余弦函数变换

Translation

Translation 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 cosine function, we will see the different ways we can translate the graph

You'll notice that for the following example graphs, the $x$ axis is in units of $\pi$! That's because cosine is a trigonometric function.

Vertical Translation

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

This moves our graph up by $2$ units! If we were to do $f(t) = cos(t) - 2$, 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: $$y = 2 + \cos(t)$$

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 $y = \cos(t+2)$.

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 a Cosine Function with a Horizontal Translation

by Brian McLogan

In this video by Brian McLogan, you will learn how to graph a cosine function with a horizontal translation.

Summary

Graph the following cosine function: $y=4 \text{ cos }(x+\frac{\pi}{4})$

As we've learned previously, we know that:

• Amplitude $= |a| = |4| = 4$

• Period $= \frac{2\pi}{b} = \frac{2\pi}{1} = 2\pi$

We will also notice that $x$ has a value after it. We are adding $\frac{\pi}{4}$ to $x$, which indicates a horizontal shift. Remember that the cosine parent graph has endpoints at $0$ and $2\pi$. So, if we shift the function, we can take whatever is inside the parentheses of the function, and set it equal to the endpoint of the function.

$$\begin{align} x+\frac{\pi}{4} &= 0 \\ x &= -\frac{\pi}{4} \\ \end{align}$$ $$\begin{align} x+\frac{\pi}{4} &= 2\pi \\ x &= 2\pi -\frac{\pi}{4} \\ x &= \frac{7\pi}{4} \\ \end{align}$$

That gives us our two new endpoints! Our graph now starts at $-\frac{\pi}{4}$ and the first cycle ends at $\frac{7\pi}{4}$.

The graph of our function looks like:

Translating Cosine Functions

Now that we've established what the parent cosine function looks like, we can look at graphs for translating cosine functions. We have seen similar translations with the sine function, so the patterns may seem familiar to you!

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 $+k$ to the $x$, inside of the parentheses.

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

In the next lesson, we will dive a little deeper into cosine equations.