Question: Vertex Form y = -0.2(x - 2.47)^2 + 3.042 Factored Form y = -0.2(x + 1.43)(x - 6.37) Standard Form y = -0.2x^2 + 0.988x + 1.82 Here are three equivalent equations. Use the equivalent equations to identify the key features. The x-intercepts: The vertex: The y-intercept:

1. **State the problem:** We are given three equivalent forms of a quadratic function: - Vertex form: $$y = -0.2(x - 2.47)^2 + 3.042$$ - Factored form: $$y = -0.2(x + 1.43)(x - 6.37)$$ - Standard form: $$y = -0.2x^2 + 0.988x + 1.82$$ We need to identify the key features: the x-intercepts, the vertex, and the y-intercept. 2. **Find the x-intercepts:** The x-intercepts occur where $$y=0$$. Using the factored form: $$0 = -0.2(x + 1.43)(x - 6.37)$$ Since $$-0.2 \neq 0$$, the solutions come from: $$x + 1.43 = 0 \implies x = -1.43$$ $$x - 6.37 = 0 \implies x = 6.37$$ So, the x-intercepts are $$x = -1.43$$ and $$x = 6.37$$. 3. **Find the vertex:** The vertex form is: $$y = a(x - h)^2 + k$$ where $$(h,k)$$ is the vertex. From the vertex form given: $$y = -0.2(x - 2.47)^2 + 3.042$$ So, the vertex is at: $$(2.47, 3.042)$$ 4. **Find the y-intercept:** The y-intercept occurs when $$x=0$$. Using the standard form: $$y = -0.2(0)^2 + 0.988(0) + 1.82 = 1.82$$ So, the y-intercept is: $$(0, 1.82)$$ 5. **Summary of key features:** - **x-intercepts:** $$-1.43$$ and $$6.37$$ - **Vertex:** $$(2.47, 3.042)$$ - **y-intercept:** $$(0, 1.82)$$ 6. **Graph/table shapes:** - Since $$a = -0.2 < 0$$, the parabola opens downward. - The vertex is a maximum point. This completes the identification of the key features.