This is a basic tutorial on how graphs are translated by various terms in the equation

James S Jun 2010

y = x3 + 3

y = x3

The “+ 3” added to x3 adds 3 to every point on the graph and thus moves the whole graph upward. This is true of any shape graph

y = x + 3

Graph shape is not important to translation

y = x2 + 3

y = ln (x) + 3

y = (x – 3)3 + 3

y = (x)3

In this equation, there is a horizontal shift due to the – 3 inside the parentheses and a vertical shift due to the + 3 outside the parentheses.

y = (x - 3)3 + 3

y = x3

The combination of horizontal and vertical is a diagonal shift.

y = (x – 3)2 + 3

Shape of the curve in not important.

y = (x – 3)2 – 3

Shifted right and down

y = (x + 3)2 – 3 Shifted left and

down

y = ln (x – 3) + 3

Shifted right and up

In any equation in the form: y = (x ± h)c ± k,the “h” inside the parens will shift the graph horizontally, and the “k” outside the parens will shift the graph vertically.

+ h shifts left, – h shifts right

+ k shifts upward, – k shifts downward

James S Jun 2010