MYP Extended Mathematics Topic 2.2 Horizontal…

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.

Active recall

Which way does $y=f(x-6)$ move the graph?
A point $(10,2)$ is on $y=f(x)$. What point is on $y=f(x+4)$?

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.

Active recall

Which way does $y=f(x-6)$ move the graph?
A point $(10,2)$ is on $y=f(x)$. What point is on $y=f(x+4)$?

MYP Extended Mathematics 2.2 Horizontal Translation Notes

MYP Extended Mathematics 2.2 Horizontal Translation Notes

Horizontal Translations Move Graphs Left Or Right

The Key Rules For $f(x-h)$ And $f(x+h)$

Why Does The Direction Feel Reversed?

Visual Summary

Horizontal And Vertical Translations Together

Worked Examples With Horizontal Transformations

Horizontal Translations Move Graphs Left Or Right

The Key Rules For $f(x-h)$ And $f(x+h)$

Why Does The Direction Feel Reversed?

Visual Summary

Horizontal And Vertical Translations Together

Worked Examples With Horizontal Transformations

1 Advanced Number and Bounds

2 Function Transformations

3 Inequalities and Modelling

4 Logarithms and Non-Linear Inequalities

5 Sequences, Rational Expressions and Equations

6 Geometric Transformations

7 Trigonometric Functions and Rules

8 Variation and Standard Deviation

9 Covariance and Correlation

10 Advanced Probability

11 Graph Theory and Vectors

1 Advanced Number and Bounds

2 Function Transformations

3 Inequalities and Modelling

4 Logarithms and Non-Linear Inequalities

5 Sequences, Rational Expressions and Equations

6 Geometric Transformations

7 Trigonometric Functions and Rules

8 Variation and Standard Deviation

9 Covariance and Correlation

10 Advanced Probability

11 Graph Theory and Vectors

MYP Extended Mathematics 2.2 Horizontal Translation Notes

MYP Extended Mathematics 2.2 Horizontal Translation Notes

Horizontal Translations Move Graphs Left Or Right

The Key Rules For $f(x-h)$ And $f(x+h)$

Why Does The Direction Feel Reversed?

Visual Summary

Horizontal And Vertical Translations Together

Worked Examples With Horizontal Transformations

Horizontal Translations Move Graphs Left Or Right

The Key Rules For $f(x-h)$ And $f(x+h)$

Why Does The Direction Feel Reversed?

Visual Summary

Horizontal And Vertical Translations Together

Worked Examples With Horizontal Transformations