Transformations

Given the curve of a given function $y=f(x)$ , they may require you to sketch transformations of the curve. Transformations can shift, stretch and flip the curve of a function. Don’t confuse these with the shape transformations in coordinate geometry at GCSE (transformations at GCSE).

y-transformations

A $y$ -transformation affects the y coordinates of a curve. You can identify a $y$ -transformation as changes are made outside the brackets of $y=f(x)$ . We examine $y$ -transformations first since they behave as expected.

Upward shift: $f(x)\rightarrow f(x)+4$ , this is a $y$ -shift. This does not affect $x$ coordinates but all the $y$ coordinates go up by 4.
Downward shift: $f(x)\rightarrow f(x)-3$ , this is also a $y$ -shift. This does not affect $x$ coordinates but all the $y$ coordinates go down by 3.
Vertical stretch: $f(x)\rightarrow 2f(x)$ , this is a $y$ -stretch. This does not affect $x$ coordinates but all the $y$ coordinates are doubled.
Reflect in x-axis: $f(x)\rightarrow -f(x)$ , this is a flip in $y$ . This does not affect $x$ coordinates but all the $y$ coordinates are flipped across the $x$ -axis.

x-transformations

$x$ -transformations always behave in the opposite way to what is expected. They can be identified when changes are made inside the brackets of $y=f(x)$ .

Left shift: $f(x)\rightarrow f(x+4)$ , this is an $x$ -shift. This does not affect $y$ coordinates but all the $x$ coordinates go to the left by 4, the opposite direction to what is expected.
Right shift: $f(x)\rightarrow f(x-3)$ , this is also an $x$ -shift. This does not affect $y$ coordinates but all the $x$ coordinates go to the right by 3, the opposite direction to what is expected.
Shrink in x: $f(x)\rightarrow f(2x)$ , this is a stretch in the $x$ direction. This does not affect $y$ coordinates but all the $x$ coordinates are halved, the opposite to what is expected.
Reflect in y-axis: $f(x)\rightarrow f(-x)$ , this is a flip in the $x$ direction. This does not affect $y$ coordinates but all the $x$ coordinates are flipped across the $y$ -axis.

Note that $y$ -transformations usually behave as expected as opposed to $x$ -transformations that seem to do the opposite. See the Transformations Questions by Topic to practice exam-style questions at the basic level.

See more on Curve Sketching.

Compound Transformations

At more advanced levels, exam questions may expect you to apply compound transformations. This means applying more than one transformation. It is very important that they are applied in the correct order – see Example 1. An exam question may expect you to apply compound transformations to a given curve or possibly even known graphs – see videos. This can also include trigonometric graphs – see trigonometry examples. A question may also ask you to apply transformations to the exponential curve – see Example 2. Finally, recall that completing the square allows you to sketch the curve for a given quadratic via transformations – see Board Example.

Examples

Sketch the graph of $y=\ln(3-2x)$ .

Solution:

First sketch the graph of $y=\ln(x)$ and on the same plot sketch the graph of $y=\ln(x+3)$ . This transformation involves replacing x with x+3 and so this is a shift to the left by 3 (see Transformations).

The next transformation is replacing x with -x which is a reflection across the y-axis:

The final transformation is obtained by replacing x with 2x. This is squeezes the graph towards the y-axis and the final result is:

Sketch the curve of $y=e^{2x+1}-3$ .

Solution:

We can sketch this curve by starting with the graph of $y=e^x$ . Firstly, we replace $x$ with $x+1$ . which shifts the graph to the left by 1. Next we replace $x$ with $2x$ which shrinks the curve by a factor of 2 in the $x$ -direction. The final transformation shifts the graph down by 3:

Videos

https://youtu.be/4NPl1c41t10

Transformations 1 (https://youtu.be/4NPl1c41t10)

Sketching a reciprocal function using transformations and performing various additional transformations to the original curve.

Extra Resources

Questions by Topic

TransformationsExamQuestions

Open Transformations Questions by Topic in New Window

Board Example

AS Maths Algebra & Functions

A2 Maths Algebra & Functions

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