Translating graphs of functions

This has the effect of making the y‑coordinate the negative of whatever it was.

Note that the x‑coordinate is untouched.

This has the effect of making the x‑coordinate the negative of whatever it was.

Note that the y‑coordinate is untouched.

This shifts the entire graph horizontally left or right.

Remember to go the opposite way to the sign:
For \((x - c)\), go right \(c\) spaces.

For \((x + c)\), go left \(c\) spaces.

Note that the y‑coordinate is untouched.

This shifts the entire graph vertically up or down.

If \(y = f(x) + d\), go up \(d\) spaces:

If \(y = f(x) - d\), go down \(d\) spaces:

Note that the x‑coordinate is untouched.

This squashes the graph horizontally when \(b > 1\) and stretches it horizontally when \(b \lt 1\).

If \(y = f(bx)\), divide the x‑coordinate by \(b\):

If \(y = f(x/b)\), multiply the x‑coordinate by \(b\):

Note that the y‑coordinate is untouched.

This stretches the graph vertically when \(a > 1\) and squashes it vertically when \(a \lt 1\).

If \(y = a f(x)\), multiply the y‑coordinate by \(a\):

If \(y = f(x)/a\), divide the y‑coordinate by \(a\):

Note that the x‑coordinate is untouched.

Apply in order: \(c\), \(b\), \(a\), and finally \(d\).

1. Shift horizontally C units
2. Squeeze horizontally b units
3. Stretch vertically a units
4. Translate vertically d units

Given \(y = f(x)\) as above, draw the graph \(y = 3f(2x - 2) + 4\).

Comparing the graph with the general form:

Order of translations

1. \(c = -2\), so shift graph right 2 units
2. \(b = 2\), so divide x‑coordinates by 2
3. \(a = 3\), so multiply y‑coordinates by 3
4. \(d = 4\), so shift the whole graph up 4 spaces

Taking point \((0, -4)\):

Point \((1, 0)\):

Point \((3, -4)\):

Given a translated function:

Work backwards to get the original \(y = f(x)\).

The points \((0, -3)\) and \((1, 6)\) lie on the graph with equation \(y = \tfrac12 f(3x + 2) - 2\).

Find their corresponding original coordinates on the graph \(y = f(x)\).

Point \((0, -3)\):

Point \((1, 6)\):

The original point \((5, 16)\) has been translated to \((1, 6)\). The original point \((2, -2)\) has been translated to \((0, -3)\).

…if it exists.

Note that the values of the x and y coordinates are swapped, since the reflection line is \(y = x\).

\(y = |f(x)|\)

The entire graph is above the x‑axis.