How do you translate a cube root function?

Translating a cube root function involves shifting its graph horizontally and/or vertically. The general form of a translated cube root function is:

f(x) = a∛(x - h) + k

Where:

• a affects the vertical stretch or compression and reflection.
• h represents the horizontal translation.
• k represents the vertical translation.

Understanding the Translations

Here's a breakdown of how h and k affect the graph of the cube root function:

• Horizontal Translation (h): The h value shifts the graph horizontally.

  • If h is positive, the graph shifts to the right by h units. (x - h)
  • If h is negative, the graph shifts to the left by |h| units. (x + h)

  As stated in the reference: "...if you get x minus h it translates to the right. And if you get x plus h it's going to translate..."

• Vertical Translation (k): The k value shifts the graph vertically.

  • If k is positive, the graph shifts up by k units.
  • If k is negative, the graph shifts down by |k| units.

Examples

Let's look at a few examples to illustrate these translations:

• Example 1: f(x) = ∛(x - 2) + 3

  • h = 2: The graph shifts 2 units to the right.
  • k = 3: The graph shifts 3 units up.

• Example 2: f(x) = ∛(x + 1) - 4

  • h = -1: The graph shifts 1 unit to the left.
  • k = -4: The graph shifts 4 units down.

• Example 3: f(x) = 2∛(x - 5)

  • h = 5: The graph shifts 5 units to the right.
  • k = 0: There is no vertical shift.
  • a = 2: There is a vertical stretch by a factor of 2.

By understanding how h and k affect the position of the cube root function, you can easily translate its graph.