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Translate a Trigonometry Function Up, Down, Left, or Right

When you translate a trig function to solve a problem, you can think of the translation as a slide. This means that the function has the same shape graphically, but the graph of the function slides up, down, left, or right on the coordinate plane.

Slide the graph of a trigonometry function up or down

The following figure shows the parabola y = x2 with a translation 5 units up and a translation 7 units down. A parabola is the graph of a second-degree polynomial, which means that the polynomial has a power of 2 for one exponent. The graph makes a nice, U-shaped curve.

Translations up and down from the function <i>y = x</i><sup>2</sup>.
Translations up and down from the function y = x2.

Think about a function that you use to determine how much money a person earns for working a certain number of hours. The amount can slide up or down if you add a bonus or subtract a penalty from the amount. Here’s what the situation may look like in function notation:

  • Translating up C units: f (x) + C

  • Translating down C units: f (x)C

Let the base pay function be

image1.png

where x is the number of units sold. This restricts the domain to 0 and positive numbers. Then, if a person is to get a bonus of $50, you add that amount on to the base pay to get

image2.png

A person being penalized for coming in late has $50 deducted:

image3.png

These two are represented with vertical slides.

Slide the graph of a trigonometry function left or right

The following figure shows the parabola y = x2 with a translation 7 units right and a translation 5 units left.

Translations right and left from the function <i>y = x</i><sup>2</sup>.
Translations right and left from the function y = x2.

And, continuing with the person working for a certain number of hours, you can adjust for taking time off or getting some vacation hours by adding or subtracting the number of hours. The functions

image5.png

would represent, respectively, a salesperson getting credit for two vacation hours before even starting and a person getting deductions for taking time off. In the second function, x has to be at least 2 for there to be any payout.

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