Questions: State the various transformations applied to the base function f(x)=x^2 to obtain a graph of the function g(x)=-2[(x-1)^2+3]. A reflection about the y-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. A reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units. A reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. A reflection about the y-axis, a vertical stretch by a factor of 2, a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units.

State the various transformations applied to the base function f(x)=x^2 to obtain a graph of the function g(x)=-2[(x-1)^2+3]. A reflection about the y-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. A reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units. A reflection about the x-axis, a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. A reflection about the y-axis, a vertical stretch by a factor of 2, a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units.
Transcript text: 8. State the various transformations applied to the base function $f(x)=x^{2}$ to obtain a graph of the function $g(x)=-2\left[(x-1)^{2}+3\right]$. A reflection about the $y$-axis, a vertical stretch by a factor of 2 , a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. A reflection about the $x$-axis, a vertical stretch by a factor of 2 , a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units. A reflection about the $x$-axis, a vertical stretch by a factor of 2 , a horizontal shift of 1 unit to the right, and a vertical shift downward of 6 units. A reflection about the $y$-axis, a vertical stretch by a factor of 2 , a horizontal shift of 2 units to the right, and a vertical shift downward of 6 units.
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Solution

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Solution Steps

Step 1: Identify the base function

The base function is \( f(x) = x^{2} \).

Step 2: Apply horizontal shift

The term \( (x-1)^{2} \) indicates a horizontal shift. Specifically, the graph of \( f(x) \) is shifted 1 unit to the right. This transformation can be written as: \[ f(x) \rightarrow f(x-1) = (x-1)^{2}. \]

Step 3: Apply vertical shift

The term \( +3 \) inside the brackets indicates a vertical shift. The graph of \( f(x-1) \) is shifted 3 units upward. This transformation can be written as: \[ f(x-1) \rightarrow f(x-1) + 3 = (x-1)^{2} + 3. \]

Step 4: Apply vertical scaling

The coefficient \( -2 \) outside the brackets indicates a vertical scaling and reflection. The graph of \( f(x-1) + 3 \) is vertically stretched by a factor of 2 and reflected across the x-axis. This transformation can be written as: \[ f(x-1) + 3 \rightarrow -2\left[(x-1)^{2} + 3\right]. \]

Final Answer

The transformations applied to the base function \( f(x) = x^{2} \) to obtain \( g(x) = -2\left[(x-1)^{2} + 3\right] \) are:

  1. Horizontal shift: 1 unit to the right.
  2. Vertical shift: 3 units upward.
  3. Vertical scaling: Stretched by a factor of 2 and reflected across the x-axis.

\[ \boxed{g(x) = -2\left[(x-1)^{2} + 3\right]} \]

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