MAT.116 2.1-2.2

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2.2 Equations of Lines

Horizontal and Vertical Lines

Vertical lines have the form x = a.

Horizontal lines have the form y = b.

General Form

The equation Ax + By + C = 0 where A, B, and C are constants and A and B are not both zero, is called the general form of a linear equation in the variables x and y.

Standard Form

The equation Ax + By = C where A, B, and C are constants and A and B are not both zero, is called the standard form of a linear equation in the variables x and y.

Slope-Intercept Form

The equation of the line that has slope m and intersects the y-axis at the point (0,b) is given by

y = mx + b

This is also the function form of linear equations

y-intercept

Points on a graph of the form (0,b) where a graph crosses the y-axis.

x-intercept

Points on a graph of the form (a,0) where a graph crosses the x-axis.

Point-Slope Form

An equation of the line that has slope m and passes through the point (x1,y1) is given by

y - y1 = m(x - x1)

This can be derived from the formula for slope.

2.1 The Cartesian Coordinate System and Straight Lines

Straight Lines

Perpendicular Lines

If L1 and L2 are distinct nonvertical lines that have slopes m1 and m2 respectively, then L1 is perpendicular to L2 if and only if

m1 = -1 / m2

or alternately

m1 * m2 = -1

'

Lines with slope m=0 are perpendicular to lines with undefined slopes; that is, horizontal lines and vertical lines are perpendicular to each other.

Parallel Lines

Two distinct lines are parallel if and only if their slopes are equal or their slopes are undefined.

m1=m2

Slope of a Line

The slope, m, of a line containing the distinct points (x1,y1) and (x2,y2) is given by

vertical change

slope = m = --------------------

horizontal change

y2-y1

= ------

x2-x1

rise

= ----

run

change in y

= -------------

change in x

m>0 implies the line increases from left to right

m<0 implies the line decreases from left to right

m=0 implies a horizontal line

m undefined imples a vertical line

Vocabulary

Cartesian coordinate system

coordinate plane

xy-plane

Axes

x-axis

y-axis

Origin

Ordered pair

coordinates

(x, y)

(distance from vertical axis, distance from horizontal axis)

Quadrants

I (+,+)

II (-,+)

III (-,-)

IV (+,-)