Reflections

In mathematics, particularly the study of functions and graphs, a reflection is a type of transformation. Reflections provide a unique way to manipulate functions and their graphs, creating a mirror image by flipping over a chosen line such as the x-axis or y-axis. Such transformations are fundamental in understanding how functions behave and are essential tools in algebra and calculus.

Let us take a deeper look at reflections - what they are and how they affect functions and their graphs - and provide plenty of examples to aid understanding.

What are images?

A reflection is a transformation that "flips" the graph over a line. This line is known as the line of reflection. We will mainly discuss two types of reflections with respect to the coordinate plane:

• Reflection about the x-axis
• Reflection about y-axis

Each type of reflection will transform the graph in a different way, as we will see in the sections below.

Reflection about the x-axis

When we reflect a function across the x-axis, each point on the graph of the function is reflected across the x-axis. In simple terms, this type of reflection changes the sign of the y-coordinate of each point. If the function is originally y = f(x), its reflection across the x-axis is given by:

y = -f(x)

The effect of this transformation is that the graph appears upside-down compared to its original form.

Example of reflection across the x-axis

Consider the simple quadratic function:

y = x^2

Below is a visual representation of what this function looks like on a graph:

Now let's reflect this function across the x-axis:

y = -x^2

The graph of this function is given below:

In the graph above, you can see that the parabola, which originally opened upward, now opens downward as a result of being reflected across the x-axis.

Reflection about y-axis

We will now look at reflections across the y-axis. In this transformation, each point on the graph of the function is reflected across the y-axis. This means that the x-coordinates of each point are negative. If the function is originally y = f(x), its reflection across the y-axis is given by:

y = f(-x)

This transformation effectively reverses the direction of the graph horizontally.

Example of reflection about the y-axis

Consider the exponential function:

y = 2^x

Let's look at the graph of this function:

Now, reflect this function across the y-axis:

y = 2^(-x)

The mirrored graph looks like this:

This graph shows the function mirrored around the y-axis, and shows how it now decreases as it moves to the right, the opposite of its original direction.

General reflection in works

Reflections are versatile tools in mathematics and apply to a variety of functions that cannot be easily plotted or that are conceptually abstract. Complex reflections can apply to entire equations or systems of equations.

Consider these types of tasks:

y = log(x)

When reflected about the y-axis it forms:

y = log(-x)

This transformation mirrors the behavior of logarithmic functions when reflected around the y-axis, which affects its domain and effectively reflects inverse symmetry in logarithmic terms.

Another example is the trigonometric functions:

y = sin(x)

Its image on the x-axis becomes:

y = -sin(x)

This transformation shows how the peaks and troughs of the sine function are reversed, yet the wave continues in its periodic nature.

Combination of reflection and other transformations

Reflections do not have to stand alone in transforming a graph. They can be combined with other transformations such as translation, stretch, and compression. It is important to understand the sequence of these operations because the order can affect the final result.

For example, a function can be stretched vertically before being reflected across the x-axis. Consider:

y = 3*(x-1)^2

This function is a simple parabola, shifted to the right by 1 and extended vertically by a factor of 3.

On reflecting across the x-axis, we get:

y = -3*(x-1)^2

Visualizing the overlay of these transformations not only helps simplify complex function behavior but also helps solve practical problems that involve symmetry and inversion in direction.

Conclusion

Reflections are powerful transformations in the math toolkit for understanding and manipulating functions. By reflecting a graph across the x-axis or y-axis, we gain information about properties and symmetries of the function, enhancing our understanding of algebraic expressions and their graphical equivalents.

Being able to visualize, interpret, and apply these transformations expands your ability to effectively solve mathematical problems, making reflection a fundamental concept in the study of functions and graphs.

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