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Section 9.1 Identifying 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'll 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 how these two questions relate to the function notation:

Table 9.4. 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 9.5.

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.

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

Checkpoint 9.6.

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. Remember that horizontal transformations work backwards from what we expect, so this a horizontal shift to the right.

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

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.

Example 9.7.

Suppose \(f(x)=2x^2-7x+4\) and \(g(x)=10(x-4)^2-35(x-4)+20\text{.}\) What transformations took \(f(x)\) to \(g(x)\text{?}\)

Comparing the original function (\(f(x)\)) to the transformed function (\(g(x)\)), we see two key differences:

  • Everywhere we originally had \(x\) in the formula for \(f(x)\text{,}\) we now have \((x-4)\text{.}\)

    • Since the change is to the inputs (\(x\)), this is a horizontal transformation.

    • Since the change is adding/subtracting, this is a shift.

    • Since horizontal transformations are backwards from what we expect, this subtraction moves the graph in the positive \(x\)-direction (to the right).

    • This is a shift right by \(4\).

  • All of the coefficients were multiplied by \(5\text{.}\)

    • Since the change happened to the entire formula, it is changing the outputs, so it is a vertical change.

    • Since the change is multiplying/dividing by a positive number, it is a scale.

    • Since vertical transformations are exactly what we expect, this multiplication is a stretch.

    • This is a vertical stretch by \(5\).

Checkpoint 9.8.

Suppose \(f(x)=x^2+3x-2\) and \(g(x)=\dfrac{-(3x)^2-3(3x)+2}{5}\text{.}\) What transformations took \(f(x)\) to \(g(x)\text{?}\)

Answer.

There were three tranformations: a horizontal compression/squish by 3, a vertical compression/stretch by 5, and a vertical reflection over the \(x\)-axis.

Solution.

Comparing the original function (\(f(x)\)) to the transformed function (\(g(x)\)), we see three key differences:

  • Everywhere we originally had \(x\) in the formula for \(f(x)\text{,}\) we now have \((3x)\text{.}\)

    • Since the change is to the inputs (\(x\)), this is a horizontal transformation.

    • Since the change is multiplying/dividing, this is a scale.

    • Since horizontal transformations are backwards from what we expect, this multiplication squishes the graph.

    • This is a horizontal comprssion by \(3\).

  • The whole formula was divided by \(5\text{.}\)

    • Since the change happened to the entire formula, it is changing the outputs, so it is a vertical change.

    • Since the change is multiplying/dividing by a positive number, it is a scale.

    • Since vertical transformations are exactly what we expect, this division squishes the graph.

    • This is a vertical compression by \(5\).

  • 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).

    • Since the change happened to the entire formula, it is changing the outputs, so it is a vertical change.

    • Since the change is multiplying by \(-1\text{,}\) it is a reflection.

    • This is a vertical reflection.