Translations of Graphs

Introduction

students, imagine taking the graph of a function and sliding it across a coordinate grid like moving a picture on a wall 🎯. The shape stays the same, but its position changes. That is the idea of a translation of a graph. In IB Mathematics Analysis and Approaches SL, translations are one of the most important ways to understand how functions are transformed and represented.

By the end of this lesson, you should be able to:

explain what a translation is and use the correct vocabulary,
describe how a graph changes when it is shifted horizontally or vertically,
connect translations to function notation such as $y=f(x)$, $y=f(x)+k$, and $y=f(x-h)$,
compare translations across different types of functions, such as linear, quadratic, rational, exponential, and logarithmic models,
use examples to interpret transformed graphs in real-life situations.

Translations are not random moves. They follow precise rules, and those rules help you predict how a function changes without redrawing the whole graph from scratch. That skill is very useful in IB because it builds your understanding of function language and representation.

What is a Translation?

A translation is a shift of a graph without changing its shape, size, or orientation. If a graph moves up, down, left, or right, it is translated. The important idea is that the graph is not stretched, reflected, or rotated. It is simply moved.

There are two main types of translations:

Vertical translations: the graph moves up or down.
Horizontal translations: the graph moves left or right.

Suppose the original function is $y=f(x)$. Then:

$y=f(x)+k$ shifts the graph up by $k$ units,
$y=f(x)-k$ shifts the graph down by $k$ units,
$y=f(x-h)$ shifts the graph right by $h$ units,
$y=f(x+h)$ shifts the graph left by $h$ units.

This is one of the first places where students notice that horizontal changes work differently from vertical ones. The sign inside the function is opposite to the direction of movement. That is a key IB idea.

For example, if $f(x)=x^2$, then $y=(x-3)^2$ is the graph of $y=x^2$ shifted $3$ units to the right. The vertex moves from $(0,0)$ to $(3,0)`. If $y=x^2+4$, the graph shifts $4 units up, so the vertex becomes $(0,4)$.

Vertical Translations: Moving Up and Down

Vertical translations are the easiest to see because the output values of the function change directly. If every $y$-value increases by the same amount, the entire graph moves upward. If every $y$-value decreases by the same amount, it moves downward.

If $y=f(x)+k$, then each point $(x,y)$ on the original graph becomes $(x,y+k)$. The $x$-coordinates stay the same, and only the $y$-coordinates change.

Example 1: Quadratic graph

Let $f(x)=x^2$. Then:

$y=f(x)+2$ is $y=x^2+2$.
The graph shifts up $2$ units.
The vertex changes from $(0,0)$ to $(0,2)$.

Now consider $y=f(x)-5$, which is $y=x^2-5$.

The graph shifts down $5$ units.
The vertex changes from $(0,0)$ to $(0,-5)$.

Example 2: Exponential graph

Let $f(x)=2^x$. Then $y=2^x-1$ shifts the graph down $1$ unit. This changes the horizontal asymptote from $y=0$ to $y=-1$. That is an important feature because translations can move asymptotes too.

In real life, vertical translations can represent a baseline change. For example, if a company’s profit model is $P(x)=1000e^{0.2x}$ and the company has fixed costs of $300$, then the adjusted model might be $P(x)=1000e^{0.2x}-300$. The graph is shifted down by $300$, reflecting the extra cost.

Horizontal Translations: Moving Left and Right

Horizontal translations are a little trickier because they affect the input variable $x$ instead of the output. If a graph moves right or left, the function rule changes inside the brackets.

If $y=f(x-h)$, then the graph shifts right by $h$ units. If $y=f(x+h)$, then the graph shifts left by $h$ units.

Why does the sign work this way? Think of the new function as asking: “What original input would produce this output now?” To get the same output as before, the $x$-value must be adjusted in the opposite direction.

Example 3: Quadratic graph

Let $f(x)=x^2$.

$y=(x-4)^2$ shifts the graph right by $4$ units.
The vertex moves to $(4,0)$.

$y=(x+2)^2$ shifts the graph left by $2$ units.
The vertex moves to $(-2,0)$.

Example 4: Rational graph

Let $f(x)=\frac{1}{x}$.

$y=\frac{1}{x-1}$ shifts the graph right by $1$ unit.
The vertical asymptote moves from $x=0$ to $x=1$.
The horizontal asymptote stays at $y=0$.

This shows that translations can move key features of graphs, not just the whole picture. In IB, you should always look for intercepts, turning points, and asymptotes after a translation.

Example 5: Logarithmic graph

Let $f(x)=\log(x)$.

$y=\log(x-3)$ shifts the graph right by $3$ units.
The vertical asymptote moves from $x=0$ to $x=3$.
The domain becomes $x>3$.

This matters because horizontal translations can change the domain. That is a major reason why transformed graphs must be checked carefully.

Using Translations in Function Language

Function notation helps describe translations clearly. If the original function is $y=f(x)$, then transformed functions can be written as:

$$y=f(x)+k$$

$$y=f(x-h)$$

students, in IB math, the notation is as important as the graph itself. A graph is a visual representation, but the equation is the rule that generates it.

You should be able to move between these forms:

verbal description: “shift right $2$ and up $3$,”
equation: $y=f(x-2)+3$,
graph: the original graph moved to a new position.

Example 6: Combining translations

If $f(x)=x^2$, then $y=(x-1)^2+4$ means:

shift right by $1$ unit,
shift up by $4$ units.

So the vertex changes from $(0,0)$ to $(1,4)$.

When there are multiple transformations, apply them carefully. In this lesson, we are focusing on translations only, so the shape of the graph remains unchanged.

Translations and Key Features of Graphs

A translation changes the position of all points equally. This means some features move, but their relationships remain the same.

Important ideas:

intercepts may change,
vertices move,
asymptotes may shift,
domain and range may change,
the overall shape stays the same.

For a linear function like $y=mx+c$, a vertical translation changes the value of $c$. For example, if $y=2x+1$ is shifted up by $3$, the new function is $y=2x+4$.

For a quadratic function like $y=ax^2+bx+c$, a translation changes the position of the parabola. For instance, $y=(x-2)^2+1$ is the graph of $y=x^2$ moved right $2$ and up $1$.

For exponential and logarithmic functions, translations often change asymptotes and domain boundaries, so they are especially useful in modeling growth, decay, and real-world limits.

In applications, translations can model situations like:

adjusting a temperature scale by adding a constant,
adding a fixed fee to a cost model,
shifting a demand curve because of a policy change,
changing the height of a tracking sensor reading above ground level.

Common Mistakes to Avoid

One common mistake is confusing the direction of horizontal shifts. Remember:

$f(x-3)$ moves right,
$f(x+3)$ moves left.

Another mistake is forgetting that translations do not alter the shape of the graph. If the graph looks stretched or reflected, then the change is not a translation alone.

A third mistake is assuming that all features move the same way in description but not checking the equations. For example, in $y=\frac{1}{x-2}$, the vertical asymptote becomes $x=2$, not $x=-2$.

A good strategy is to identify the base function first, then determine how the new equation changes it. This works well for every major function family in the IB syllabus.

Conclusion

Translations of graphs are a core part of function understanding in IB Mathematics Analysis and Approaches SL. They allow you to move graphs vertically or horizontally while keeping the same shape. Vertical translations change the output by adding or subtracting a constant, while horizontal translations change the input and work in the opposite direction.

students, if you can read an equation like $y=f(x-h)+k$ and explain exactly how the graph moves, you are building a strong foundation for transformations, composite functions, inverse functions, and graph analysis. Translations are not just about drawing pictures; they are about understanding how algebra and geometry describe the same function in different ways 📈.

Study Notes

A translation moves a graph without changing its shape, size, or orientation.
$y=f(x)+k$ shifts the graph up by $k$ units.
$y=f(x)-k$ shifts the graph down by $k$ units.
$y=f(x-h)$ shifts the graph right by $h$ units.
$y=f(x+h)$ shifts the graph left by $h$ units.
Horizontal shifts use the opposite sign inside the function.
Translating a graph moves all points the same amount.
For quadratics, the vertex moves with the translation.
For rational functions, asymptotes may shift.
For logarithmic functions, the domain and vertical asymptote may change.
Translations are a key part of function language, graph interpretation, and IB modeling.