So in each case, the $y$-coordinate stays the same, but $3$ becomes $-5$, $-2$ becomes $0$, $0$ becomes $-2$, and $13$ becomes $-15$. Similar reasoning shows that, for example, When you reflect this point, you should end up at the same "height" ($y$-coordinate) of $-5$, but this time four units to the left of your axis of symmetry. (You should follow along and draw things out on a sheet of graph paper or on your computer, in order to make them clear.) Therefore, if you have a point at $(3, -5)$, it is three units to the right of the $y$-axis, but four units to the right of your axis of symmetry. The line $x = -1$ is a vertical line one unit to the left of the $y$-axis. The best way to practice finding the axis of symmetry is to do an example problem.įind the axis of symmetry for the two functions shown in the images below.Rather than think about transformation rules symbolically, and trying to generalize them, try thinking about them visually. This is because, by it's definition, an axis of symmetry is exactly in the middle of the function and its reflection. In this case, all we have to do is pick the same point on both the function and its reflection, count the distance between them, divide that by 2, and count that distance away from one of the graphs.
#Reflection over x axis how to#
How to Find the Axis of Symmetry:įinding the axis of symmetry, like plotting the reflections themselves, is also a simple process. It can be the x-axis, or any horizontal line with the equation y y y = constant, like y y y = 2, y y y = -16, etc. The axis of symmetry is simply the horizontal line that we are performing the reflection across. But before we go into how to solve this, it's important to know what we mean by "axis of symmetry". In some cases, you will be asked to perform horizontal reflections across an axis of symmetry that isn't the x-axis. Plot new points after dividing y values by -1Īnd that's it! Simple, right? What is the Axis of Symmetry: Remember, pick some points (3 is usually enough) that are easy to pick out, meaning you know exactly what the x and y values are. Step 2: Identify easy-to-determine points So, make sure you take a moment before solving any reflection problem to confirm you know what you're being asked to do.
When drawing reflections across the x x x and y y y axis, it is very easy to get confused by some of the notations. Since we were asked to plot the – f ( x ) f(x) f ( x ) reflection, is it very important that you recognize this means we are being asked to plot the reflection over the x-axis.
Step 1: Know that we're reflecting across the x-axis Below are several images to help you visualize how to solve this problem. Don't pick points where you need to estimate values, as this makes the problem unnecessarily hard. When we say "easy-to-determine points" what this refers to is just points for which you know the x and y values exactly. Remember, the only step we have to do before plotting the − f ( x ) -f(x) − f ( x ) reflection is simply divide the y-coordinates of easy-to-determine points on our graph above by (-1). Given the graph of y = f ( x ) y = f(x) y = f ( x ) as shown, sketch y = − f ( x ) y = -f(x) y = − f ( x ).
The best way to practice drawing reflections across the y-axis is to do an example problem: In order to do this, the process is extremely simple: For any function, no matter how complicated it is, simply pick out easy-to-determine coordinates, divide the y-coordinate by (-1), and then re-plot those coordinates. In a potential test question, this can be phrased in many different ways, so make sure you recognize the following terms as just another way of saying "perform a reflection across the x-axis":ġ) Graph y = − f ( x ) y = -f(x) y = − f ( x ) One of the most basic transformations you can make with simple functions is to reflect it across the x-axis or another horizontal axis. Before we get into reflections across the y-axis, make sure you've refreshed your memory on how to do simple vertical and horizontal translations.
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