vertex coordinates, zeros, max/min, axis of symmetry, optimal value, y-intercept, and step pattern. 2. **Identify vertex form:** The given function is already in vertex form: $$y = a(x - h)^2 + k$$ where $a = -\frac{1}{2}$, $h = -6$, and $k = 2$. 3. **Vertex coordinates:** The vertex is at $(h, k) = (-6, 2)$. 4. **Axis of symmetry:** The axis of symmetry is the vertical line $x = h = -6$. 5. **Max or min:** Since $a = -\frac{1}{2} < 0$, the parabola opens downward, so the vertex is a maximum point. 6. **Optimal value:** The maximum value of $y$ is the $y$-coordinate of the vertex, which is $2$. 7. **Convert to standard form:** Expand the vertex form: $$y = -\frac{1}{2}(x + 6)^2 + 2 = -\frac{1}{2}(x^2 + 12x + 36) + 2$$ $$= -\frac{1}{2}x^2 - 6x - 18 + 2$$ $$= -\frac{1}{2}x^2 - 6x - 16$$ 8. **Convert to zero form (factored form):** Set $y=0$ and solve for $x$: $$0 = -\frac{1}{2}(x + 6)^2 + 2$$ $$\frac{1}{2}(x + 6)^2 = 2$$ $$ (x + 6)^2 = 4$$ $$x + 6 = \pm 2$$ So zeros are: $$x = -6 + 2 = -4$$ $$x = -6 - 2 = -8$$ Zero form: $$y = -\frac{1}{2}(x + 4)(x + 8)$$ 9. **Y-intercept:** Set $x=0$ in standard form: $$y = -\frac{1}{2}(0)^2 - 6(0) - 16 = -16$$ So the y-intercept is $(0, -16)$. 10. **Step pattern:** Since $a = -\frac{1}{2}$, the parabola opens downward and the step pattern from the vertex is: - Move 1 unit right or left in $x$, $y$ changes by $-\frac{1}{2} \times 1^2 = -\frac{1}{2}$ - Move 2 units right or left in $x$, $y$ changes by $-\frac{1}{2} \times 2^2 = -2$ - Move 3 units right or left in $x$, $y$ changes by $-\frac{1}{2} \times 3^2 = -\frac{9}{2} = -4.5$ This means from the vertex at $(-6, 2)$, the points go down by these amounts as you move horizontally. **Final answers:** - Vertex form: $y = -\frac{1}{2}(x + 6)^2 + 2$ - Standard form: $y = -\frac{1}{2}x^2 - 6x - 16$ - Zero form: $y = -\frac{1}{2}(x + 4)(x + 8)$ - Vertex: $(-6, 2)$ - Zeros: $x = -8, -4$ - Max value: $2$ at $x = -6$ - Axis of symmetry: $x = -6$ - Y-intercept: $(0, -16)$ - Step pattern: down $\frac{1}{2}$, down $2$, down $4.5$ for steps 1, 2, 3 from vertex