Horizontal Shift
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
A horizontal shift refers to the movement of a graph left or right along the x-axis. This shift occurs when a function is modified by adding or subtracting a constant to its input variable (usually x).
If you have a function f(x), and it is transformed to f(x + h), the graph shifts left by h units.
If the function is transformed to f(x - h), the graph shifts right by h units.
For example, if the function is f(x)=x^2, a horizontal shift would look like this:
f(x + 3) = (x + 3)^2, which shifts the graph 3 units to the left.
f(x - 2) = (x - 2)^2, which shifts the graph 2 units to the right.
In summary, a horizontal shift changes the position of the graph along the x-axis without altering its shape.
Why does this happen?
The reason why horizontal shifts work like this has to do with how the input values (x-values) are changed in relation to the graph of the function.
When you modify the function by adding or subtracting a constant inside the parentheses (like f(x + h) or f(x - h)), you are essentially changing the x-values that you input into the function.
Here's a breakdown of how it works:
Shifting Left with f(x + h):
If you have f(x), and then you change it to f(x + h), the graph shifts left.
Why? Because, when you input values of x, you're now using x + h instead of just x. For this new function to give the same output as the original function, you need to input a smaller value of x than you would for the original function.
In other words, to get the same output from f(x + h) as you would from f(x), you must "reach" the same point on the graph earlier, so the graph appears shifted to the left by h units.
Example:
If f(x)=x^2, then:
For f(x + 3) = (x + 3)^2, the point x = -3 would now give you the same value as x = 0 would in the original function. So, the entire graph shifts 3 units to the left.
Shifting Right with f(x - h):
If the transformation is f(x - h), the graph shifts to the right.
Why? This time, the input is x - h. To get the same output as the original function f(x), you now need to input a larger value of x than you would in the original graph.
In simpler terms, for the same output to occur, you need to "reach" the same point on the graph later, so the graph appears shifted to the right by h units.
Example:
If f(x)=x^2, then:
For f(x - 2) = (x - 2)^2, the point x = 2 would give you the same output as x = 0 did in the original function, causing the graph to shift 2 units to the right.
Key Idea:
The basic idea is that a horizontal shift affects how you move through the values of x. A transformation like f(x + h) moves everything left because it makes you reach a certain output earlier on the x-axis, while f(x - h) moves everything right because it makes you reach that output later.
By adjusting the values of x this way, the entire graph moves horizontally without changing its shape!