$f(x+c)$ vs $f(x)$

Graphs of $f(x+c)$

Pretend you had some nasty function $f(x)$ and you just finished figuring out how it works. But then the teacher tells everybody, "Graph $f(x+3)$!!!"

Oh no! Do we have to plug in $x+3$ to every $x$ and expand everything out?

Hopefully not. Let's see if there is any pattern.

The blue functions are the original $f(x)$s, and the red functions are the changed $f(x+3)$s.

So for any $f(x+c)$, it is simply $f(x)$ translated left $c$ units.

But why does this happen? And why left?

If we analyze the x-values, we can see a pattern. Which new $x$ do we need to get $f(0)$? The one that makes $f(x+c)=f(0)$, or $-c$.

We can extend this for any $f(x)$: $$f(x_{new}+c)=f(x)\rightarrow x_{new}+c=x\rightarrow x_{new}=x-c$$

So the new x-values are simply the old ones translated $-c$ units, or $c$ units to the left, since that direction is the negative direction.

When moving a function right or left, you change the $x$ coordinate. For example, take $f(x) = x^2$ and move it to the left:

The only thing changing, is the $x$ coordinate, so in your equation, you only change the $x$ value. When the function moves to the right, you change $x$ to $(x - a)$, and when it moves to the left, you change $x$ to $(x + a)$.

As an example: $y(x) = \frac{x^2}{3}$ is moved 6 to the right. What'll be the new formula?

$y_{new}(x) = \frac{(x-6)^2}{3}$

$y_{new}(x) = \frac{x^2 -12x + 36}{3} = \frac{1}{3}x^2 - 4x + 12$

In Context

In this series of function transformations, we are starting off with vertical and horizontal translations. This lesson will consider the horizontal transformations.

Horizontal Translation

The next in our series of function transformations is the horizontal translation!

A horizontal translation allows us to shift a function horizontally, or left and right, along the $x$-axis.

Example

Let's start with the parent function of the quadratic function, $y=x^2$.

Let's say we want to shift this function over to the right by one unit. Our new graph would look like:

The equation for this new graph would need $1$ added to $x$, inside the parentheses. If we want to shift a function to the left, we add a constant. If we want to shift a function to the right, we subtract a constant.

Our new equation is: $y=(x-1)^2$

(Video) Horizontal Function Transformations

by Anusha Rahman

In this lesson by Anusha Rahman, you will learn about shifting a function to the left or right.

Summary

A horizontal function transformation will move a function to the left or right. We can also call this type of transformation a horizontal translation. We are typically given the number of units we will be translating a function, and the size and shape of the function will stay the same.

$$g(x) = f(x-k)$$

$k$ is the number of units that we will horizontally translate the function. If $k>0$, the function will shift to the right by $k$ units. If $k<0$, the function will shift to the left by $k$ units. Watch your signs!