2.12: Vertical Translations
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?
Keep reading, and at the end of this Concept, you'll know how to do exactly that using a "vertical shift".
Watch This
In the second portion of this video you will learn how to perform vertical translations of the sine and cosine functions.
James Sousa: Horizontal and Vertical Translations of Sine and Cosine
Guidance
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
This process worked the same way for functions. Since the value of a function corresponds to the
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
To avoid confusion, this translation is usually written in front of the function:
Various texts use different notation, but we will use
Another way to think of this is to view sine or cosine curves “wrapped” around a horizontal line. For
For
Either method works for the translation of a sine or cosine curve. Pick the thought process that works best for you.
Example A
Find the minimum and maximum of
Solution: This is a cosine wave that has been shifted down 6 units, or is now wrapped around the line
Example B
Graph
Solution: This will be the basic cosine curve, shifted up 4 units.
Example C
Find the minimum and maximum of
Solution: 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.
Vocabulary
Vertical Translation: A vertical translation is a shift in a graph up or down along the "y" axis, generated by adding a constant to the original function.
Guided Practice
1. Which of the following is true for the equation:
The minimum value is 0.
The maximum value is 3.
The
The
This is the same graph as
2. Which of the following is true for the equation:
The minimum value is 0.
The maximum value is 3.
The
The
This is the same graph as
3. Which of the following is true for the equation:
The minimum value is 0.
The maximum value is 3.
The
The
This is the same graph as
Solutions:
1. "This is the same graph as
2. "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.
3. "The maximum value is 3." is the answer to this question, since the is a cosine graph (which normally ranges between 1 and 1) shifted upward by two units. Therefore its new range is from 1 to 3.
Concept Problem Solution
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
Practice
Use vertical translations to graph each of the following functions.

y=x3+4 
y=x2−3 
y=sin(x)−4 
y=cos(x)+7 
y=sec(x)−3 
y=tan(x)+2 
y=3+sin(x)  \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 yintercept 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.
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Amplitude
The amplitude of a wave is onehalf of the difference between the minimum and maximum values of the wave, it can be related to the radius of a circle.Vertical shift
A vertical shift is the result of adding a constant term to the value of a function. A positive term results in an upward shift, and a negative term in a downward shift.Vertical Translation
A vertical translation is a shift in a graph up or down along the axis, generated by adding a constant to the original function.Image Attributions
Here you'll learn how to express vertical translations of graphs algebraically.