Translating Sine Functions

Translation

Translating a graph means that you are moving it along the $x$ or $y$ axis. This goes for any graph or function, including those of trigonometric functions. If you want to refresh or learn about translations of figures, click here!

For a graph, adding a constant to the equation will move it in the positive direction, and subtracting a constant will move it in the negative direction. So, for a sine function, we will see the different ways we can translate the graph

Vertical Translation

In order to shift the graph in the $y$ direction, we will simply add or subtract a constant from the original sine function. For example, let's look at $f(t) = sin(t) + 2$.

This moves our graph up by $2$ units! If we were to do $f(t) = sin(t) - 2$, then that would move our graph down $2$ units, like so.

To avoid any confusion, the translation is usually written in front of the function like so: $$y = 2 + \sin(x)$$

Horizontal Translation

In order to shift the graph in the $x$ direction, we will simply add or subtract a constant from the $x$-value in the original sine function. However, the graph may act the opposite way than you think. Let’s look at the graph of $y = \sin(x+2)$.

Note that adding $2$ to the $x$ value actually moves our graph in the negative direction! This is the opposite of what we think. So, conversely, if we subtract $2$ from the $x$ value in the sine function, it will actually move the graph $2$ units in the positive direction!

Horizontal translations are also known as phase shifts, that’s because two sine waves that are identical, but have been moved horizontally are said to be out of phase with each other.

Here is a table to keep your facts straight!

(Video) Graphing Sine Functions with Shifting and Translations

by Brian McLogan

In this video by Brian McLogan, you will learn to graph a sine function by applying a phase shift and translation.

Summary

$$f(x)=sin(x+ \frac{\pi}{6}) + 1$$

In order to graph this function, we need to understand what transformations are being applied to this graph. We know that the general form of the sine function is $f(x)=a \text{ sin }(bx-c) +d$.

We need to find the following information before we can graph the sine function: amplitude, period, $x$-scale, phase shift, and the vertical transformation.

We learned how to find the amplitude and period last lesson. The amplitude is $|a|$, and the period is $\frac{2\pi}{b}$. The $x$-scale is $\frac{p}{4}$. The phase shift is when $bx-c=0$. The vertical transformation is $d$.

For our above example, the amplitude is $1$. The period is $2\pi$. Our $x$-scale is $\frac{2\pi}{4} = \frac{\pi}{2}$. So far, all of these transformations are the same as our parent sine function.

The phase shift is $x + \frac{\pi}{6} = 0 \rightarrow x=-\frac{\pi}{6}$. The vertical translation is $d=1$, meaning we will shift the graph up by one unit.

Now that we have all of that information, we can graph our function! The sine graph generally starts at $0$, but because we have a phase shift, we will start our graph at $-\frac{\pi}{6}$.

From here, watch the video to see a step-by-step of how to graph the function. The function looks like:

Translating Sine Functions

The previous lesson taught you how to graph sine functions, which is a sinusoidal function, using the period and amplitude.

Let's take a look at what sine function translations look like! As we've seen before, the basic sine and cosine functions are typical periodic functions. This means that the graph repeats itself over one period.

The top-left box shows a horizontal translation to the right. The top-right box shows a horizontal translation to the left. The bottom-left box shows a vertical translation down. The bottom-right box shows a vertical translation up. This lesson will not cover the horizontal or vertical stretch. Therefore, in these set of translations, the amplitude of the graph stays the same. The period of the graph also stays the same.

We've seen this before with function transformations! Most of it makes intuitive sense, but keep an eye on the horizontal translating - to move the function to the left, toward the negative $x$, you are actually going to add $+h$ to the $x$, inside of the parentheses.

$$\text{shifts to the left = add h}$$ $$\text{shifts to the right = subtract h}$$

With trigonometric functions, we can also call horizontal translations phase shifts because you are simply shifting the phase of the function.