Numbers and Their Application - Lesson 19?

Quadratic Relations and Conic Sections

Lesson Overview

To differentiate between quadratic relations and quadratic functions, the general equation of a quadratic function follows:



y = ax2 + bx + c.

The above formula, is in the shape of a parabola. We might want to check to see if it passes the vertical line test and actually is a function. To perform such a test, simply pick a value of "x" and draw a vertical line through it. If any such line crosses the graph more than once then the vertical line test is said to have failed and the relation is not a function. Since all polynomials are functions, and this is a polynomial, we expect it to pass the vertical line test.

Consider next the relation:



y2 = x

Keeping this in mind, we can now view a quadratic relation, which is specified by the general equation (or inequality) of the form:



Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Distance Formula:

The distance formula is derived from the pythagorean theorem which says that the sum of the squares of the two sides of right triangle is equal to the square of the hypotenuse. Thus the distance (d) between two known points, (x 1 ,y 1 ) and (x 2 ,y 2 ) is the square root of the following:



d2 = (x 2 - x 1 )2 + (y 2 - y 1 )2.

Completing the Square:

If the coefficient of the quadratic term is equal to one, as in x2+bx, then the number that will complete the square can be found by halving the linear coefficient, (b), squaring it, and adding the result: x2 + bx + (b/2)2=(x + b/2)2. When the coefficient of the quadratic term is not equal to one you must factor if out first as illustrated in some examples below.

Vertex:

In a parabola, the x coordinate of the vertex is given by: h = -(b/2a). The y coordinate is given by k=y(h) or k=c-b2/(4a). The x coordinate equation should be easy to remember since the roots (zeroes, x-intercepts, solutions) of a quadratic are symmetric about the vertex and these roots are given by the quadratic formula. h = -(b/2a) is thus the portion of the quadratic formula without the ± portion. The y coordinate formula can be derived by substituting this h as x into y(x).

Inequalities:

" ">" indicates the region outside the conic section.

A circle is a collection of points (x,y) in a coordinate plane, such that each point is equidistant from a fixed point (h,k) known as the center. For circles the coefficients for the x2 and y2 terms in the general quadratic relationship are equal (i.e. A=C).

Graph of a circle x2 + y2 = 25 Using a square view window

The equation of a circle in standard form is as follows:

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

Remember:

( h,k ) is the center point.

) is the center point. r is the radius from the center to the circle's (x,y) coordinates.

Example:

x2 + y2 + 6x - 4y - 12 = 0

Step 1 - Commute and associate the x and y terms; additive inverse the -12:

(x2 + 6x) + (y2 - 4y) = 12

Step 2 - Complete the squares, (what you do to one side be sure to do to the other side):

(x2 + 6x + 9) + (y2 - 4y + 4) = 12 + 9 + 4

Step 3 - Factor:

(x + 3)2 + (y - 2)2 = 25 = 52

Observations The conic section will be a circle since the x2 and y2 terms have the same sign and equal coefficients. The center, (h,k), is (-3,2). Note, these are the values of x and y which make the corresponding term equal to zero. The radius of the circle will be 5 units, since the square root of 25 is 5. A circle can be drawn with a compass or one thumbtack and a string. The eccentricity of a circle is zero (e=0).

An ellipse is also a collection of points (x,y) in a coordinate plane. It is very similar to a circle, but somewhat "out of round" or oval. For an ellipse, the x2 and y2 terms have unequal coefficients, but the same sign (A C, and AC > 0). (The plural of ellipse is ellipses, which is also: .... Both stem from the same basic root meaning to leave out.)

Graph of an ellipse 2x2 + y2 = 25 Using a square view window

Ellipses have the following standard form:



((x-h)/r x )2 + ((y-k)/r y )2 = 1

Remember: ( h,k ) is the center point.

) is the center point. r x is the length of the radius in the ± x -direction.

is the length of the radius in the ± -direction. r y is the length of the radius in the ± y-direction.

Example:



x2 + 4y2 = 16

(x2)/16 + (4y2)/16 = 1.

Step 2 - Simplify the second term:

(x2)/16 + (y2)/4 = 1.

Step 3 - Factor/rewrite in standard form:

(x/4)2 + (y/2)2 = 1.

Observations The SAME sign but DIFFERENT coefficients for the x2 and y2 terms tell us that the graph will be an ellipse. The two denominators, 4 and 2, (located in step 3) tell us that the ellipse's vertices are found 4 units from the center (0,0) in the ± x-direction and two other critical points are located 2 units from the center in the ± y-direction. r x =4 is called the x-radius and is the distance from the center to the ellipse in the x-direction. r y =2 is called the y-radius and is the distance from the center to the ellipse in the y-direction. The semi-major axis is the larger of r x and r y , in this case 4. The semi-minor axis is the smaller of r x and r y , in this case 2. Semi- means half. Thus the major and minor axes are twice the semi-major and semi-minor axes. An ellipse can also be described as the set of points in a plane such that the sum of each point's distance, d 1 + d 2 , from two fixed points F 1 and F 2 is constant. Thus an ellipse may be drawn using two thumbtacks and a string. F 1 and F 2 are foci, that is each is a focus. They are located at (h±c,k) or (h,k±c) The distance from the center to a focus is the focal radius. If a is the semi-major axis and b is the semi-minor axis, then c is the focal radius, where d 1 + d 2 = 2a, and c2=a2-b2. In this case, c2=16-4=12. The eccentricity e of an ellipse is given by the ratio: e=c/a. Since c < a and both are positive this will be between 0 and 1. An eccentricity close to zero corresponds to an ellipse shaped like a circle, whereas an eccentricity close to one corresponds more to a cigar. The area of an ellipse is: A= ab. The circumference must generally be approximated. The latus recta of an ellipse are line segments through a focus with endpoints on the ellipse and perpendicular to the major axis. Their length is 2b2/a.

A Parabola has an equation that contains only one squared term. If the x2 term is excluded, then the graph will open in an x-direction. If the y2 term is excluded, then the graph will open in a y-direction. Only graphs which open in the ±y-direction are quadratic functions, thus those which open in the ±x-direction are quadratic relations.

Graph of a parabola x=y2 - 25 In a distorted view window

Parabolic functions have the general equation:



y = ax2 + bx + c

A general parabolic relation has the general quadratic relation equation located on the opening page, except either A=0 or C=0.

Example:



x = -2y2 +12y -10

Observations The conic section will be a parabola because there is only one squared term, y2. Since the x2 term is missing, the graph will open in an x-direction, specifically the -x since C < 0. The y-coordinate of the vertex is found by the formula: k = -b/2a. So that k = -12/2(-2) = 3. The x-coordinate of the vertex is: h = -2(32) + 12(3) - 10 = 8. The eccentricity of a parabola is one (e=1). A parabola can be described as the set of coplanar points each of which is the same distance from a fixed focus as it is from a fixed straight line called the directrix. The midpoint between the focus and the directrix is the vertex. The line passing through the focus and the vertex is the axis of the parabola. A focal chord is a line segment passing through the focus with endpoints on the parabola. The latus rectum is the focal chord perpendicular to the axis of the parabola. Another standard form for a parabola is: (x-h)2=4p(y-k) or (y-k)2=4p(x-h) The focus lies on the axis p units from the vertex: (h,k+p) or (h+p,k). The directrix is the line y=k-p or x=h-p

An Hyperbola has two symmetric, disconnected branches. Each branch approaches diagonal asymptotes*. Hyperbolas can be detected by the opposite signs of the x2 and y2 terms. (AC < 0).

*(Asymptotes are lines which a graph gets arbitrarily close to, but never actually touches as the variable continues to move in the positive or negative direction.)

Graph of a hyperbola -x2+y2 = 25 In a distorted view window

Hyperbolas have the specific equations:

((x-h)/r x )2 - ((y-k)/r y )2 = 1 OR -((x-h)/r x )2 + ((y-k)/r y )2 = 1

Remember: ( h,k ) is the center point.

) is the center point. r x is the distance from the center to the hyperbola's ± x -direction vertex (or asymptote).

is the distance from the center to the hyperbola's ± -direction vertex (or asymptote). r y is the distance from the center to the hyperbola's ± y -direction vertex (or asymptote).

is the distance from the center to the hyperbola's ± -direction vertex (or asymptote). asymptotes have slopes of r y /r x and -( r y /r x )

and -( ) A hyperbola is the set of points in a plane such that for each point ( x,y ) on the hyperbola, the difference between its distance from two fixed foci is a constant.

) on the hyperbola, the difference between its distance from two fixed foci is a constant. The semi-major axis, a , is the larger of r x and r y .

, is the larger of and . The semi-minor axis, b is the smaller of r x and r y .

is the smaller of and . The transverse axis connects the two vertices.

connects the two vertices. The conjugate axis is perpendicular to the transverse axis.

is perpendicular to the transverse axis. Thus the transverse axis is twice the semi-major, a = abs( d 1 - d 2 ).

= abs( - ). c 2 =a 2 +b 2 .

. The eccentricity e of a hyperbola is given by the ratio: e=c/a. Since c > a and both are positive this will be greater than 1. If e is close to one, the hyperbola will be narrow and pointed; whereas if e is large, the hyperbola will be nearly flat.

Example:



-(x/4)2 + (y/3)2 = 1

Observations The conic section will be a hyperbola since the x2 and y2 terms have different signs. The graphs open in the ±y-direction since the sign before the y-term is positive. The asymptotes would have a slope of 3/4 or -(3/4).

22((x-3)2+(y-0)2)=(x--3)2+(y-0)2

This comes from applying the distance formula, but both sides have been squared. This leads to the following relationships:



4(x2-6x+9+y2)=x2+6x+9+y2

3x2-30x+27+3y2=0 or (x-5)2+y2=42.

We thus have a circle centered at (5,0) with radius 4.

Another example might be as follows: Each point is equidistant from the point (3,-4) and the line y=2. Thus:

(x-3)2+(y+4)2=(x-x)2+(y-2)2.

x2-6x+9+y2+8y+16=0+y2-4y+4.

x2-6x+12y+21=0 or y+1=-(x-3)2/12.

We thus have a parabola with vertex at (3,-1) opening in the -y direction.

One last example is as follows: For each point, its distance from the point (0,3) is 3/2 times its distance from the line y=-3.

4((x-0)2+(y-3)2)=9((x-x)2+(y+3)2)

4(x2+y2-6y+9)=9(y2+6y+9)

4x2-5y2-78y-45=0

We thus have a hyperbola opening in the x direction.

Several steps have been omitted from the above derivations and it behooves the student to check these over, verify, and master them, because several common algebraic mistakes often occur.

If the discriminant is less than zero we have a circle (if A = C) or an ellipse;

if the discriminant is equal to zero we have a parabola;

if the discriminant is greater than zero we have a hyperbola.

Our equation can be rewritten with B' = 0 by rotating the coordinate axes through an angle 0 , where cot(2 0 )=(A-C)/B. Note that F = F' is invariant under rotation. Note also A + C = A' + C' and B2-4AC = (B')2-4A'C'. We choose B' = 0.

The simplest hyperbola comes from the graph: y=1/x or xy=1. For this relationship we note that A=0, B=1, and C=0. Thus cot 2 0 =0 or 0 = /4. Thus in a x'y' coordinate system, which is rotated by 45° from our normal xy coordinate system, our equation would be:

(x'/ )2- (y'/ )2=1.