You are working on a graphing project in your math class, where you are supposed to graph several functions. You are working on graphing a cosine function, and things seem to be going well, until you realize that there is a bold, horizontal line two units above where you placed your "x" axis! As it turns out, you've accidentally shifted your entire graph. You didn't notice that your instructor had placed a bold line where the "x" axis was supposed to be. And now, all of the points for your graph of the cosine function are two points lower than they are supposed to be along the "y" axis.

You might be able to keep all of your work, if you can find a way to rewrite the equation so that it takes into account the change in your graph.

Can you think of a way to rewrite the function so that the graph is correct the way you plotted it?

### Vertical Translations

When you first learned about vertical translations in a coordinate grid, you started with simple shapes. Here is a rectangle:

To translate this rectangle vertically, move all points and lines up by a specified number of units. We do this by adjusting the \begin{align*}y-\end{align*}coordinate of the points. So to translate this rectangle 5 units up, add 5 to every \begin{align*}y-\end{align*}coordinate.

This process worked the same way for functions. Since the value of a function corresponds to the \begin{align*}y-\end{align*}value on its graph, to move a function up 5 units, we would increase the value of *the function* by 5. Therefore, to translate \begin{align*}y=x^2\end{align*} up five units, you would increase the \begin{align*}y-\end{align*}value by 5. Because \begin{align*}y\end{align*} is equal to \begin{align*}x^2\end{align*}, then the equation \begin{align*}y = x^2 + 5\end{align*}, will show this translation.

Hence, for any graph, adding a constant to the equation will move it up, and subtracting a constant will move it down. From this, we can conclude that the graphs of \begin{align*}y = \sin x\end{align*} and \begin{align*}y = \cos x\end{align*} will follow the same rules. That is, the graph of \begin{align*}y = \sin(x) + 2\end{align*} will be the same as \begin{align*}y = \sin x\end{align*}, only it will be translated, or shifted, 2 units up.

To avoid confusion, this translation is usually written *in front of* the function: \begin{align*}y = 2 + \sin x\end{align*}.

Various texts use different notation, but we will use \begin{align*}D\end{align*} as the constant for vertical translations. This would lead to the following equations: \begin{align*}y=D \pm \sin x\end{align*} and \begin{align*}y=D \pm \cos x\end{align*} where \begin{align*}D\end{align*} is the vertical translation. \begin{align*}D\end{align*} *can be positive or negative*.

Another way to think of this is to view sine or cosine curves “wrapped” around a horizontal line. For \begin{align*}y = \sin x\end{align*} and \begin{align*}y = \cos x\end{align*}, the graphs are wrapped around the \begin{align*}x-\end{align*}axis, or the horizontal line, \begin{align*}y = 0\end{align*}.

For \begin{align*}y = 3 + \sin x\end{align*}, we know the curve is translated up 3 units. In this context, think of the sine curve as being “wrapped” around the line, \begin{align*}y = 3\end{align*}.

Either method works for the translation of a sine or cosine curve. Pick the thought process that works best for you.

Let's take a look at a few problems involving vertical translations.

1. Find the minimum and maximum of \begin{align*}y = -6 + \cos x\end{align*}

This is a cosine wave that has been shifted down 6 units, or is now wrapped around the line \begin{align*}y = -6\end{align*}. Because the graph still rises and falls one unit in either direction, the cosine curve will extend one unit above the “wrapping line” and one unit below it. The minimum is \begin{align*}-7\end{align*} and the maximum is\begin{align*} -5\end{align*}.

2. Graph \begin{align*}y=4+\cos x\end{align*}.

This will be the basic cosine curve, shifted up 4 units.

3. Find the minimum and maximum of \begin{align*}y = \sin x + 3\end{align*}

This is a sine wave that has been shifted up 3 units, so now instead of going up and down around the 'x' axis, it will go up and down around the line y = 3. Since the sine function rises and falls one unit in each direction, the new minimum is 2 and the new maximum is 4.

### Examples

#### Example 1

Earlier, you were asked if you can think of a way to rewrite the function so that the graph is correct the way you plotted it.

Since you now know how to shift a graph vertically by adding or removing a constant after the function, you can keep your graph by changing the equation to \begin{align*}y = \cos(x) - 2\end{align*}

#### Example 2

Which of the following is true for the equation: \begin{align*}y=\sin \left(x-\frac{\pi}{2} \right)\end{align*}

The minimum value is 0.

The maximum value is 3.

The \begin{align*}y-\end{align*}intercept is -2.

The \begin{align*}y-\end{align*}intercept is -1.

This is the same graph as \begin{align*}y = \cos (x)\end{align*}.

"This is the same graph as \begin{align*}y = \cos (x)\end{align*}." is the answer to this question, since the \begin{align*}\frac{\pi}{2}\end{align*} is a shift to the graph which makes it the same as a cosine graph.

#### Example 3

Which of the following is true for the equation: \begin{align*}y = 1 + \sin x\end{align*}

The minimum value is 0.

The maximum value is 3.

The \begin{align*}y-\end{align*}intercept is -2.

The \begin{align*}y-\end{align*}intercept is -1.

This is the same graph as \begin{align*}y = \cos (x)\end{align*}.

"The minimum value is 0." is the answer to this question, since this graph is the same as a regular sine graph, which ranges from -1 to 1, but shifted upward one unit on the "y" axis, so it ranges from 0 to 2.

#### Example 4

Which of the following is true for the equation: \begin{align*}y = 2 + \cos x\end{align*}

The minimum value is 0.

The maximum value is 3.

The \begin{align*}y-\end{align*}intercept is -2.

The \begin{align*}y-\end{align*}intercept is -1.

This is the same graph as \begin{align*}y = \cos (x)\end{align*}.

"The maximum value is 3." is the answer to this question, since a cosine graph (which normally ranges between -1 and 1) shifted upward by two units. Therefore its new range is from 1 to 3.

### Review

Use vertical translations to graph each of the following functions.

- \begin{align*}y=x^3+4\end{align*}
- \begin{align*}y=x^2-3\end{align*}
- \begin{align*}y=\sin(x)-4\end{align*}
- \begin{align*}y=\cos(x)+7\end{align*}
- \begin{align*}y=\sec(x)-3\end{align*}
- \begin{align*}y=\tan(x)+2\end{align*}
- \begin{align*}y=3+\sin(x)\end{align*}
- \begin{align*}y=\cos(x)+1\end{align*}
- \begin{align*}y=6+\sec(x)\end{align*}
- \begin{align*}y=\tan(x)-4\end{align*}

Find the minimum and maximum value of each of the following functions.

- \begin{align*}y=\sin(x)+6\end{align*}
- \begin{align*}y=\cos(x)-1\end{align*}
- \begin{align*}y=\sin(x)-4\end{align*}
- \begin{align*}y=-3+\cos(x)\end{align*}
- \begin{align*}y=2+\cos(x)\end{align*}
- Give an example of a sine function with a y-intercept of 6.
- Give an example of a cosine function with a maximum of -1.
- Give an example of a sine function with a minimum of 0.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.12.