Graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function \( y=x^{2} \) and show all stages. Be sure to identify at least three key points. Find the domain and the range of the function. \( f(x)=2(x+2)^{2}-1 \) Which transformations are needed to graph the function \( f(x)=2(x+2)^{2}-1 \) ? Choose the correct answer below. A. une graph of \( y=x^{2} \) should be horizontally shifted to the left by 2 1 units. The graph of \( y=x^{2} \) should be horizontally shifted to the left by 2 B. units, vertically stretched by a factor of 2 , and shifted vertically down by a factor of 2 , and shifted vertically up by by 1 units. The graph of \( y=x^{2} \) should be horizontally shifted to the right by 2 c. units, vertically compressed by a factor of 2 , and shifted vertically down by 1 units. The graph of \( y=x^{2} \) should be horizontally shifted to the right by 2 D. units, vertically compressed by a factor of 2 , and shifted vertically up by 1 units.

We start with the base function y = x² and then perform the following transformations in order: 1. Horizontal Shift: Replace x with (x + 2). This shifts the graph 2 units to the left. (The vertex moves from (0, 0) to (–2, 0).) 2. Vertical Stretch: Multiply the function by 2, which stretches it vertically by a factor of 2. 3. Vertical Shift: Subtract 1 from the function, shifting it downward by 1 unit. (Now the vertex is at (–2, –1).) Key points on the final graph include: • Vertex: (–2, –1) • A point originally at (1, 1) on y = x² moves as follows:   - After the left shift: (–1, 1)   - After the vertical stretch: (–1, 2)   - After the vertical shift: (–1, 1) • Another point originally at (–1, 1) on y = x² moves to:   - After the left shift: (–3, 1)   - After the vertical stretch: (–3, 2)   - After the vertical shift: (–3, 1) The domain of f(x) = 2(x + 2)² – 1 is all real numbers, (–∞, ∞), and the range is [–1, ∞) (since the minimum value is achieved at the vertex). Of the answer choices given, the correct transformation description is: A. The graph of y=x² should be horizontally shifted to the left by 2 units, vertically stretched by a factor of 2, and shifted vertically down by 1 unit. Thus, the correct answer is option A.