| How far a function moves, or is translated, sideways from its natural position. Also known as the phase shift. | How far a function moves, or is translated, up or down from its natural position. |
How to Find it in an Equation
Simply put:
- Vertical – outside the function.
- Horizontal – inside the function.
Vertical Shift
If \(y=f(x)\) then the vertical shift is caused by adding a constant outside the function, \(f(x)\).
Adding 10, like this \(y=f(x)+10\) causes a movement of \(+10\) in the y-axis. Pay attention to the sign…
Outside the function, a positive constant moves the function in the positive x-direction.
In the same way, a negative constant moves the function in the negative x-direction.
The sign is the same as the change in the y-direction.
Horizontal Shift
If \(y=f(x)\) then the horizontal shift is caused by adding a constant inside the function, \(f(x)\).
Subtracting 5, like this \(y=f(x-5)\) causes a movement of \(+5\) in the x-axis. Pay attention to the sign…
Inside the function, a positive constant moves the function in the negative x-direction.
In the same way, a negative constant moves the function in the positive x-direction.
The shift is the opposite of the sign inside the function. Remember it by “getting your left and right mixed up.”
A Multiplier Inside the Function
Okay, so you’re given the equation \(y=(2x+6)^3\) in a test, and the question asks:
What is the translation in the x-axis from \(y=x^3\)?
The answer is a translation of \(3\) in the negative \(x\)-direction.
But why?
The \(2\) that multiplies the \(x\) changes how steep the curve is. For every 1-unit change in \(x\), there is a 2-unit change in \(y\).
So, what would’ve been a shift of \(6\) is now a shift of \(\large\frac{6}{2}\normalsize=3\) because of the multiplier.
This applies to all types of functions.
Examples
1) What does the +3 do in the equation \(y=(x+3)^2\)?
The +3 is inside the function, and there is no multiplier of x.
So, it is a translation of 3 in the negative x-direction.
2) What does +3 do in the equation \(y=x^2+3\)?
The +3 is outside the function, so it is a translation of 3 in the positive y-direction.
3) What is the difference between \(y=\sqrt{x}\) and \(y=\sqrt{x+5}\)
The +5 is inside the function and there is no x-multiplier.
The second equation has a horizontal shift of 5 in the negative x-direction.
4) What changes have been made in the equation \(y=\sqrt{5x+3}-5\)?
The +3 is inside the function and the -5 is outside the function. The x-multiplier is 5.
That means it is a translation of \(\)\large\frac{3}{5} in the negative x-direction and 5 in the negative y-direction.
Graphing: What it Looks Like
In this example, the equation is \(y=f(x)\) and the natural function is \(f(x)=|x|\).
When the changes are made so that \(y=f(x-5)\) and \(y=f(x)+10\), it has the following effect.
If you put them together, so that \(y=f(x-5)+10\) then you’ll see this.
Experiment With Vertical and Horizontal Shift
Use the sliders on the left of this Desmos graph to experiment with changing constants.
Notice the last one, the straight-line function \(y=x+E\)? What happens to the graph when you set the value of \(E\) to +5?
Adding 5 translates, or moves, the straight line graph either 5 in the positive y-direction or 5 in the negative x-direction.
Not both.
You can call it either a vertical shift or a horizontal shift. Go back to the interactive graph and look at what happens again.
This constant has the same effect either way because there is no way to include a constant inside the function. It’s is both inside and outside the function \(f(x)=x\) at the same time.
Trig Functions: Horizontal Shift In Radians
This is where you are most likely to come across a horizontal shift or a vertical shift in real life – in waves.
The trigonometric functions \(\sin\) and \(\cos\) are used to represent waves as a mathematical function. This is probably where you’ll have heard of phase shift before.
Thankfully, both horizontal and vertical shifts work in the same way as other functions.
Remember, trig functions are periodic so a horizontal shift in the positive x-direction can also be written as a shift in the negative x-direction.
In the case of \(y=\sin(x)\) above, the period of the function is π. That means that a phase shift of \(\pm\pi\) leads to \(y=\sin(x)\) all over again.
Tan is also a periodic function, that also has a natural period of π, so the answer is yes!
Do you have any questions? You can contact Jesse or use the comment box below.
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