3. Given: \( f(x)=\left(\frac{1}{4}\right)^{x} \) Describe the transformation: 4. Given: \( f(x)=\left(\frac{1}{4}\right)^{x} \) Describe the transformation: Transformation: \( g(x)=\left(\frac{1}{4}\right)^{x}-3 \) 5. Given: \( f(x)=\left(\frac{1}{4}\right)^{x} \) Describe the transformation:

To describe the transformations of the function \( f(x) = \left(\frac{1}{4}\right)^{x} \), we will analyze the changes made to the function in each case. ### 3. Transformation of \( f(x) = \left(\frac{1}{4}\right)^{x} \) The function \( f(x) = \left(\frac{1}{4}\right)^{x} \) is an exponential decay function. The base \( \frac{1}{4} \) is less than 1, which means that as \( x \) increases, \( f(x) \) decreases towards 0. ### 4. Transformation: \( g(x) = \left(\frac{1}{4}\right)^{x} - 3 \) In this case, the transformation from \( f(x) \) to \( g(x) \) involves a vertical shift. - **Vertical Shift Downward**: The function \( g(x) = f(x) - 3 \) indicates that the entire graph of \( f(x) \) is shifted downward by 3 units. This means that every point on the graph of \( f(x) \) is moved down 3 units. For example, if \( f(0) = 1 \), then \( g(0) = 1 - 3 = -2 \). ### 5. General Description of Transformation If we consider a general transformation of the form \( g(x) = f(x) + k \), where \( k \) is a constant: - If \( k > 0 \), the graph of \( f(x) \) shifts upward by \( k \) units. - If \( k < 0 \), the graph of \( f(x) \) shifts downward by \( |k| \) units. In the specific case of \( g(x) = \left(\frac{1}{4}\right)^{x} - 3 \), we have \( k = -3 \), indicating a downward shift of 3 units. In summary, the transformation from \( f(x) \) to \( g(x) \) is a vertical shift downward by 3 units.