**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 $y=a\mathrm{sin}b\left(\theta -{\mathit{h}}\right)$, 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 $y=\mathrm{sin}\left(\theta +\frac{\mathrm{\pi}}{2}\right)$ 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.

Notice that, when the graph of sine is shifted to the left by $\frac{\mathrm{\pi}}{2}$, the resulting graph is the cosine graph.

If

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

The graph of $y=\mathrm{sin}\left(\theta -\frac{\mathrm{\pi}}{2}\right)$ looks like this:

### Vertical translations

If you add a constant to a trigonometric function $y=a\mathrm{sin}b(\theta -h)+{\mathit{k}}$, 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 $y={\mathit{k}}$, 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 $y=\mathrm{sin}\theta +2$ 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.

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

The graph of $y=\mathrm{sin}\theta -2$ shows that when the constant is negative, then the graph is moved downwards 2 units. The new midline is $y=-2$.

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

$y=a\mathrm{sin}b\left(\theta -h\right)+k\phantom{\rule{0ex}{0ex}}y=a\mathrm{cos}b\left(\theta -h\right)+k\phantom{\rule{0ex}{0ex}}y=a\mathrm{tan}b\left(\theta -h\right)+k$

Remember that from the expressions above, you can calculate the **amplitude**, as $\left|a\right|$ for sine and cosine. The tangent function has no amplitude. Also, the **period** of the function is $\frac{2\mathrm{\pi}}{\left|b\right|}$ for sine and cosine, and $\frac{\mathrm{\pi}}{\left|b\right|}$for 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:

Find the

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

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

**period**of the function.Plot a few points and joint them with a smooth and continuous curve.

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) $y=2\mathrm{cos}2\left(\theta +\frac{\mathrm{\pi}}{2}\right)+1$

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

$a=2\Rightarrow $ the amplitude is $\left|2\right|=2$

$b=2\Rightarrow $ the period is $\frac{2\mathrm{\pi}}{\left|2\right|}=\frac{\overline{)2}\mathrm{\pi}}{\overline{)2}}=\mathrm{\pi}$

$h=-\frac{\mathrm{\pi}}{2}\Rightarrow $the horizontal shift is $\frac{\mathrm{\pi}}{2}$ to the left

b) $y=-4\mathrm{tan}\theta -2$

$k=-2\Rightarrow $ 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\Rightarrow $ the period is$\frac{\mathrm{\pi}}{\left|1\right|}=\frac{\mathrm{\pi}}{1}=\mathrm{\pi}$

$h=0\Rightarrow $ there is no horizontal shift

## 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.

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##### 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 *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.

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?

Find the vertical shift, if there is one, and graph the midline

*y = k*.Find the amplitude, if applicable. Draw dashed lines to represent the maximum and minimum

Calculate the period of the function.

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

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

*h*.

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