A translation is a slide, which means that the function has the same shape graphically, but the graph of the function slides up or down or slides left or right to a different position on the coordinate plane.
Sliding up or down
The 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
A person who makes $8 an hour but gets a $50 bonus has a pay function for h hours that looks like P(h) = 8h + 50. If that same person were penalized $6 for being late, the pay function would look like P(h) = 8h – 6.
Sliding left or right
The figure shows the parabola y = x2 with a translation 5 units right and a translation 7 units left.
If you use a function to determine how much commission a person earns for selling a certain number of computers, the commission can be affected when you add or subtract the number of units the person needs to sell. Here's what the situation looks like in function notation:
Translating left C units: f(x + C)
Translating right C units: f(x – C)
A person who makes $50 commission for every computer sold but gets upfront credit for two computers as an incentive has a commission function for x computers that looks like P(x) = 50(x + 2). On the other hand, a person who has the same commission schedule but had two computers returned and starts with a deficit has a commission function that looks like P(x) = 50(x – 2).