Given the curve of a given function y=f(x), you may be required to sketch transformations of the curve. Transformations can shift, stretch and flip the curve of a function.

## 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).

• Upward shift: $f(x)\rightarrow f(x)+4$, this is a shift in y. The x coordinates are unaffected but all the y coordinates go up by 4.
• Downward shift: $f(x)\rightarrow f(x)-3$, this is a shift in y. The x coordinates are unaffected but all the y coordinates go down by 3.
• Vertical stretch: $f(x)\rightarrow 2f(x)$, this is a stretch in y. The x coordinates are unaffected but all the y coordinates are doubled.
• Reflect in x-axis: $f(x)\rightarrow -f(x)$, this is a flip in y. The x coordinates are unaffected 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 a shift in the x direction. The y coordinates are unaffected 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 a shift in the x direction The y coordinates are unaffected 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. The y coordinates are unaffected 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. The y coordinates are unaffected 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.

## Example

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

First draw the graph of ln(x)

Now replace x with x+3

Then replace x with -x

Finally, replace x with 2x

### Transformations Exam Question

This exam question involves completing the square and using the transformations to sketch the curve of the quadratic.

See Curve Sketching for some more practice questions on Transformations.

JOKE: I’ll do algebra, geometry, trigonometry and probability…. but graphing is where I draw the line.

More graph jokes. 🙂

Also see other Things to Remember.

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