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