1.3 Transforming Function

Today we learned about transforming functions,translating, reflecting, and stretching/compressing.

Here is an original problem:

y = f (x)

Translating Functions
Translation occurs when we move the graph units up/down or right/left

Moving a function up or down:

y = f (x) + c 

where c is the number of units moved.  If c is positive (as it is above), the function will move up c units; however, if c is negative, the function will move down c units.

If we were to translate the original problem down two units, the end formula would look like:

y = f (x) - 2

Moving a function left or right:

y = f (x - c)

C is still the number of units moved, however the graph is now moved to the left or to the right.  If c is positive, the function will move c units to the left.  If c is negative, the function will move c units to the right.

If we were to translate the original problem to the left 4 units, the end formula would look like:

y = f (x + 4)

A combination of movements left/right and up/down would look like this:

y = f (x - 3) + 2

where the original function will be translated 3 units to the right and up 2 units.

Stretching or compressing functions

Stretching/Compressing occurs when we move the graph c times away from the x or y axis.

Stretching a function:

y = c * f (x)

where c is the number of units moved.  The graph will be pulled from the x-axis c times.  The  x-intercepts stay the same.

If we were to stretch the original problem by 2, the end formula would look like:

y = 2 f (x)

which would result in a 2x distance from the x-axis.

Compressing a function:

y = f (c * x)

C is still the number of units moved, however the graph is now compressed.  If c is a large number (like 42), the function compresses more closely to the y-axis.  If c is a small number (like 0.2), the function stretches from the y-axis.

If we were to compress the original problem by 4, the end formula would look like:

y = f (4x)

which would result in a 4x distance from the y-axis.

A combination of stretching from the x and y axis would look like this:

y = 3 f (0.1 x)

where the original function will be 3x away from the x-axis and 0.1x away from the y-axis [which would actually be closer because it is less than the original function (which was 1)].

Reflecting Functions

Reflecting occurs when we reflect the graph across the x or y axis.

Reflecting a graph across the x-axis looks like:

y = - f (x)

where the function's x-coordinates remain the same; however, all the y-coordinates become negative, and reflect across the x-axis.

Reflecting a graph across the y-axis:

y = f (-x)

where the function 's x-coordinates become negative; however, all the y-coordinates remain the same, and reflect across the y-axis.

Combination of the Functions

A combination of all three types of transforming functions would look like:

y = -3 f (x + 2) - 8

where the graph is:

• reflected across the x-axis
• stretched 3 units away from the x-axis
• translated 2 units to the left
• and translated 8 units down.