Questions: 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-3)^2+1 Which transformations are needed to graph the function f(x)=2(x-3)^2+1? Choose the correct answer below. A. The graph of y=x^2 should be horizontally shifted to the left by 3 units, vertically compressed by a factor of 2, and shifted vertically up by 1 units. B. The graph of y=x^2 should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2, and shifted vertically down by 1 units. C. The graph of y=x^2 should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2, and shifted vertically up by 1 units. D. The graph of y=x^2 should be horizontally shifted to the left by 3 units, vertically compressed by a factor of 2, and shifted vertically down by 1 units.

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-3)^2+1 Which transformations are needed to graph the function f(x)=2(x-3)^2+1? Choose the correct answer below. A. The graph of y=x^2 should be horizontally shifted to the left by 3 units, vertically compressed by a factor of 2, and shifted vertically up by 1 units. B. The graph of y=x^2 should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2, and shifted vertically down by 1 units. C. The graph of y=x^2 should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2, and shifted vertically up by 1 units. D. The graph of y=x^2 should be horizontally shifted to the left by 3 units, vertically compressed by a factor of 2, and shifted vertically down by 1 units.
Transcript text: 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. \[ f(x)=2(x-3)^{2}+1 \] Which transformations are needed to graph the function $f(x)=2(x-3)^{2}+1$ ? Choose the correct answer below. A. The graph of $y=x^{2}$ should be horizontally shifted to the left by 3 units, vertically compressed by a factor of 2 , and shifted vertically up by 1 units. B. The graph of $y=x^{2}$ should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2 , and shifted vertically down by 1 units. C. The graph of $y=x^{2}$ should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2 , and shifted vertically up by 1 units. D. The graph of $y=x^{2}$ should be horizontally shifted to the left by 3 units, vertically compressed by a factor of 2 , and shifted vertically down by 1 units.
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Solution

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Solution Steps

Step 1: Identify the transformations

The function \( f(x) = 2(x-3)^2 + 1 \) can be derived from the basic function \( y = x^2 \) through the following transformations:

  • Horizontally shift to the right by 3 units.
  • Vertically stretch by a factor of 2.
  • Vertically shift up by 1 unit.
Step 2: Choose the correct answer

The correct answer is: C. The graph of \( y = x^2 \) should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2, and shifted vertically up by 1 unit.

Step 3: Identify key points

Key points on the graph of \( y = x^2 \) are:

  • (0, 0)
  • (1, 1)
  • (-1, 1)

Applying the transformations to these points:

  • (0, 0) becomes (3, 1)
  • (1, 1) becomes (4, 3)
  • (-1, 1) becomes (2, 3)
Step 4: Determine the domain and range

The domain of \( f(x) = 2(x-3)^2 + 1 \) is all real numbers, \( (-\infty, \infty) \).

The range of \( f(x) = 2(x-3)^2 + 1 \) is \( [1, \infty) \).

Final Answer

The correct answer is C. The graph of \( y = x^2 \) should be horizontally shifted to the right by 3 units, vertically stretched by a factor of 2, and shifted vertically up by 1 unit.

The key points after transformation are:

  • (3, 1)
  • (4, 3)
  • (2, 3)

The domain is \( (-\infty, \infty) \).

The range is \( [1, \infty) \).

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