What is a reflection over the x-axis? Firstly, a reflection is a type of transformation representing the flip of a point, line, or curve. The different figures in mathematics can be reflected about points, lines, or even planes, but for this lesson, the only reflections considered will be about lines.
A reflection over the x-axis is a reflection in which the line of reflection is the x-axis. The figure in question is reflected over the x-axis. That is, the points of the figure are reflected over the x-axis and no other line.
How do you reflect over the x-axis? The formula for reflecting a point or line segment is as follows:
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In the image in Figure 1, the x-coordinates of the point are fixed throughout the reflection, and the y-coordinate changes signs. This formula will work for any number of points, as shown by the reflection of the line segment. The reflection of the segment requires the same process, but it is applied to every point on the segment. Therefore, it is possible to reflect any figure (line, curve, etc.) across the x-axis using this simple formula.
Finally, does a reflection across the x-axis make intuitive sense? Any point being reflected across the x-axis will only change the y-coordinate since the reflection is only a vertical transformation. Therefore, the x-coordinate should remain unchanged. The distance the point is from the x-axis is the y-value of the coordinate, so reflecting the point should just change the sign of the y-coordinate.
Given an equation or function, how does one reflect it over the x-axis? Since the x-coordinate of a point on the curve will remain unchanged through the reflection, but the y-coordinate of a point on the curve will change sign, simply change the sign of the y-variable in the equation. This will produce a curve that, when graphed, is a reflection of the original curve across the x-axis.
Here are the general rules for the reflection over x-axis equation:
The outputs of the equation are multiplied by -1, since only the y-coordinate is changed and it represents the outputs.
Here are the general rules for the reflection over the x-axis of a linear equation and a quadratic equation:
Ultimately, in order to reflect an equation over the x-axis, multiply the equation by {eq}-1 {/eq}: {eq}y=f(x) \rightarrow y=-f(x) {/eq}.
Here are some examples of how to reflect different equations across the x-axis:
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Reflection over the y-axis is very similar to reflection over the x-axis. In this case, the reflection is performed about the y-axis. So, the points of the figure being reflected are reflected over the y-axis and no other lines.
Here is the general formula for reflecting across the y-axis:
The y-coordinate in this case is unchanged, but the x-coordinate changes sign. The intuition behind this reflection is similar to the case of the x-axis reflection since the distance to the y-axis is represented by the x-coordinate and that is the only coordinate to be changed.
Here is an image in Figure 2 showing a point and a line segment being reflected across the y-axis:
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In the image in Figure 2, only the x-coordinate is changed, while the y-coordinate remains unchanged. This is because the reflection is purely horizontal, so it will only affect the x-coordinate.
How to reflect over y-axis given an equation or function? In the case of the x-axis reflection, the equation is simply multiplied by {eq}-1 {/eq} in order to produce the new reflection equation. This is because the y-variable is the one being transformed. In this case, the x-variable is the one being transformed. Therefore, the x-variable must be manipulated in order to produce the reflection over y-axis equation.
Here is the general rule for reflection across the y-axis:
The inputs of the equation are being multiplied by {eq}-1 {/eq} since the x-coordinates represent the inputs of the equation.
Here are the general rules for reflecting linear equations and quadratic equations over the y-axis:
Ultimately, when reflecting over the y-axis, replace the x-variable in the equation with {eq}-x {/eq}. This will produce a new equation that is reflected over the y-axis.
Here are some examples of different equations being reflected across the y-axis:
Two more commonly used lines of reflection that arise in mathematics are the lines {eq}y=x {/eq} and the line {eq}y=-x {/eq}. These are the main diagonals of the plane. Here is how to reflect over the line {eq}y=x {/eq}:
The above formula works for reflecting points, but the process for reflecting an equation is more difficult:
The act of reflecting over the line {eq}y=x {/eq} is also known as finding the inverse of the equation. Therefore, reflecting over this line and finding the inverse of an equation are synonymous. Here is an example showing how an equation is reflected over the line {eq}y=x {/eq}:
Consider the equation {eq}y=x^3 {/eq}. What would be its image after reflecting over {eq}y=x {/eq}?
Solution:
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Here is how to reflect over the line {eq}y=-x {/eq}:
This formula works for points, but the method for reflecting an equation is even more difficult.
Here is an example showing how an equation is reflected over the line {eq}y=-x {/eq}:
Consider the same equation as before, {eq}y=x^3 {/eq}. What would be its equation after being reflected over the line {eq}y=-x {/eq}?
Solution:
Here is an image in Figure 4 showing the original equation and its reflected equation:
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Reflections over these types of rigid lines are a type of transformation. They all rely on the principle of rigid motion and flipping over a line of reflection. The distance between a point and a line of reflection is the same as the distance between the line of reflection and the reflected image point. No matter what line is used, this principle will hold.
A reflection is a transformation representing a flip of a point, curve, or some figure. The two primary reflections covered in this lesson are:
Finally, reflections over the lines {eq}y=x {/eq} and {eq}y=-x {/eq} were also covered. Reflections over the line {eq}y=x {/eq} involve switching the coordinates: {eq}(x,y) \rightarrow (y,x) {/eq}. In order to reflect an equation over this line, one must switch the variables so that x and y are interchanged in the equation and then solve for the dependent variable y. This new equation will be of the reflected curve. This is also the action of taking an inverse of an equation or function. Reflections over the line {eq}y=-x {/eq} involve switching the variables as in the previous case and then switching the signs of both of the variables: {eq}(x,y) \rightarrow (-y,-x) {/eq}. The process for reflecting an equation is similar to the {eq}y=x {/eq} case. First, switch the variables and solve the equation as in the previous case. Then, change the signs of the input and output variables to produce the reflection equation.
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The formula for reflecting over the line y=-x first involves switching the variables: (x,y) becomes (y,x). Then, one must change the signs of each of the variables: (y,x) then becomes (-y,-x). For an equation, first switch the variables in the equation so that x becomes y and y becomes x. Then solve for the dependent variable y. Finally, multiply the input and output variables (x and y) by -1: y=f(x) becomes y=-f(-x).
The formula for reflection over the x-axis is to change the sign of the y-variable of the coordinate point. The point (x,y) is sent to (x,-y). For an equation, the output variable is multiplied by -1: y=f(x) becomes y=-f(x).
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