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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 24: CIRCLES

 

SECTION 24.1: THE EQUATION OF A CIRCLE

This chapter will be a very brief introduction to circles. All of you have encountered circles in geometry. Notice the circle below.


1

The line segment that you see inside the circle and starting from the center of the circle is called the "radius." The distance from the center of the circle to any point the boundary of the circle is constant.  The center of a circle is not considered part of the circle: the center is a reference point to determine the radius of a circle.   The line segment that goes through the center of a circle and touches the boundary of the circle is called the "diameter." The diameter is equal to twice the radius. The length around a circle is called the "circumference." See the circle below.

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A circle is not a function! Remember the Vertical Line Test? You can use that test to see that a circle is not a function. Look below at the circle. This circle would have two points on it that would have the same x-values but different y-values. Hence, the circle is NOT a function.

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My goal is to teach you how to recognize the equation of a circle and how to graph a circle. Consider the general rule below:

THE EQUATION OF THE GENERAL FORM OF A CIRCLE


(x - h)2 + (y - k)2 = r2

where h, k, and r are real numbers.
The center of the circle is (h, k) and the radius is r.

Please note that there is addition between the two binomials
and
subtraction between the binomials that are squared.

Below are some examples of equations of circles:

 

ex. 1) Rewrite (x - 3)2 + (y + 2)2 = 81 in the form of a circle. Then neatly sketch the circle.

(x - 3)2 + (y - -2)2 = 92

I rewrote the binomials as subtraction and rewrote the numerical part on the right hand side of the equation as a real number squared. I can now draw some conclusions about the circle.

The center is (3, -2) and the radius is 9.

3

ex. 2) Rewrite x2 + y2 = 25 in the form of a circle. Then neatly sketch the circle.

(x - 0)2 + (y - 0)2 = 52

Notice how I rewrote x 2 + y2 = 25 above! I put x2 + y2 = 25 into the form of an equation of a circle.

The center is (0, 0) and the radius is 5.

Figure

Again, the center is NOT part of the circle, it is a reference point used to graph the circle.

 

ex. 3) Rewrite (x + 4)2 + (y - 4)2 = 64 in the form of a circle. Then neatly sketch the circle.

(x - - 4)2 + (y - 4)2 = 82

Center (-4, 4) with radius 8.

 

Figure

 

Here are some examples of equations that are NOT circles:

 

ex. 1) Is (x + 4)2 - (y + 5)2 = 4 an equation of a circle?

ans:
The above is NOT a circle since subtraction exists between the two binomials squared, but the operation needs to be addition!

 

ex. 2) Is -(x + 4)2 + (y - 5)2 = 4 an equation of a circle?

ans:
The above is NOT a circle because, in this case, you cannot have a negative symbol in front of one of the binomials squared. Why?
HINT: Multiply both sides of the equation by -1 and see what happens.

 

SECTION 24.2: MORE ON GRAPHING THE EQUATION OF A CIRCLE

I will go over an example on graphing a circle using the general form of an equation of a circle. I will explain what I'm doing as I go through the recipe that I use to graph the circle.

 

ex 1) Graph x2 + y2 = 4 using 4 points.

ans:

Step 1) (x - 0) 2 + (y - 0)2 = 4

Step 2) I am able to reveal the center of the circle, (h, k) and the radius r.

Figure

The center of the circle is (0, 0) and the radius is 2.

 

Step 3) Plot center point determined by (0, 0) to use as a reference to graph your circle. Since you know the radius is 2, we can easily get 4 points relative to the center (0, 0).


Figure


Step 4) Remove the center of circle and sketch a circle around the 4 points
.


Figure

 

ex. 2) Graph (x - 3) 2 + (y + 2)2 = 16 using 4 points.

ans:

Step 1) (x - 3) 2 + (y - -2)2 = 42

I rewrote the original equation in the circle form to reveal the center, (3, -2), and the radius 4.

 

Step 2) Plot the center point (3, -2) to use as a reference to graph your circle.

Since you know the radius is 4, we can easily get 4 points relative to the center (3, -2).


Figure

Step 3) Remove the center of circle and sketch a circle around the 4 points.


Figure

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

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RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES