Questions: The function f(x)=-2 * 4^(x-6)+2, has which of the following transformations? Select all that apply. Compressed by a factor of 2 Stretched by a factor of 2 Reflected over the x-axis Translated left 6 units Translated down 2 units Reflected over the y-axis Translated up 2 units Translated right 6 units

The function f(x)=-2 * 4^(x-6)+2, has which of the following transformations? Select all that apply. Compressed by a factor of 2 Stretched by a factor of 2 Reflected over the x-axis Translated left 6 units Translated down 2 units Reflected over the y-axis Translated up 2 units Translated right 6 units
Transcript text: The function $f(x)=-2 \cdot 4^{x-6}+2$, has which of the following transformations? Select all that apply. Compressed by a factor of 2 Stretched by a factor of 2 Reflected over the $x$-axis Translated left 6 units Translated down 2 units Reflected over the $y$-axis Translated up 2 units Translated right 6 units
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Solution

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Solution Steps

To determine the transformations of the function \( f(x) = -2 \cdot 4^{x-6} + 2 \), we need to analyze each component of the function:

  1. The negative sign in front of the coefficient indicates a reflection over the \( x \)-axis.
  2. The coefficient \( -2 \) indicates a vertical stretch by a factor of 2.
  3. The term \( x-6 \) inside the exponent indicates a horizontal translation to the right by 6 units.
  4. The constant term \( +2 \) outside the exponential function indicates a vertical translation up by 2 units.
Step 1: Identify the Transformations

To determine the transformations of the function \( f(x) = -2 \cdot 4^{x-6} + 2 \), we analyze each component of the function:

  1. The negative sign in front of the coefficient indicates a reflection over the \( x \)-axis.
  2. The coefficient \( -2 \) indicates a vertical stretch by a factor of 2.
  3. The term \( x-6 \) inside the exponent indicates a horizontal translation to the right by 6 units.
  4. The constant term \( +2 \) outside the exponential function indicates a vertical translation up by 2 units.
Step 2: List the Transformations

Based on the analysis, we can list the transformations as follows:

  • Compressed by a factor of 2: False
  • Stretched by a factor of 2: True
  • Reflected over the \( x \)-axis: True
  • Translated left 6 units: False
  • Translated down 2 units: False
  • Reflected over the \( y \)-axis: False
  • Translated up 2 units: True
  • Translated right 6 units: True

Final Answer

The function \( f(x) = -2 \cdot 4^{x-6} + 2 \) has the following transformations:

  • Stretched by a factor of 2
  • Reflected over the \( x \)-axis
  • Translated up 2 units
  • Translated right 6 units

\(\boxed{\text{Stretched by a factor of 2, Reflected over the } x \text{-axis, Translated up 2 units, Translated right 6 units}}\)

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