Recall that transformations can be applied to any function. So long as we know it is a function and can identify where the input of the function belongs, we can apply our knowledge of transformations to that function. This example takes a function you *might* be familiar with, but we can apply our transformations whether you know what the function is, what it looks like, or not! As long as you can identify the input portion of the function, you can write a newly transformed function by applying the described transformations.
Using the description, rewrite the function equation after applying all the transformations. Your final answer will have \(\sin\) in it, *not* \(f\).
\[ f(x) = \sin(x) \]Reflect the function over the \(x\)-axis, vertically stretch by a factor of 10, shift right by \(\frac{\pi}{{2}}\) units and up by 5 units.
Solution
I will note each transformation in order of operations:
\[ \begin{{array}}{{rclr}} y &=&-f(x)&\text{ Reflection over }x\text{-axis.}\\ y &=&-10f(x)&\text{Vertical Stretch by factor of 10}\\ y &=&-10f\left(x-\frac{\pi}{{2}}\right)&\text{Shift right by }\frac{\pi}{{2}}\text{ units}\\ y &=&-10f\left(x-\frac{\pi}{{2}}\right)+5&\text{Shift up by }5 \text{ units} \end{{array}} \]