1. **State the problem:** Convert the quadratic function $y = -2x^2 + 20x - 42$ into vertex form, identify the axis of symmetry, extrema (maximum or minimum), and zeros (x-intercepts). Then use these features to graph the function. 2. **Recall the vertex form:** The vertex form of a quadratic is $$y = a(x-h)^2 + k$$ where $(h,k)$ is the vertex. 3. **Complete the square to convert to vertex form:** Start with $$y = -2x^2 + 20x - 42$$ Factor out $-2$ from the first two terms: $$y = -2(x^2 - 10x) - 42$$ 4. **Complete the square inside the parentheses:** Take half of $-10$, which is $-5$, and square it: $(-5)^2 = 25$ Add and subtract 25 inside the parentheses: $$y = -2(x^2 - 10x + 25 - 25) - 42$$ Rewrite: $$y = -2((x - 5)^2 - 25) - 42$$ 5. **Distribute $-2$ and simplify:** $$y = -2(x - 5)^2 + 50 - 42$$ $$y = -2(x - 5)^2 + 8$$ 6. **Identify the vertex:** The vertex is at $(h, k) = (5, 8)$. 7. **Axis of symmetry:** The axis of symmetry is the vertical line $x = 5$. 8. **Extrema:** Since $a = -2 < 0$, the parabola opens downward, so the vertex is a maximum point at $(5, 8)$. 9. **Find zeros (x-intercepts):** Set $y=0$: $$0 = -2x^2 + 20x - 42$$ Divide both sides by $-2$: $$0 = \cancel{-2}x^2 - \cancel{20}x + \cancel{42} \Rightarrow 0 = x^2 - 10x + 21$$ 10. **Factor the quadratic:** $$x^2 - 10x + 21 = (x - 3)(x - 7)$$ 11. **Zeros:** $x = 3$ and $x = 7$ 12. **Y-intercept:** Set $x=0$: $$y = -2(0)^2 + 20(0) - 42 = -42$$ 13. **Summary of key features:** - Vertex: $(5, 8)$ - Axis of symmetry: $x = 5$ - Maximum value: $8$ at $x=5$ - Zeros: $x=3$ and $x=7$ - Y-intercept: $(0, -42)$ These features can be used to graph the parabola opening downward with vertex at $(5,8)$, crossing the x-axis near $3$ and $7$, and y-axis at $-42$.