Elementary Functions
Part 2, Polynomials
Lecture 2.1c, Equations of circles
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU)
Elementary Functions
2013
29 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
We digress for a moment from our study of polynomials to consider another
important quadratic equation which appears often in calculus. This is an equation
in which both x and y are squared, the equation for a circle.
Suppose we have a circle centered at the point (a, b) with radius r.
Let (x, y) be a point on the circle.
We can draw a right triangle with short sides of lengths (x − a) and (y − b) and
hypotenuse of length r.
Smith (SHSU)
Elementary Functions
2013
30 / 33
The equation of a circle
By the Pythagorean Theorem,
(x − a)2 + (y − b)2 = r2 .
(3)
This is the general equation for a circle.
Smith (SHSU)
Elementary Functions
2013
31 / 33
The equation of a circle
By the Pythagorean Theorem,
(x − a)2 + (y − b)2 = r2 .
(3)
This is the general equation for a circle.
Smith (SHSU)
Elementary Functions
2013
31 / 33
The equation of a circle
By the Pythagorean Theorem,
(x − a)2 + (y − b)2 = r2 .
(3)
This is the general equation for a circle.
Smith (SHSU)
Elementary Functions
2013
31 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
The equation of a circle
If we are given a quadratic equation in which x2 and y 2 both occur together with
coefficient 1 then we can recover the equation for a circle by completing the
square. For example, suppose we are given the equation
x2 + 4x + y 2 + 2y = 6.
We can complete the square on x2 + 4x, rewriting that as (x + 2)2 − 4 and
complete the square on y 2 + 2y writing (y + 1)2 − 1. So our equation becomes
(x + 2)2 − 4 + (y + 1)2 − 1 = 6 or
(x + 2)2 + (y + 1)2 = 11.
This is an equation for a circle with center (−2, −1) and radius
Smith (SHSU)
Elementary Functions
√
11.
2013
32 / 33
Summary
We have used the concept of “completing the square” to
1
2
find the standard form for a quadratic polynomial,
identify the transformations that turn the general parabola y = x2 into any
other parabola,
3
create the quadratic formula,
4
and briefly analyze the equation of a circle.
In the next presentation, we explore polynomials in general.
(END)
Smith (SHSU)
Elementary Functions
2013
33 / 33
Summary
We have used the concept of “completing the square” to
1
2
find the standard form for a quadratic polynomial,
identify the transformations that turn the general parabola y = x2 into any
other parabola,
3
create the quadratic formula,
4
and briefly analyze the equation of a circle.
In the next presentation, we explore polynomials in general.
(END)
Smith (SHSU)
Elementary Functions
2013
33 / 33
Summary
We have used the concept of “completing the square” to
1
2
find the standard form for a quadratic polynomial,
identify the transformations that turn the general parabola y = x2 into any
other parabola,
3
create the quadratic formula,
4
and briefly analyze the equation of a circle.
In the next presentation, we explore polynomials in general.
(END)
Smith (SHSU)
Elementary Functions
2013
33 / 33
Summary
We have used the concept of “completing the square” to
1
2
find the standard form for a quadratic polynomial,
identify the transformations that turn the general parabola y = x2 into any
other parabola,
3
create the quadratic formula,
4
and briefly analyze the equation of a circle.
In the next presentation, we explore polynomials in general.
(END)
Smith (SHSU)
Elementary Functions
2013
33 / 33
Summary
We have used the concept of “completing the square” to
1
2
find the standard form for a quadratic polynomial,
identify the transformations that turn the general parabola y = x2 into any
other parabola,
3
create the quadratic formula,
4
and briefly analyze the equation of a circle.
In the next presentation, we explore polynomials in general.
(END)
Smith (SHSU)
Elementary Functions
2013
33 / 33
Summary
We have used the concept of “completing the square” to
1
2
find the standard form for a quadratic polynomial,
identify the transformations that turn the general parabola y = x2 into any
other parabola,
3
create the quadratic formula,
4
and briefly analyze the equation of a circle.
In the next presentation, we explore polynomials in general.
(END)
Smith (SHSU)
Elementary Functions
2013
33 / 33
Summary
We have used the concept of “completing the square” to
1
2
find the standard form for a quadratic polynomial,
identify the transformations that turn the general parabola y = x2 into any
other parabola,
3
create the quadratic formula,
4
and briefly analyze the equation of a circle.
In the next presentation, we explore polynomials in general.
(END)
Smith (SHSU)
Elementary Functions
2013
33 / 33