- Horizontal transformations change a graph by altering the input (the $x$-values) of a function.
- Because the change happens inside the function, horizontal transformations often feel "backwards" at first, especially translations.
Horizontal Translations Move Graphs Left Or Right
A horizontal translation shifts a graph left or right.
Definition
Horizontal translation
A transformation that moves the graph of $y=f(x)$ left or right by changing the input to $f$.
The Key Rules For $f(x-h)$ And $f(x+h)$
- Right shift by $h$: $y=f(x-h)$ (when $h>0$)
- Left shift by $h$: $y=f(x+h)$ (when $h>0$)
So, the sign inside the bracket is the opposite of the direction you move.
Common Mistake
- A common mistake is to think $f(x-h)$ moves left because of the minus sign.
- It actually moves the graph right by $h$.
Why Does The Direction Feel Reversed?
- Think in terms of matching the same output value.
- Suppose a point $(a,b)$ is on the original graph $y=f(x)$, meaning $f(a)=b$.
- Now look at $y=f(x-h)$. To get the same output $b$, you need $$f(x-h)=b$$
- This happens when $x-h=a$, so $x=a+h$.
- So the point $(a,b)$ on $f(x)$ becomes $(a+h,b)$ on $f(x-h)$, which is a shift to the right.
Analogy
- Consider the function as a machine that gives output $b$ when you feed it input $a$.
- If you change the rule to $f(x-h)$, you now have to feed the machine the old input $a$ by choosing a new $x$ that is $h$ bigger (so that $x-h=a$).
- That pushes the whole graph to the right.
Visual Summary
Horizontal And Vertical Translations Together
Translations can happen in both directions:
- $y=f(x-h)$ shifts horizontally by $h$
- $y=f(x)+k$ shifts vertically by $k$
- $y=f(x-h)+k$ shifts by $h$ horizontally and $k$ vertically
Note
- Vertical translation was covered in the previous article.
- Horizontal and vertical shifts are both called translations, but they affect different coordinates: horizontal translations change $x$-coordinates, vertical translations change $y$-coordinates.
Worked Examples With Horizontal Transformations
Example question
The graph of $y=f(x)$ passes through $(2,5)$. Find the corresponding point on $y=f(x-3)$.
Solution
Since $f(x-3)$ shifts the graph right by $3$, add $3$ to the $x$-coordinate: $$(2,5)\to (5,5)$$
Example question
Describe the horizontal transformation from $y=x^2$ to $y=(x+3)^2$.
Solution
Because $(x+3)=(x-(-3))$, the graph shifts left 3 units.
Self review
- Which way does $y=f(x-6)$ move the graph?
- A point $(10,2)$ is on $y=f(x)$. What point is on $y=f(x+4)$?
