Linear Functions and Slope

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Linear Functions and Slope
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• We call function with straight line graphs linear
  functions.

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• A major difference between two straight lines is
  the direction and steepness of the lines. This is
  measured with slope.
• Given two distinct points (x1, y1) and (x2, y2),
  the slope of the line through these points is
  given by

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• Example 1 Find the slope of the line through the
  given points
• (-1, -1) and (3, 11)
• (2, -2) and (5, 4)
• Be careful to make sure that you have the ys and
  xs in the right order. If you start with the y
  value of one point on top, you must start with
  the x value of the same point on the bottom. If
  you reverse them, you will get the incorrect
  sign.

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Point-Slope Form of the Equation of a Line

• If (x1, y1) is on a line, and (x, y) is any other
  point on that line, then the slope is
• m (y y1)/(x-x1).
• If we multiply both sides by (x-x1), we get the
  point-slope equation
• y y1 m(x x1)

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Example 2 Write a point-slope form of the
equation of the line passing through (-1, -2)
with a slope of 6. Then solve the equation for
y. Example 3 Write a point-slope form of the
line passing through (-2, 7) and (4, 5).
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• When we solve our equation of a line for y, we
  get the slope-intercept form
• y mx b
• where m is the slope and b is the y-intercept

• To graph y mx b using the slope and b
• Plot the y-intercept, (0, b)
• Write the slope as a fraction (rise/run).
• Using the rise/run find a second point.
• Draw a straight line through these points.

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Example 4 Graph the line whose equation is y
1/3 x - 2. Example 5 Graph the line whose
equation is y -1/3 x - 2. Example 6 Graph
the line whose equation is y 3 x - 2.
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Vertical and Horizontal Lines

• Example 7 Graph y 2 and x 3 and find their
  slopes
• For any constant k,
• y k is a horizontal line through (0,k) with
  slope 0.
• x k is a vertical line through (k, 0) with
  undefined slope.

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General Form of a Line

• If we move everything to one side of the
  equation, we get the general form of a line
• Ax By C 0.
• A, B, and C are real numbers with at least one of
  A and B nonzero.
• (Note Any line can be written in this form, but
  vertical lines cannot be written in point-slope
  or slope-intercept form.)

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Example 8 Find the slope and the y-intercept of
the line whose equation is 5x 2y 8 0.
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Equations of Lines

• Point-slope form y y1 m(x x1)
• Slope-intercept form y m x b
• Horizontal line y b
• Vertical line x a
• General form Ax By C 0

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What does slope tell us

• If m is positive, the lines rises from left to
  right.
• If m is negative, the line falls from left to
  right.
• The larger m, the steeper the line.
• If m 0, the line is horizontal.
• If m is undefined, the line is vertical.

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• Example 9 Match the equations to the graphs
• f(x) -4x1 b) f(x) -1/2 x 1
• c) f(x) x 1 d) f(x) 3x 1

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Parallel Lines

• Two lines which never intersect are called
  parallel. These lines will have the same slope.
• Example 10 Write an equation of the line passing
  through (-4, 3) and parallel to the line whose
  equation is y 2x 9. Express the equation in
  point-slope form and slope-intercept form.

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Perpendicular Lines

• Two lines are perpendicular if they intersect at
  right (90 degree) angles.
• If y m1xb1 is perpendicular to y m2xb2,
  then m1m2 -1.
• That is m1 -1/m2 and m2 -1/m1.
• Such numbers are called negative reciprocals.

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Example 11 Find the slope of any line that is
perpendicular to the line whose equation is 3x
2y 7 0. Then find the equation of a
particular perpendicular line through (3,6).