Question

The graph below is a transformation of f(x)= \frac{1}{4} ^x . Write an equation describing the transformation.exponential. f(x) approaches negative infinity as x approaches negative infinity. As x approaches infinity f(x) approaches -2The coefficient on our transformed function is AnswerThe exponent on our transformed function is AnswerThe constant we are adding to our function f(x)= \frac{1}{4} ^x is Answer

Answer 1

The original function undergoing the transformation is:

$f(x)=\frac{1}{4}^x$

This function is shown below:

The function is transformed by reflecting it over the x-axis and shifting downwards by 2 units.

Reflection over the x-axis has the rule:

$f(x)\to-f(x)$

Therefore, the function becomes:

$f^{\prime}(x)=-\frac{1}{4}^x$

Shifting downwards by 2 units has the rule:

$f(x)\to f(x)-2$

The new function becomes:

$f^{\prime}(x)=-\frac{1}{4}^x-2$

To check if the function represents the transformation, we can use the provided point:

$\begin{gathered} (x,y)=(-1,-6) \\ \therefore \\ -6=-\frac{1}{4}^{-1}-2 \\ -6=-4-2 \\ -6=-6(True\text{)} \end{gathered}$

Therefore, the transformed function is:

$f^{\prime}(x)=-\frac{1}{4}^x-2$

$-2$