What is a Shifting Function
By Kathleen Cantor, 03 Apr 2021
Most people have seen some basic graphs before. Graphs are pictorial representations of data and values along axes. By understanding basic graphs and how to apply translations to them, you'll realize that each new graph is a variation of the old one. It is not a completely different graph than you've ever seen before. Understanding these translations will allow you to quickly recognize and sketch a new function without resorting to drawing points.
This brings us to the meaning of shifting functions. Shifting functions don't change the size and shape of the graph but rather its position.
A shift is an addition or subtraction to the
f(x) component. When you shift a function, you're basically changing the position of the graph of the function. A vertical shift raises or lowers the function as it adds or subtracts a constant to each
y coordinate, while the
x coordinate remains the same. A horizontal shift moves the function right or left since it adds or subtracts a constant to each
x coordinate while keeping the Y coordinate unchanged. You can combine vertical and horizontal shifts in a single expression. If the constants are grouped with
x, then the shift is horizontal; otherwise, it is vertical.
Common functions include:
- Constant functions:
y = c
- Linear functions:
y = x
- Quadratic functions:
y = x^2
- Cubic function:
y = x^3
- Absolute value function:
y = |x|
- Square root function:
y = sqrt(x)
Consider quadratic functions and their associated parabolas. When you first graph quadratic functions, you start with the basic equation.
f(x) = x^2
Then, you make some related graphs such as
g(x) = -x^2 - 4x + 5
h(x) = x^2 - 3x - 4
k(x) = (x + 4)^2
In each case, the basic parabolic shape is the same. The only difference is where the vertex is and whether it is right-side up or upside down. If you've been doing hands-on graphing, you've probably started to notice some relationships between the equation and the graph. The topic of function transformations makes these relationships more explicit.
Moving Up and Down
Let's start by looking at the function symbols for the basic quadratic equation.
f(x) = x^2
A function transformation, or translation, is a fancy way of saying that you change the equation a bit so that the graph moves.
To move the function up, add to the function:
f(x) + b is
f(x) moved up by
b units. The same is true for shifting the function downward;
f(x) - b is
f(x) shifted downward by
We can add a 3 to the basic quadratic equation
f(x) = x^2, going from the basic quadratic function
x^2 to the transformed function
x^2 + 3.
This moves the function up three units.
Shifting to the Left and the Right
To move a function left or right, a constant is added or subtracted from
Let’s look at the equation
y = (x + 3)^2
In this graph,
f(x) shifts three units to the left. Now, instead of graphing
f(x) we are graphing
f(x + 3). This means the equation is now
y = (x + 3)^2, and the original graph is shifted three units to the left.
When moving a function to the left, always add to the function's argument:
f(x + b) shifts
f(x) b units to the left. The reverse is true for moving to the right,
f(x - b) is always
b units to the right.
Functions flipped over the x-axis and mirrored across the y-axis are called reflections. You find the equation for functions reflected over the x-axis by taking the original function's negative value.
If you reflect
f(x) = x^2 + 2x - 3 over the x-axis, it becomes
f(x) = -(x^2) - 2x + 3. This always works for flipping a function upside down.
To illustrate how this transformation works, remember that
f(x) is the same as
y. By adding a minus sign to everything, you change all positive (upper axis)
y values to negative (lower axis)
y values, and vice versa. Any points on the x-axis will stay as they are, and only the off-axis points will move.
To reflect a function across the y-axis, let's consider the cubic function
g(x) = x^3 + x^2 - 3x 1.
If you replace the
x from the original function with
-x, you get
g(-x) = (-x)^3 + (-x)^2 - 3(-x) – 1
g(-x) = -x^3 + x^2 - (-3x) – 1
g(-x) = -x^3 + x^2 + 3x – 1
This transformation will flip the original graph across the y-axis. Any point on the y-axis stays on the y-axis; only off-axis points change sides.
Function Conversion/Transformation Rules
f(x) + b shifts the function up by
f(x) – b shifts the function down by
f(x + b) shifts the function to the left by
f(x - b) shifts the function to the right by
-f(x) reflects the position of the function on the x-axis (i.e., inverted).
f(-x) reflects the position of the function on the y-axis (i.e., left and right swapped).
Real-Life Examples of Shifting Functions
If the sun is shining, water tends to reflect everything. If you look at the mountains behind the lake from a distance, you will see that the mountains are reflected downward. In mathematical terms is reflected over the x-axis creating the same image, only flipped.
Changes in mirrors. When you look into a mirror, you see an accurate reflection of yourself. Most of us look in the mirror every day and are not even aware of this reflection.
A moving car. The car demonstrates transformation because it is moving, but it does not change in size or shape. It also does not flip sideways or upside down.
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