Lesson 14.2: Help for Exercises 50-55 on page 845
It is helpful to review translations and reflections of graphs. If y=asinbx then:
- y=asinb(x-h) represents a horizontal translation of h units. If h is positive, the graph is translated right h units, and if h is negative, the graph is translated to the left h units.
- y=asinbx+k represents a vertical translation of k units. If k is positive, the graph is translated up k units, and if k is negative, the graph is translated down k units.
- y=-asinbx represents a reflection over the x-axis.
Any combination of the above translations can be applied to a graph. For example, y=-asinb(x-h)+k represents a horizontal translation of h units, a vertical translation of k units, and a reflection over the line y = k.
There are several common errors that can be made when applying transformations. For example, consider the function y=3sin5x . Look at some mistakes made when applying a horizontal translation of 2 units left:
- y=3sin5xy =3sin(5x+2)You must factor out the five before applying the shift.
- y=3sin5xy=3sin5(x-2) The expression "x - 2" represents a shift right of 2 units instead of 2 units to the left.
The correct answer is 3sin5(x+2)
Another common error occurs when a reflection is made over the line y = k. Consider the same function given above with a vertical shift of 3 units up and a reflection over the line y = 3. An incorrect answer is as follows:
- y=3sin5x y = -(3sin5x+3) This expression represents a vertical translation of 3 units down and a reflection over the line y = 3, which is just the opposite of the desired result.
The correct answer is y = -3sin5x+3