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Section 5.6 Combining Transformations

Each transformation on its own isn't too bad, but the hard part about this topic for most students is keeping track of all the different ones. We'll discuss that in this section.

Each of the transformations we've discussed in this chapter have two properties:

  • The Direction of Transformation: Is it vertial or horizontal?

  • The Kind of Transformation: Is it a shift, scale, or reflection?

When we are working with multiple transformations at once, those are the two key questions we need to ask. Here is a summary of what we know about each transformation we have talked about:

Table 5.71. Summary of Transformations
Vertical (outside) Horizontal (inside)
Add Shift up by \(a\text{:}\) \(f(x)+a\) Shift left by \(a\text{:}\) \(f(x+a)\)
Subtract Shift down by \(a\text{:}\) \(f(x)-a\) Shift right by by \(a\text{:}\) \(f(x-a)\)
Multiply (\(b\gt 1\)) Vertical stretch by \(b\text{:}\) \(bf(x)\) Horizontal squish by \(b\text{:}\)  \(f(bx)\)
Divide by \(b\gt 1\) 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: \(f(-x)\)

Example 5.72.

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. We know from Section 5.3 that a vertical scale by a number greater than \(1\) is 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. We know from Section 5.2 that a horizontal shift by a positive number is a horizontal shift to the left.

Therefore, the answer is that there is a vertical stretch by 2 and a horizontal shift left by 3.

Checkpoint 5.73.

Suppose \(g(x)=-f(x-7)\text{.}\) What transformations took \(f(x)\) to \(g(x)\text{?}\)

Answer.

There is a vertical reflection and a horizontal shift right by 7.

Solution.

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. We know from Section 5.2 that a horizontal shift by a negative number is a horizontal shift to the right.

Therefore, the answer is that there is a vertical reflection and a horizontal shift right by 7.

Example 5.74.

Suppose the solid blue graph below is the graph of \(f(x)\) and the dotted red graph is the graph of \(g(x)\text{:}\)

We want to write a formula for \(g(x)\) in terms of \(f(x)\text{.}\) In other words, we start with \(f(x)\) and figure out how the graph changed to get to \(g(x)\text{,}\) then write how that changes the formula. Since the question asks for the formula "in terms of \(f(x)\)", that means we will write \(f(x)\) somewhere in our answer. Looking at the graph, we see that \(g(x)\) is the graph of \(f(x)\text{,}\) but it has been flipped vertically (over the \(x\)-axis) and shifted to the right (in the positive \(x\)-direction) by \(5\text{.}\) The vertical flip means we are multiplying by \(-1\) outside of the function \(f(x)\text{,}\) and the shift to the right means we are subtracting \(5\) from the inside of the function. So, our final answer is \(g(x)=-f\left(x-5\right)\text{.}\)

Checkpoint 5.75.

Suppose the solid blue graph below is the graph of \(f(x)\) and the dotted red graph is the graph of \(g(x)\text{:}\)

Write a formula for \(g(x)\) in terms of \(f(x)\text{.}\)

Answer.

\(g(x)=f\left(x+2\right)-5\)

Solution.

We want to write a formula for \(g(x)\) in terms of \(f(x)\text{.}\) In other words, we start with \(f(x)\) and figure out how the graph changed to get to \(g(x)\text{,}\) then write how that changes the formula. Since the question asks for the formula "in terms of \(f(x)\)", that means we will write \(f(x)\) somewhere in our answer. Looking at the graph, we see that \(g(x)\) is the graph of \(f(x)\text{,}\) but it has been shifted down by \(5\) and shifted to the left (in the negative \(x\)-direction) by \(2\text{.}\) The vertical shift down means we are subtracting \(5\) from the outside of the function \(f(x)\text{,}\) and the shift to the left means we are adding \(2\) to the inside of the function. So, our final answer is \(g(x)=f\left(x+2\right)-5\text{.}\)