Horizontal Translation

• y = f(x - h)
• when x in a function is replaced with (x - h), it shifts that function h units horizontally
• if h is positive, shift right (in the positive x direction)
• if h is negative, shift left (in the negative x direction)
• when y = f(x) becomes y = f(x - h), (x, y) becomes (x + h, y)

e.g.

• y = f(x - 3) = f(x - (3)), h = 3, shift 3 units right
• y = f(x - 5) = f(x - (5)), h = 5, shift 5 units right
• y = f(x + 2) = f(x - (-2)), h = -2, shift 2 units left
• y = f(x + 1) = f(x - (-1)), h = -1, shift 1 unit left

Vertical Translation

• y = f(x) + k, or y - k = f(x)
• when k is added outside of f(x), it shifts that function k units vertically
• if k is positive, shift up (in the positive y direction)
• if k is negative, shift down (in the negative y direction)
• when y = f(x) becomes y = f(x) + k, (x, y) becomes (x, y + k)

e.g.

• y = f(x) + 4, k = 4, shift 4 units up
• y = f(x) - 2, k = -2, shift 2 units down
• y - 3 = f(x) can be written as y = f(x) + 3, k = 3, shift 3 units up
• y + 5 = f(x) can be written as y = f(x) - 5, k = -5, shift 5 units down

Combining Horizontal and Vertical Translations

• y = f(x - h) + k
• can be written as y - k = f(x - h)
• when y = f(x) becomes y = f(x - h) + k, (x, y) becomes (x + h, y + k)

Translating a Graph

• use the equation to recognize the transformations y = f(x - h) + k
• transform each point
• connect points

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Finding a Translated Equation Example #1

• translated equation will have equation y = f(x - h) + k
• in any equation, replace x with (x - h), add k at the end
• use the transformations to find the values of h and k
• plug into equation

Finding a Translated Equation Example #2

Finding a Translated Equation Example #3

Finding the Equation of a Translated Graph

• choose one point on y = f(x) and find it's corresponding point on the translated graph
• find the horizontal and vertical shift between points to find the value of h and k
• plug into y = f(x - h) + k

How Asymptotes Change With Translations

• when a graph is shifted horizontally, the vertical asymptotes shift with it
• when a graph is shifted vertically, the horizontal asymptotes shift with it