Techniques in Graphing

Functions you should be able to graph without even thinking about it (We will sometimes refer to these as parent functions.):

· Y = mx+b

· Y = x²

· Y =

· Y = x³

· y = |x|

· y = 1/x

Translation of a graph: A shift in the graphs location such that every point of the graph is moved in the same distance in the same direction. The size and shape of the graph are unaffected by translation.

Translations and Reflections (Property Summary)

Note: c denotes a positive constant in items 1 – 4.

EQUATION HOW TO OBTAIN THE GRAPH

FROM THAT OF Y = F(X).

1. y = f(x) +c Translate c units vertically upward

2. y = f(x) – c Translate c units vertically downward

3. y = f(x+c) Translate c units to the left

4. y = f(x-c) Translate c units to the right

5. y = -f(x) Reflect in the x-axis

6. y = f(-x) Reflect in the y-axis.

* Continued *

Multiple translations where order of translation is important:

Case 1: translation in the y-direction (up or down, i.e. vertical) coupled with reflection in the x-direction. => Reflect first, shift up or down second!

Case 2: translation in the x-direction (left or right, i.e. horizontal) coupled with reflection in the y-direction. => Shift left or right first, reflect second!

Cases where order of translation is not important :

1. Two translations horizontal and vertical.

2. Two reflections, one in the x-axis, one in the y-axis.

3. Translation in the x-direction (horizontal) and reflection in the x-axis.

4. Translation in the y-direction (vertical) and reflection in the y-axis.