Programmed tutorial: Find the Graph and Equation of a Circle from a Description: Example 1

If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.

Please try to do this problem without a graphing calculator.

Find the graph and general equation of a circle with center at (-3, 2)
and radius R = 5.

We’ll work first on the graph by determining some points on the circle. Then we’ll find its equation. Draw axes on paper and show the center of the circle at the
point (-3, 2). Then click "Next" to check your work.

alt="Java applet graph of the center of the circle." Your browser is completely ignoring the <APPLET> tag!

To sketch the circle (freehand), it helps to locate some points on the circle. Let’s consider the point furthest to the right. How can we find x-coordinate, x_r?

It has to be a distance R to the right of the center, which has an x-coordinate x_c

Write an equation which describes this relationship among x_c, x_r, and R.

x_r = x_c + R

Substitute values to get a numerical value for x_r.

x_r = –3 + 5 = 2

What is the y-coordinate at this extreme right point?

Since this point is “directly” to the right of the enter point, it will have the same y-coordinate as the center.

What is that value?

State the coordinates of this extreme right point
as (x, y).

The extreme right point is (2, 2).

Add this point to your diagram. Then check your work by clicking “Next”.

alt="Java applet graph of a point on the circle." Your browser is completely ignoring the <APPLET> tag!

In a similar way, let’s find the coordinates of the extreme left point. To get x_l, should we measure to the right or the left from the center?

To the left.

Should we add R to x_c or subtract R from x_c?

Subtract.

Determine the value of x_l.

x_l = x_c – R = – 3 – 5 = –8

What is the value of the y-coordinate for this point?

This point also has the same y-coordinate as the center of the circle.

State the coordinates of the extreme left point in (x, y) form.

The point is (–8, 2).

Add this point to your diagram. Then check your work by clicking “Next”.

alt="Java applet graph of points on the circle." Your browser is completely ignoring the <APPLET> tag!

Next we’ll work on the top” and “bottom” of the circle. These two points have one of their coordinates the same as a coordinate of the center. Which coordinate is that?

Since these points are above and below the center, they must have the same x-coordinate as the center.

How do we find their y-coordinates?

In a way similar to what we did before, but involving the y-coordinate. We add the radius, R, to y_c and subtract it from y_c.

Determine the coordinates of the “top” of the circle.

y_top = y_c + R, or
y_top = 2 + 5 = 7
Hence the coordinates at the top are (–3, 7)

Add this point to your diagram. Then check your work by clicking “Next”.

alt="Java applet graph of points on the circle." Your browser is completely ignoring the <APPLET> tag!

Determine the coordinates of the “bottom” of the circle.

y_bottom = y_c – R, or
y_bottom = 2 – 5 = –3
Hence the coordinates at the bottom are (–3, –3).

Add this point to your diagram. Then check your work by clicking “Next”.

alt="Java applet graph of points on the circle." Your browser is completely ignoring the <APPLET> tag!

Now, let’s work on the equation of the circle. State the standard equation.

What is the meaning of h and k?

They are the x- and y-coordinates of the center.

What are their values here?

h = –3 and k = 2

Substitute these values into the standard equation.

We get

Simplify this equation.

The equation becomes

Finally, plot the circle on your diagram. Then check your graph by clicking “Next”.

alt="Java applet graph of a circle." Your browser is completely ignoring the <APPLET> tag!

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor