When we are working with multiple transformations at once, those are the two key questions we need to ask. Here is a summary of how these two questions relate to the function notation:
Vertical transformations do exactly what we expect: adding 3 outside moves the graph up 3, while subtracting would move it down. However, horizontal transformations are backwards from what we expect:
To shift left (the negative \(x\)-direction), you have to add inside the function.
Vertical squish by \(b\text{:}\)\(\frac{1}{b}f(x)\) or \(\frac{f(x)}{b}\)
Horizontal stretch by \(b\text{:}\)\(f\left(\frac{1}{b}x\right)\) or \(f\left(\frac{x}{b}\right)\)
Multiply by \(-1\)
Vertical flip: \(-f(x)\)
Horizontal flip: \(f(-x)\)
Example9.1.5.
Suppose \(g(x)=2f(x+3)\) and we want to identify what tranformations took \(f(x)\) to \(g(x)\text{.}\) We see that there are two numbers we need to look at: the \(2\) and the \(3\text{:}\)
In the formula \(g(x)={\color{blue}{2}}f(x+3)\text{,}\) the \(2\) is on the outside of the function, so it’s a vertical transformation. Since it’s being multiplied, we know that it is a scale. Since vertical transformations behave the way we expect, this a vertical stretch.
In the formula \(g(x)=2f(x+{\color{blue}{3}})\text{,}\) the \(3\) is on the inside of the function, so it’s a horizontal transformation. Since it’s being added, we know that it is a shift. Since horizontal transformations behave backwards from what we expect, this a horizontal shift to the left.
We see that there are two numbers we need to look at: the \(-1\) and the \(7\text{:}\)
In the formula \(g(x)={\color{blue}{-}}f(x-7)\text{,}\) the \(-1\) is on the outside of the function, so it’s a vertical transformation. Since it’s multiplying by \(-1\text{,}\) we know that it is vertical reflection.
In the formula \(g(x)=-f(x-{\color{blue}{7}})\text{,}\) the \(7\) is on the inside of the function, so it’s a horizontal transformation. Since it’s being subtracted, we know that it is a shift. Remember that horizontal transformations work backwards from what we expect, so this a horizontal shift to the right.
Sometimes, we aren’t looking directly at the function notation. Instead, we are given the actual formula for the original function and the transformed function. In this case, the key is to look for what changed and where. Then, consult the table above to see what transformation that corresponds to.
There were three tranformations: a horizontal compression/squish by 3, a vertical compression/stretch by 5, and a vertical reflection over the \(x\)-axis.
All of the coefficients were multiplied by \(-1\text{.}\) You can see this because all of them flipped their sign (positive became negative and negative became positive).
