Translations of Trigonometric Functions

When graphing trigonometric functions, you can find cases where the graphs are shifted on the coordinate plane, either to the right or left, or up or down. This type of transformation is called a translation. In this article, we will define the different types of trigonometric function translations, and describe the rules that you need to follow in each case using practical examples.

Translations of Trigonometric Functions Translations of Trigonometric Functions

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    Translations of trigonometric functions are transformations of trigonometric function graphs that involve shifting them horizontally or vertically.

    What are the types of trigonometric function translations?

    The different types of trigonometric functions translations include horizontal translation, where the graph shifts either to the left or to the right, and vertical translation, where the graph moves up or down on the coordinate plane. Let's see in more detail how to work out each one of these translations.

    Horizontal translations

    If you have a trigonometric function in the form y=a sin bθ-h, where b>0, then the graph of sine, in this case, will be shifted h units to the left or right, depending on whether h is positive or negative. This type of translation is also called a phase shift. These are the two possible cases that you will find:

    • If h is negative, then the graph will be shifted to the left.

    The graph of y=sin θ+π2 is shown below. The sine graph is represented with the dashed green line, so you can clearly see that by adding π/2 inside the parentheses, the whole graph shifted π/2 to the left.

    Translations of Trigonometric Functions Horizontal translation h negative StudySmarterHorizontal translation when h is negative - StudySmarter Originals

    Notice that, when the graph of sine is shifted to the left by π2, the resulting graph is the cosine graph.

    • If h is positive, then the graph will be shifted to the right.

    The graph of y=sin θ-π2 looks like this:

    Translations of Trigonometric Functions Horizontal translation h positive StudySmarterHorizontal translation when h is positive - StudySmarter Originals

    Vertical translations

    If you add a constant to a trigonometric function y=a sin b(θ-h)+k, it will move its graph up or down along the y-axis as many units as the value of the constant. This type of translation is also known as vertical shift. In this case, you will also get a new midline, which is y=k, and you will use it as your new reference horizontal axis.

    • If k is positive, then the graph will be shifted upwards.

    The graph of y=sin θ+2 is shown below. The midline y=2 is represented with a dashed red line. As you can see, the graph has been moved upwards 2 units.

    Translations of Trigonometric Functions Vertical translation k positive StudySmarterVertical translation when k is positive - StudySmarter Originals

    • If k is negative, then the graph will be shifted downwards.

    The graph of y=sin θ-2 shows that when the constant is negative, then the graph is moved downwards 2 units. The new midline is y=-2.

    Translations of Trigonometric Functions Vertical translation k negative StudySmarterVertical translation when k is negative - StudySmarter Originals

    In general, trigonometric functions can be written in the form:

    y=a sin bθ-h+ky=a cos bθ-h+ky=a tan bθ-h+k

    Remember that from the expressions above, you can calculate the amplitude, as a for sine and cosine. The tangent function has no amplitude. Also, the period of the function is 2πb for sine and cosine, and πbfor the tangent function. If you need to refresh the basics about amplitude and period, please read about Graphing Trigonometric Functions.

    All the horizontal and vertical translations explained above can be applied in the same way to cosine and tangent graphs. Also, the reciprocal graphs of trigonometric functions (cosecant, secant and cotangent) can also be translated vertically and horizontally.

    What are the rules in trigonometric functions translation?

    The different rules that you need to keep in mind when translating trigonometric functions are as follows:

    1. Find the vertical shift, if there is one, and graph the midline y=k.

    2. Find the amplitude, if applicable. Draw dashed lines to represent the maximum and minimum values of the function.

    3. Calculate the period of the function.

    4. Plot a few points and joint them with a smooth and continuous curve.

    5. Determine if there is a phase shift, and translate the graph according to the value of h.

    If the value of a<0, then the graph will be reflected over the x-axis.

    Examples of translations of trigonometric functions

    Find the amplitude, period, vertical and horizontal shift of the following trigonometric functions, and then graph them:

    a) y=2 cos 2θ+π2+1

    k=1the vertical shift is 1 (upwards), so the midline is y=1

    a=2 the amplitude is 2=2

    b=2 the period is 2π2=2π2=π

    h=-π2the horizontal shift is π2 to the left

    Translations of Trigonometric Functions Cosine translation example StudySmarterTranslation of cosine function example - StudySmarter Originals

    b) y=-4 tan θ-2

    k=-2 the vertical shift is -2 (downwards), so the midline is y=-2

    The tangent function has no amplitude. However, a<0, therefore the graph is reflected over the x-axis

    b=1 the period isπ1=π1=π

    h=0 there is no horizontal shift

    Translations of Trigonometric Functions Tangent translation example StudySmarterTranslation of tangent function example - StudySmarter Originals

    Translations of Trigonometric Functions - Key takeaways

    • Translation of trigonometric functions are transformations of trigonometric function graphs that involve shifting them horizontally or vertically.
    • Horizontal translation means that the graph shifts either to the left or to the right, and vertical translation, is where the graph moves up or down on the coordinate plane.
    • All trigonometric functions, including their reciprocals, can be translated horizontally or vertically.
    • If the value of a is negative, then the graph will be reflected over the x-axis.
    Frequently Asked Questions about Translations of Trigonometric Functions

    What is a translation in Trigonometry?

    Translation in Trigonometry refers to the vertical or horizontal shift of trigonometric functions graphs.

    How do you translate trigonometric functions?

    Horizontal translation:

    • If h is negative, then the graph will be shifted to the left.
    • If h is positive, then the graph will be shifted to the right.

    Vertical translation:

    • If k is positive, then the graph will be shifted upwards.
    • If k is negative, then the graph will be shifted downwards.

    What is a horizontal translation of trigonometric functions?

    If you have a trigonometric function in the form y = a sin b (θ - h), then the graph of sine, in this case, will be shifted units to the left or right, depending on whether h is positive or negative. This type of translation is also called a phase shift. 

    What is a vertical translation of trigonometric functions?

    If you add a constant to a trigonometric function y = a sin b(θ - h) + k, it will move its graph up or down along the y-axis as many units as the value of the constant. This type of translation is also known as vertical shift. In this case, you will also get a new midline, which is y = k, and you will use it as your new reference horizontal axis. 

    What are the rules in trigonometric functions translations?

    1. Find the vertical shift, if there is one, and graph the midline y = k.    

    2. Find the amplitude, if applicable. Draw dashed lines to represent the maximum and minimum values of the function.

    3. Calculate the period of the function.

    4. Plot a few points and joint them with a smooth and continuous curve.

    5. Determine if there is a phase shift, and translate the graph according to the value of h.

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