Name:
Date: ______________
Algebra 2
Equations of Circles
Equations of Circles
Objectives: Given a circle’s graph, you will write its equation.
Given an equation for a circle, you will draw its graph.
How can a circle be described by an equation? Circles always
fail to be functions (do you know why?) so we can’t write a
single f(x) formula that produces a circle. There are two ways to
deal with this problem: (1) writing an x-and-y equation that isn’t
a function formula; (2) writing two f(x) formulas, one describing
the top half of the circle and the other describing the bottom half.
Circle equations using center and radius
Here’s the general equation for a circle that has center (h, k) and
radius r:
(x – h)2 + (y – k)2 = r2.
For example, the circle pictured above has this equation: (x – 5)2 + (y – 4)2 = 9.
For many circles, the h and/or k are negative; remember that a double-negative is often written as a +.
You try it
1. Write equations for these circles.
a.
2. Draw the circles that have these equations.
a. (x + 3)2 + (y – 1)2 = 36
b.
b. x2 + (y + 5)2 = 4
Name:
Date: ______________
Algebra 2
Equations of Circles
Getting function equations for a circle
The main disadvantage of the equation form (x – h)2 + (y – k)2 = r2 is that it’s not solved for y.
This shortcoming makes it harder to create an x-and-y table for the function, and impossible to
graph the function using the Y= screen on the calculator.
But what would happen if we tried solving for y? Here’s how it goes for the example circle
from the top of page 1:
(x – 5)2 + (y – 4)2 = 9
(y – 4)2 = 9 – (x – 5)2
y – 4 =  9  ( x  5) 2
note the ± from the square-root step
y
= 4  9  ( x  5) 2
We end up with two functions: one from choosing the + of the ±, the other from choosing the –.
Here are the graphs of these two functions separately:
f(x) = 4 +
9  ( x  5) 2
g(x) = 4 –
9  ( x  5) 2
The formula with the + gives the top half of the circle, and the formula with the – gives the
bottom half! And things work exactly this way for any circle.
You try it
3. Your answer to 1a should have been: (x – 3)2 + (y + 2)2 = 25. Solve this equation for y.
Name:
Date: ______________
Algebra 2
Equations of Circles
4. On your calculator, enter the two functions of your answer to
problem 3 on the Y= screen as Y1 and Y2, then graph. Each
function’s graph should be a semi-circle, and together they should
make a circle centered at (3, –2) with radius 5. Make sure that’s
what you’re seeing, then sketch your screen.
Homework Problems
5. Write equations for these circles.
a.
b.
c.
d.
Name:
Date: ______________
e.
Algebra 2
Equations of Circles
f.
6. Draw the circles that have these equations. Also draw a point at the center and label it with
the coordinates of the center. Make sure that the center, top, bottom, left, and right points are
located accurately; you may sketch the rest of the graph.
a. (x – 2)2 + (y – 3)2 = 49
b. (x + 4)2 + (y + 1)2 = 16
c. (x – 3)2 + (y + 5)2 = 4
d. (x + 4)2 + (y – 5)2 = 6.25
Name:
Date: ______________
e. x2 + (y – 5)2 = 1
Algebra 2
Equations of Circles
f. x2 + y2 = 100
Name:
Date: ______________
7. a. Solve the equation (x – 2)2 + (y – 3)2 = 49 for y.
b. Separate your answer into two parts that are functions, and graph
the functions on your calculator. Together they should form a
circle that matches your answer to problem 6a. Sketch what you
see on the screen.
8. a. Solve the equation (x + 4)2 + (y – 5)2 = 6.25 for y.
b. Separate your answer into two parts that are functions, and graph
the functions on your calculator. Together they should form a
circle that matches your answer to problem 6d. Sketch what you
see on the screen.
9. a. Solve the equation x2 + y2 = 100 for y.
b. Separate your answer into two parts that are functions, and graph
the functions on your calculator. Together they should form a
circle that matches your answer to problem 6f. Sketch what you
see on the screen.
Algebra 2
Equations of Circles
Name:
Date: ______________
Algebra 2
Equations of Circles
10. Here are two functions. Each function’s graph is a semi-circle, and together the graphs make
a circle.
f(x) = –3 +
49  ( x  2) 2
g(x) = –3 –
49  ( x  2) 2
Write a single x-and-y equation for this circle. (You’ll need to think about how your solving
method from problems 3, 7a, 8a, 9a would work going in the reverse direction.)
11. Here are two functions that together form a circle:
f(x) = 2 + 16  ( x  7) 2
g(x) = 2 – 16  ( x  7) 2
Write a single x-and-y equation for this circle.
12. Suppose someone graphed a circle on a calculator by entering
the functions shown in the screen-shot. Write a single x-and-y
equation for this circle.