Graphing Radical Functions
Purplemath
In the example on the previous page, it was fairly simple to find some nice neat plot points for the square-root function. This will not always be the case. Fractions may be helpful sometimes, but often we are stuck working with decimal approximations. In such situations, it becomes even more important to be using a ruler for drawing a graph's axes and scales, and to be plotting points as carefully as we can.
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Graph
First, I'll find the domain of this function (that is, I'll find the allowable x-values) by solving the inequality for where the argument (that is, the expression inside the square root) is non-negative:
Now I know that the "first" point on the graph will occur at , and will head off to the right from there. But I'll find additional plot points in order to draw a good graph. Here is my T-chart:
How did I come up with the x-values above that gave me such nice y-values in my T-chart? On scratch-paper [that is, by doing work off to the side, that isn't included in my hand-in answer], I started with the desired end values (that is, with the nice y-values), and I worked backwards to find the corresponding x-values.
For instance, 25 is a perfect square, so the square root can be simplified to just 5. So I set the argument equal to 25, and solved 3x – 2 = 25 to get 3x = 27, or x = 9, as my starting x-value. Then I showed my work, going forwards, in my T-chart.
Usually, this work need not be shown, either forwards or backwards, when doing graphs. Just make sure that you're comfortable with the process so, when possible, you can get these good plot points.
I'll plot the six points from my T-chart (above), and then I'll sketch my graph:
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Graph
First, I'll find the domain by setting the argument of the square root to be greater than or equal to zero, and then solving this inequality to find the restrictions on the x-values:
Next, I'll find some plot points — at least five — for my graph. (Why "at least five"? Because I've learned, the hard way, that I need more than two or three points for curvy graphs if I'm to have much hope of getting the graph right, and I want to get full points on this question!) I'll start with the end-point of the domain, being :
As before, I found my neat plot-points by setting the argument of the square root equal to a perfect square number that is within the domain, such as 4 or 9, and then solving for the corresponding x-value. Other x-values will work just as well; the choice is up to you.
Now I'll do my graph:
Note that, since the domain starts at , the blue line of my graph could not go to the left of, nor below, the beginning point, . But the function continues forever in the other direction (that is, off to the upper right of my drawing area), so my graph needed to continue to the end of my drawing area on the top right-hand side of my picture.
Make sure you're careful about these domain and range issues for graphing with square roots.
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Graph
As usual, I'll start by finding the domain. In order to do this, I have to solve the quadratic inside the square root; it may be easier just to look at the graph of the quadratic.
Since the parabola is always above the x-axis, then x2 + 1 is always positive). Since the argument of the square root is always positive, then the values for x can be anything; there is no restriction on the domain of this particular square-root function.
If I have to "show my work" for the domain, then I would show that the inequality has a solution of "all real numbers":
x2 + 1 ≥ 0
x2 ≥ –1
Since any squared value is going to be zero or greater, then x2 will certainly always be greater than –1, for all x.
Continuing on, I do my T-chart. In this case, setting the argument of the square root equal to a perfect square number does not work out nicely, so I gave up and used decimal approximations from my calculator for plotting my points. I've listed the y-values accurate to three decimal places, which is more than sufficient for graphing purposes.
Finally, I'll do my graph:
Note that, unlike an absolute-value graph, this graph does not have a sharp turn (that is, an "elbow") at the bottom; the turn is gently curved. Don't go too quickly when graphing with square roots; take the time to notice details like this.
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