Module 3 - Functions and Transformations | ||||||||||||||||||||||||||||||||||||
Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Self Test | ||||||||||||||||||||||||||||||||||||
Lesson 3.2: Translations and Combined Transformations | ||||||||||||||||||||||||||||||||||||
Lesson 3.1 discussed two types of transformations: stretches and reflections. This lesson will introduce translations, a third type of transformation, and discuss the effects of combining several types of transformations. Translations A translation sends all points of a graph the same distance in the same direction. Vertical Translations A vertical translation, or vertical shift, moves every point on a graph up or down the same distance. Explore the effect of adding three to the absolute value function.
The transformed graph is a vertical shift of
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3.2.1 Display the graphs of
Horizontal Shifts We have examined vertical transformations that were created by stretching or shrinking vertically, by reflecting across the x-axis, or by shifting upward or downward. In each case, we described how the transformed graph's y-values were related to the corresponding y-values of the original function. We will now examine the effect of adding to (or subtracting from) the x-values before applying the function. Recall that when a constant was added to the y-values, the transformation was a vertical shift. It should seem reasonable to conclude that when you add a constant to the x-values, the transformation will be a horizontal shift. Describing Horizontal Shifts
When describing a horizontal shift, it is helpful to see which x-values produce the same y-value. Display the function
The fixed output of y = 0 was produced by x = 0 in Y1 and was produced by x = -3 in Y2. ![]()
3.2.2 Use your calculator to experiment and find the function that will shift
Horizontal Stretches and Compressions Just as multiplying a function by a constant stretches or shrinks the graph vertically, multiplying the x-value by a constant before applying the function will stretch or shrink the graph horizontally. In many functions like the absolute value function, the horizontal compression appears to be a vertical stretch.
From the table it appears that the values of Y2 are twice that of the values of Y1. This is true because ![]() The graphs of y = sin x and y = sin 2x better illustrate the effect of multiplying x by a constant. The Graph of y = sin x
Characteristics of the Sine Wave The graph of y = sin x has the characteristics listed below.
The graph is periodic and has a period of 2
The graph has amplitude of 1. That is, the y-values rise to 1 unit above the centerline and fall to 1 unit below the centerline. The domain of the sine function is the set of all real numbers.
The range of the sine function is
Special Points on y = sin x
It is often useful to identify the five special points on the sine wave whose x-values are
![]() The Graph of y = sin 2x Now display the graph of y = sin 2x and compare it to the graph of y = sin x.
Combined Transformations
The function
The graphs below illustrate the above sequence of combined transformations. The graph is reflected across the x-axis.
The graph is then stretched by a factor of 2.
The graph is then translated right 4 units and up 3 units.
Recognizing the basic shape of a graph and then applying the indicated transformations will help you sketch the graph before displaying it with your calculator.
3.2.3 Describe the transformations in a correct order for the equation
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