A 100 -gallon fish tank fills at a rate of \( x \) gallons per minute. The tank has already been filling for 5 minutes. The function \( f(x)=\frac{100}{x}-5 \) represents the remaining time in minutes needed to fill the tank. How is the graph of the paren function \( f(x)=\frac{1}{x} \) transformed to create the graph of the function \( f(x)=\frac{100}{x}-5 \) ? It is a vertical stretch with a factor of 100 and a translation 5 units right. It is a vertical stretch with a factor of 95 . It is a vertical stretch with a factor of 100 and a translation 5 units down. It is a vertical stretch with a factor of 500 .

The function \( f(x) = \frac{100}{x} - 5 \) is derived from the basic function \( f(x) = \frac{1}{x} \) through two transformations: a vertical stretch and a vertical translation. The stretching happens because the numerator is multiplied by 100, leading to a vertical stretch by a factor of 100. Then, the subtraction of 5 shifts the entire graph down by 5 units. So the correct transformation is indeed a vertical stretch with a factor of 100 and a translation 5 units down. As for common mistakes in interpreting transformations, it's easy to confuse horizontal and vertical shifts. Remember, subtracting from the function value affects the vertical position of the graph, whereas adjustments to the input (inside the function) would shift it horizontally. Keep an eye on those details!