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\( f(x)=-(x+1)^{2}+2 \) Make sure that you graph the vertex, the AOS, the 2 extra points and refelect them across the AOS.

Ask by Boone Ayala. in the United States
Mar 14,2025

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To graph the function \( f(x) = -(x+1)^2 + 2 \), follow these steps: 1. **Vertex**: \((-1, 2)\) 2. **Axis of Symmetry (AOS)**: \(x = -1\) 3. **Extra Points**: - \((0, 1)\) and its reflection \((-2, 1)\) - \((1, -2)\) and its reflection \((-3, -2)\) Plot these points and draw the parabola opening downward with the vertex at \((-1, 2)\) and the AOS at \(x = -1\).

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To graph the function \( f(x)=- (x+1)^{2}+2 \), we start by identifying its vertex, which is found at the point \( (-1, 2) \). The axis of symmetry (AOS) is the vertical line that runs through the vertex, given by \( x = -1 \). For additional points, we can select values of \( x \) around the vertex. Let's pick \( x = -2 \) and \( x = 0 \). Calculating these: 1. For \( x = -2 \): \[ f(-2) = -(-2 + 1)^{2} + 2 = -(-1)^{2} + 2 = -1 + 2 = 1 \Rightarrow (-2, 1) \] 2. For \( x = 0 \): \[ f(0) = -(0 + 1)^{2} + 2 = -(1)^{2} + 2 = -1 + 2 = 1 \Rightarrow (0, 1) \] Now, reflecting these points across the AOS: - The point \( (-2, 1) \) reflects to \( (0, 1) \). - The point \( (0, 1) \) reflects to \( (-2, 1) \). To summarize the key graph elements: - Vertex: \( (-1, 2) \) - Axis of Symmetry: \( x = -1 \) - Extra Points: \( (-2, 1) \) and \( (0, 1) \) So your graph will look like a downward-opening parabola with the vertex at the top, gracefully arching downwards and symmetrical around the AOS!

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