Reflections of Graphs

Reflections of graphs help you understand how functions change when their graphs are flipped over a line. In IB Mathematics: Analysis and Approaches HL, this idea is part of the wider study of transformations of functions, and it appears in many places: interpreting data, solving equations, comparing models, and using inverse functions. students, by the end of this lesson you should be able to explain what a reflection is, identify the line of reflection, and predict how the equation of a function changes when its graph is reflected. ✨

Learning objectives

• Explain the main ideas and terminology behind reflections of graphs.
• Apply IB Mathematics: Analysis and Approaches HL reasoning or procedures related to reflections of graphs.
• Connect reflections of graphs to the broader topic of functions.
• Summarize how reflections of graphs fit within functions.
• Use evidence or examples related to reflections of graphs in IB Mathematics: Analysis and Approaches HL.

What a reflection means

A reflection is a transformation that produces a mirror image of a graph across a line called the line of reflection. If a point on the original graph is $(x, y)$, then its reflected image will be the same distance from the line of reflection, but on the opposite side. This preserves shape and size, but reverses orientation.

The most common reflections in function work are across the $x$-axis, the $y$-axis, and the lines $y=x$ and $y=-x$. These reflections are especially important because they connect graph behaviour with algebraic rules.

For a function $y=f(x)$:

• Reflection in the $x$-axis gives $y=-f(x)$.
• Reflection in the $y$-axis gives $y=f(-x)$.

These two formulas are some of the most important transformation rules in IB Maths. They allow you to describe reflections without drawing every point by hand. 📈

A key idea is that reflections change the graph in a predictable way:

• Reflecting over the $x$-axis changes every output value $f(x)$ to $-f(x)$.
• Reflecting over the $y$-axis changes every input value $x$ to $-x$.

Notice the difference between these two. In $y=-f(x)$, the output changes sign. In $y=f(-x)$, the input changes sign. This is one of the most common places where students make mistakes, so students, it is worth being very careful.

Example 1: Reflection over the $x$-axis

Suppose $f(x)=x^2$. Then the reflected graph in the $x$-axis is

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

The point $(2,4)$ on $y=x^2$ becomes $(2,-4)$ after reflection. Similarly, $(-3,9)$ becomes $(-3,-9)$.

This is useful when comparing a positive parabola and a negative parabola. The graph of $y=-x^2$ opens downward, while $y=x^2$ opens upward.

Reflections in the coordinate axes

Reflections across the coordinate axes are the most direct transformations in function notation.

Reflection in the $x$-axis

If $y=f(x)$, then the reflected graph is $y=-f(x)$.

This can be interpreted as a vertical reflection. All $y$-values are multiplied by $-1$.

Real-world meaning: if a graph represents profit, then $-f(x)$ could represent loss. If it represents height above a reference line, then the reflection might describe the same shape below that line.

Reflection in the $y$-axis

If $y=f(x)$, then the reflected graph is $y=f(-x)$.

This is a horizontal reflection. The graph is flipped left to right.

A good way to think about it is this: each point $(a,b)$ on the original graph becomes $(-a,b)$ on the reflected graph.

Example 2: Reflection in the $y$-axis

Let $f(x)=x^3-2x$.

Then

$$f(-x)=(-x)^3-2(-x)=-x^3+2x.$$

So the reflection in the $y$-axis is the graph of

$$y=-x^3+2x.$$

If the original graph passes through $(1,-1)$, then the reflected graph passes through $(-1,-1)$.

This reflection is especially interesting for odd and even functions. If a function satisfies $f(-x)=f(x)$, it is even and symmetric about the $y$-axis. If a function satisfies $f(-x)=-f(x)$, it is odd and symmetric about the origin. These symmetry ideas are closely linked to reflections. 🔁

Reflections and other important lines

While the $x$-axis and $y$-axis are the main axes of reflection in function transformations, IB Maths also uses reflection across the lines $y=x$ and $y=-x$. These lines matter a lot when studying inverse functions.

Reflection in the line $y=x$

The graph of an inverse function is the reflection of the original graph in the line $y=x$, provided the function has an inverse.

If a point $(a,b)$ is on the graph of $y=f(x)$, then the point $(b,a)$ is on the graph of $y=f^{-1}(x)$.

For example, if $f(x)=2x+3$, then to find the inverse:

$$y=2x+3$$

Swap $x$ and $y$:

$$x=2y+3$$

Solve for $y$:

$$y=\frac{x-3}{2}.$$

So the inverse function is

$$f^{-1}(x)=\frac{x-3}{2}.$$

The graph of $y=2x+3$ and the graph of $y=\frac{x-3}{2}$ are reflections of each other in the line $y=x$.

This is a powerful way to check whether an inverse is correct. If the two graphs are mirror images across $y=x$, the inverse relationship is confirmed. ✅

Reflection in the line $y=-x$

Reflection in $y=-x$ is less common in basic function transformations, but it is useful in coordinate geometry and advanced graph analysis. A point $(a,b)$ maps to $(-b,-a)$.

This reflection combines a swap of coordinates with sign changes. It can appear in more advanced contexts involving relations, inverse relationships, or geometry-based transformations.

How reflections connect to function notation

Function notation makes reflections easier to describe clearly.

If $f(x)$ is a function, then:

• $-f(x)$ means the outputs are reflected over the $x$-axis.
• $f(-x)$ means the inputs are reflected over the $y$-axis.
• $f^{-1}(x)$ means the graph is reflected in the line $y=x$ when the inverse exists.

It is important not to confuse $-f(x)$ with $f(-x)$. These expressions do different things.

Example 3: Compare $-f(x)$ and $f(-x)$

Let $f(x)=x^2+1$.

Then

$$-f(x)=-(x^2+1)=-x^2-1,$$

while

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

Here, $f(-x)=f(x)$ because the function is even. But $-f(x)$ is very different: it reflects the graph across the $x$-axis.

This comparison shows why symbols matter. A small change in notation can produce a completely different graph.

Reflections in polynomial, rational, exponential, and logarithmic graphs

Reflections can be used with many families of functions in the IB syllabus.

For polynomial functions, reflection changes whether the graph opens up or down, or whether it is flipped left to right. For example, $y=x^4$ reflected in the $x$-axis becomes $y=-x^4$.

For rational functions, reflection can change the sign of the branches. If $f(x)=\frac{1}{x}$, then reflecting in the $x$-axis gives

$$y=-\frac{1}{x}.$$

Reflecting in the $y$-axis gives

$$y=\frac{1}{-x}=-\frac{1}{x},$$

which is the same graph because $\frac{1}{x}$ is odd. This is a useful example showing how symmetry can make reflections produce the same equation.

For exponential functions, such as $f(x)=2^x$, a reflection in the $y$-axis gives

$$f(-x)=2^{-x}=\left(\frac{1}{2}\right)^x.$$

This turns growth into decay. That is a major modelling idea. A quantity that grows as $2^x$ when time moves forward may behave like $\left(\frac{1}{2}\right)^x$ when the direction is reversed.

For logarithmic functions, reflections also appear through inverses. Since logarithmic and exponential functions are inverses, their graphs are reflections of each other in the line $y=x$.

For example, if $f(x)=e^x$, then its inverse is

$$f^{-1}(x)=\ln x.$$

These graphs are mirror images across $y=x$.

Common reasoning and exam-style thinking

In IB Mathematics, you should be able to reason from a graph, an equation, or a description.

If you are given a graph and asked for its reflection:

1. Identify the line of reflection.
2. Track a few clear points.
3. Reflect each point the same distance across the line.
4. Write the new equation if needed.

If you are given an equation, determine the transformation rule:

• Across the $x$-axis: multiply the function by $-1$.
• Across the $y$-axis: replace $x$ with $-x$.

Example 4: A full reflection question

If $f(x)=\sqrt{x}$, what is the graph of its reflection in the $y$-axis?

Replace $x$ with $-x$:

$$y=f(-x)=\sqrt{-x}.$$

This graph exists only when $-x\ge 0$, so $x\le 0$.

The reflected graph starts at $(0,0)$ and extends to the left instead of the right. This is a classic example of how reflections can change the domain of a function.

That domain shift is important in IB Maths because transformations affect not only the picture of the graph but also the set of allowable inputs.

Conclusion

Reflections of graphs are a core part of function transformations in IB Mathematics: Analysis and Approaches HL. They show how the same function shape can appear in different positions or orientations, and they connect directly to symmetry, inverse functions, and modelling. students, if you can move confidently between graphs, coordinates, and equations, then you have a strong understanding of this topic. Remember the key rules: $y=-f(x)$ reflects in the $x$-axis, $y=f(-x)$ reflects in the $y$-axis, and inverse functions reflect in the line $y=x$. These ideas are central to the language of functions and are used throughout the syllabus. 🌟

Study Notes

• A reflection creates a mirror image of a graph across a line called the line of reflection.
• For $y=f(x)$, the reflection in the $x$-axis is $y=-f(x)$.
• For $y=f(x)$, the reflection in the $y$-axis is $y=f(-x)$.
• A point $(a,b)$ reflected in the $x$-axis becomes $(a,-b)$.
• A point $(a,b)$ reflected in the $y$-axis becomes $(-a,b)$.
• The graph of an inverse function is reflected in the line $y=x$.
• Even functions satisfy $f(-x)=f(x)$ and are symmetric about the $y$-axis.
• Odd functions satisfy $f(-x)=-f(x)$ and are symmetric about the origin.
• Reflections apply to polynomial, rational, exponential, and logarithmic functions.
• Always check whether a transformation changes the domain or range.