lines and parabolas

7/28/2019 Lines and Parabolas

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Lines and Parabolas

Math 11 College Algebra
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Definition

SlopeIf P1(x1,y1) and P2(x2,y2) are two points not having the sameabscissa (ie x1 = x2), the slope m of the line segment P1P2 isgiven by

m =y2 y1x2 x1

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Definition

LineA line is the set of all points in the plane any two of whichhave the same slope m.

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Equation of a Line

Point-Slope Form

Let P1(x1,

y1) be a point on a line having slope m, then anequation of the line is given by

(yy1) = m(x x1)

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Example

Find an equation of the line passing through the points(1,2) and (6, 7)

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Equation of a Line

Slope-Intercept Form

Let m be the slope of a line and b be its y-intercept, then anequation of the line is given by

y = mx + b

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elsewhere...

US, AUS, CAN, EGY, MEX, PRT, PHL y = mx + b

UK, AUS, BHS, BGD, BEL, BRN, CYP,GER, GHA, IND, INA, IRL, JAM, KEN,

KUW, MAS, MAW, MLT, NEP, NED, NZL, y = mx + c

NGR, PAK, SIN, SOL, RSA, SRI, TUR,

UAE, ZMB, ZWE

ALB, BRA, CAT, CZE, DEN, ETH, FRA,LBN, HOL, KGZ, VIE y = ax + b

AZE, CHN, FIN, RUS, UKR y = kx + b

Greece = x +

Italy y = mx + q

Japan y = mx + dLatvia y = jx + t

Romania y = gA + C

Sweden y = kx + m

Slovenia y = kx + n

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Equation of a Line

Horizontal/vertical LineAn equation of a vertical line having x-intercept a is given by

x = a

An equation of a horizontal line having y-intercept b is given

by

y = b



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Equation of a Line

General FormThe graph of the equation

Ax + By + C = 0

where A, B, and C are constants and A and B are not both

zero, is a line.

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Theorem

If l1 and l2 are two distinct non-vertical lines having slopes m1and m2 respectively. Then l1 and l2 are parallel if an only if

m1 = m2



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Theorem

If l1 and l2 are two distinct non-vertical lines having slopes m1

and m2 respectively. Then l1 and l2 are perpendicular if an

only if m1 = 1

m2

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Example

1. Find an equation of the line whose x-intercept is -3 andy-intercept is 4.

2. Find an equation of the line through the point (-2,3) andparallel to the line whose equation is 2xy 2 = 0.

3. Find an equation of the line through the origin andperpendicular to the line whose equation is

2x 5

y+ 6 = 0.

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Definition

Parabola

A parabola is the set of all points in the plane equidistantfrom a fixed point called focus and a fixed line calleddirectrix.



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Equation of a ParabolaAn equation of the parabola having its focus at (0, p) andhaving as its directrix the line y = p is

x

2

= 4pyThe line through the focus perpendicular to the directrix is

called the axis of the parabola. The intersection of theparabola with its axis is called the vertex. The chord through

the focus perpendicular with the axis is called the latusrectum which is of length |4p|.

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Example

1. Sketch the graph of the parabola having an equation

a. x2 = 8y

b. x2

= 12y

c. y2 = x

d. 3y2

= 4x

2. Find an equation then sketch the parabola having focus at(0,2) and directrix the line y = 2.

3. Find an equation then sketch the parabola having vertexat then origin and directrix the line 2y + 5 = 0.

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