Translations of Graphs

Welcome, students 👋 In this lesson, you will learn how graphs move on the coordinate plane without changing their overall shape. These moves are called translations. Understanding translations helps you read, sketch, and compare functions more accurately, which is essential in IB Mathematics: Analysis and Approaches HL.

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

Explain what a translation is and use the correct terminology.
Describe how a graph changes when it is shifted horizontally or vertically.
Apply translation rules to functions such as $f(x)$, $f(x)+k$, and $f(x-h)$.
Connect translations to other function ideas like composite functions, inverses, and transformations.
Use examples to interpret translated graphs in real-world contexts 🌍

A translation is one of the simplest transformations, but it is also one of the most important. It helps us model changes such as a rise in temperature, a change in starting point, or a shift in time. For example, if a bus route begins later than usual, the graph of distance against time may shift to the right. If a company’s profit increases by a fixed amount, the graph may shift upward.

What is a Translation?

A translation moves every point on a graph the same distance in the same direction. It does not stretch, reflect, or rotate the graph. The shape stays exactly the same.

If a point $(x, y)$ is translated by the vector $\begin{pmatrix} a \\ b \end{pmatrix},$ then the new point is $(x+a, y+b)$.

This means:

$a$ controls the horizontal shift.
$b$ controls the vertical shift.

For graphs of functions, translations are often written using function notation. If $y=f(x)$, 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.

Notice something important: the sign in the function changes compared with the direction of the movement. This is a common source of mistakes, so students, pay close attention 👀

Example 1: Vertical Translation

Suppose $f(x)=x^2$. Then the graph of $y=x^2+3$ is the graph of $y=x^2$ shifted up by $3$ units.

Key points move like this:

$(0,0)$ becomes $(0,3)$
$(1,1)$ becomes $(1,4)$
$(-1,1)$ becomes $(-1,4)$

The parabola still opens upward and has the same width. Only its position changes.

Example 2: Horizontal Translation

Suppose $f(x)=|x|$. Then $y=|x-2|$ is the graph of $y=|x|$ shifted right by $2$ units.

The vertex moves from $(0,0)$ to $(2,0)$. The arms of the graph still make the same angles, because the shape is unchanged.

How to Read and Write Translation Rules

In IB Mathematics, you should be comfortable moving between a graph description and an equation description.

If the graph of $y=f(x)$ is translated by the vector $\begin{pmatrix} h \\ k \end{pmatrix},$ then the new graph is

$$y=f(x-h)+k.$$

This is one of the most useful formulas in the topic of functions. It tells you that the input changes first, then the output changes.

Let’s break it down:

Replacing $x$ with $x-h$ shifts the graph right by $h$ units.
Adding $k$ shifts the graph up by $k$ units.

If $h$ is negative, the graph moves left. If $k$ is negative, it moves down.

Example 3: Combined Translation

If $f(x)=\sqrt{x}$, then $y=\sqrt{x-4}-1$ is the graph of $y=\sqrt{x}$ shifted right by $4$ units and down by $1$ unit.

The starting point of the square root graph changes from $(0,0)$ to $(4,-1)$.

This is useful for sketching quickly. Rather than redrawing the entire graph from scratch, you can start from a known graph and translate it.

Translating Different Types of Functions

Translations work on many kinds of functions: polynomial, rational, exponential, and logarithmic models.

Polynomials

For a polynomial like $f(x)=x^3$, the graph of $y=(x-1)^3+2$ shifts right by $1$ and up by $2$.

This changes the location of the turning point or inflection point, but not the general shape of the polynomial.

For example, if $f(x)=x^2-4x+3$, then completing the square gives

$$f(x)=(x-2)^2-1.$$

This shows the vertex at $(2,-1)$. In this form, the graph is clearly a translated version of $y=x^2$.

Rational Functions

For $f(x)=\frac{1}{x}$, the graph of $y=\frac{1}{x-3}+2$ shifts right by $3$ and up by $2$.

The asymptotes also move:

Vertical asymptote: $x=3$
Horizontal asymptote: $y=2$

This is important because translations affect the asymptotes of rational functions while keeping the overall hyperbola shape.

Exponential Functions

For $f(x)=2^x$, the graph of $y=2^{x+1}-5$ shifts left by $1$ and down by $5$.

This is useful in models of growth and decay. For example, if a population model starts one year earlier in the timeline, the graph may shift horizontally.

Logarithmic Functions

For $f(x)=\ln x$, the graph of $y=\ln(x-2)+4$ shifts right by $2$ and up by $4$.

The vertical asymptote moves from $x=0$ to $x=2$. The domain also changes from $x>0$ to $x>2$.

This shows that translations can change the domain and range of a function. That is very important in HL work, because domain restrictions matter when solving equations and inequalities involving functions.

Real-World Meaning of Translations

Translations are not just abstract graph moves. They can represent real changes in a situation.

Imagine a school fundraiser where profit in dollars is modeled by $P(x)$. If the school receives an extra fixed donation of $200$, the new model may be $P(x)+200$. That is a vertical translation upward.

Now imagine a mobile game where scoring begins after a delay. If the original score model is $S(t)$, and the game starts $5$ seconds later, the model may become $S(t-5)$. That is a horizontal translation to the right.

These examples show why horizontal and vertical translations mean different things:

Vertical shifts change the output values.
Horizontal shifts change when the input values occur.

In a graph of distance versus time, shifting right often means a later start time. In a graph of temperature versus time, shifting up may mean the entire day is warmer by a fixed amount.

Common Mistakes and How to Avoid Them

Many students mix up left/right shifts. students, here is the safest way to remember it:

$f(x-3)$ means shift right $3$.
$f(x+3)$ means shift left $3$.

A good strategy is to test a key point. If $(0,0)$ is on the original graph of $y=f(x)$, then:

On $y=f(x)+4$, that point becomes $(0,4)$.
On $y=f(x-2)$, the corresponding point becomes $(2,0)$.

Another mistake is thinking that the graph changes shape during a translation. It does not. If the shape changes, then the transformation is probably a stretch, compression, reflection, or a combination of transformations.

Also remember that translations can be combined. For example, $y=-(f(x-1))+2$ is not just a translation. It includes a reflection because of the negative sign outside the function.

Conclusion

Translations of graphs are a core idea in the study of functions because they show how a graph can move without changing shape. In IB Mathematics: Analysis and Approaches HL, you need to recognize both the visual effect and the algebraic form of translations.

The most important rules are:

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

Being able to describe translations clearly helps with sketching, solving equations, analyzing domains and ranges, and interpreting function models in real situations. With practice, you will spot these shifts quickly and use them confidently across many kinds of functions 🚀

Study Notes

A translation moves every point of a graph the same distance in the same direction.
Translations do not change the shape of a graph.
A point $(x, y)$ translated by $\begin{pmatrix} a \\ b \end{pmatrix}$ becomes $(x+a, y+b)$.
$y=f(x)+k$ shifts a graph up by $k$ units.
$y=f(x)-k$ shifts a graph down by $k$ units.
$y=f(x-h)$ shifts a graph right by $h$ units.
$y=f(x+h)$ shifts a graph left by $h$ units.
Horizontal shifts affect the input, while vertical shifts affect the output.
Translations can change the domain, range, and asymptotes of some functions.
Important function families to know with translations include polynomials, rational functions, exponential functions, and logarithmic functions.
In exam questions, check whether a transformation is only a translation or part of a larger combination of transformations.
A quick way to verify a translation is to track one or two key points on the original graph.