Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
Steps to Solve
In order to graph y = sqrt(x), we first want to get a better idea of how the graph is going to look. We can do this by making some simple observations. The first thing we should take notice of is the domain and range of y = sqrt(x).
The domain consists of all the inputs, or numbers, we plug in for x that make sqrt(x) defined. Since we can't take the square root of negative numbers without getting a non-real number out, and we can't graph non-real numbers on the coordinate axis, the domain of y = sqrt(x) is all real numbers greater than or equal to 0.
The range consists of all of the outputs, or the values y takes on. Based on the fact that our domain consists of all real numbers greater than or equal to 0, the numbers we will get back out will be real numbers greater than or equal to 0. This leads to the following two facts regarding the domain and range of y = sqrt(x):
- The domain of y=sqrt(x) is all real numbers greater than or equal to 0.
- The range of y=sqrt(x) is all real numbers greater than or equal to 0.
This information gives us an idea of where our graph will be on the coordinate axis as is illustrated in this image:
Now let's plug in some strategic numbers for x and find the corresponding y values. This will give us an idea of the shape of the graph.
Our chart displays different points that will be on the graph. We can tell that the graph will be rising from left to right since y increases as x increases. We also see that the graph starts off rising fairly quickly, but then rises more and more slowly since the y values go up by 1 as the x values get more and more spaced out.
We now have a general idea of how the graph is behaving. The next thing we need to do is plot some of the points in our chart on a graph.
We see the points take on the pattern we expected. The graph will rise from left to right, and it starts off rising quickly, but then slows down. The last thing we need to do is to connect the dots in a smooth line.
We've found the following graph to be the graph of y = sqrt(x).
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There is another way to go about graphing y = sqrt(x), and it revolves around the fact that if we take the function y = x2 and restrict its domain to all real numbers greater than or equal to 0, then we have the inverse function of y = sqrt(x).
Inverse functions are functions that basically undo each other. That is, if we take an input a and plug it into a function f, then we get an output of b. If we plug that output b into the inverse function of f, then we get a back out. This gives way to the following property of inverse functions:
If (a,b) is on the graph of the function f, then (b,a) is on the graph of the inverse function of f.
This is a lot of information, and you are probably wondering how this will help us to graph y = sqrt(x). Well, as it turns out, the graphs of inverse functions are mirror images of one another over the line y = x. We can use this to help us derive the graph of y = sqrt(x) from the graph of y = x2, where x is greater than or equal to 0.
We start with the graph of y = x2 with the restricted domain.
Next, we draw the line y = x and reflect the graph over that line. For accuracy, we can use our fact to take points on y = x2 and interchange the coordinates to find points on y = sqrt(x).
We can see that graph y = sqrt(x) using its inverse function y = x2 with x greater than or equal to 0. Pretty handy, huh?
The square root function of y = sqrt(x) is a function that shows up often in many different areas, so it is useful to have an idea of what the graph of this function looks like. The great thing is, if we forget what it looks like, we now have a couple of ways to go about graphing it, including using the inverse function.
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How to Graph y=sqrt(x)
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