MAT.116
2.1-2.2

2.1 The Cartesian Coordinate System and Straight Lines

Vocabulary

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Cartesian coordinate system coordinate plane xy-planeAxes x-axis y-axisOriginOrdered pair coordinates (x, y) (distance from vertical axis, distance from horizontal axis)Quadrants I (+,+) II (-,+) III (-,-) IV (+,-)

Straight Lines

Slope of a Line

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The slope, m, of a line containing the distinct points (x1,y1) and (x2,y2) is given by vertical changeslope = m = -------------------- horizontal change y2-y1 = ------ x2-x1 rise = ---- run change in y = ------------- change in xm>0 implies the line increases from left to rightm<0 implies the line decreases from left to rightm=0 implies a horizontal linem undefined imples a vertical line

Parallel Lines

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Two distinct lines are parallel if and only if their slopes are equal or their slopes are undefined.m1=m2

Perpendicular Lines

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If L1 and L2 are distinct nonvertical lines that have slopes m1 and m2 respectively, then L1 is perpendicular to L2 if and only ifm1 = -1 / m2or alternatelym1 * 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.

2.2 Equations of Lines

Point-Slope Form

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An equation of the line that has slope m and passes through the point (x1,y1) is given byy - y1 = m(x - x1)This can be derived from the formula for slope.

Slope-Intercept Form

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The equation of the line that has slope m and intersects the y-axis at the point (0,b) is given byy = mx + bThis is also the function form of linear equations

x-intercept

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

y-intercept

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

Standard Form

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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.

General Form

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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.

Horizontal and Vertical Lines

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Vertical lines have the form x = a.Horizontal lines have the form y = b.