2.4 Slopes and Intercepts

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2.4 Slopes and Intercepts

• Algebra II
• Mrs. Spitz
• Fall 2006

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Objectives

• Determine the slope and intercepts of a line
• Use the slope and intercepts to graph a linear
  equation, and
• Determine if two lines are parallel,
  perpendicular, or neither.

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Assignment

• Pgs. 70-71 8-13, 15-40

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Application

• Caryn and Brad used a long board and bricks to
  build a ramp for their radio-controlled model
  car. At every 24 inches, they placed bricks to
  create an incline of 4 more inches. The
  steepness, or slope, of the ramp is the ratio of
  the vertical change to the horizontal change.
• Slope

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Compute slope

• The slope of the car ramp is
• Slope is also defined for the graphs of linear
  functions. In the graph at the right, look for a
  pattern in the relationship of the change in the
  y-coordinates to the change in the x coordinates
  of the points on the graph.

or
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Now what?

• The y-coordinates increase 8 units for each
  2-unit increase in the x-coordinates. The slope
  of the line whose equation is f(x)4x is or
  4. The vertical change is the difference between
  the y-coordinates of any two points on the graph
  the horizontal change is the difference between
  the the corresponding x coordinates. You can use
  the following formula to find the slope of a line
  if you know the coordinates of two points on the
  line.

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Definition of slope

• The slope m of a line passing through points (x1,
  y1) and (x2, y2) is given by

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Ex. 1 Determine the slope of the line that
passes through points (1, -3) and (0, -5). Then
graph the line.
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Ex. 2 Determine the slope of the line that
passes through point (-4, -1) with a slope of ?.
Describe the manner in which the line rises or
falls.

• Graph the ordered pair (-4, -1). Since the slope
  of the line is -?, the vertical change is -2 and
  the horizontal change is 3. From (-4, -1), move
  2 units down and 3 units to the right. This
  point is (-1, -3).
• Connect the points to draw the line.
• Notice that the line falls to the right. Its a
  negative slope.

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Note

• The slope of a line tells the direction in which
  it rises or falls. In example 2, the slope is
  negative and the line falls to the right. The
  graphs on the next slide show the three other
  possibilities for a linear graph.

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Positive, horizontal, vertical slopes.
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Ex. 3 Graph f(x) 3x 2, g(x) 3x, and h(x)
3x 5 on the same coordinate plane. Find the
slope of each line and describe what you notice
about the graphs.

• Find ordered pairs to satisfy each function.
  Connect the ordered pairs to draw each line.
• The slope of each line is 3. The lines appear to
  be parallel.
• Equations whose graphs have similar
  characteristics are often called families of
  equations.

Definition of parallel lines lines with the
same slopes. All vertical lines are parallel.
All horizontal lines are parallel.
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Ex. 4 Graph the line that goes through the
point (0, 3) and is parallel to that line.

• Graph the equation 6y 10x30. Find the slope
  of the line. The slope of the line is
• Now use the slope and the point to graph the line
  parallel to the graph of 6y 10x 30.

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Perpendicular lines

• The figure at the bottom shows the graphs of two
  lines that are perpendicular. We found that
  parallel lines have the same slope. Is there a
  special relationship between the slopes of two
  perpendicular lines? Of course there is.

• The slopes are negative reciprocals of each
  other. That is, when you multiply the slopes of
  two perpendicular lines, the product is ALWAYS -1.

slope of
slope of
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Ex. 5 Without graphing, find the y-intercept and
the x-intercept of the graph of 3x 5y30.

• Since all points on the y-axis have an
  x-coordinate of 0, you can find the y-intercept
  by substituting 0 for x in the equation.

• Likewise to find the x-intercept, you substitute
  0 for y, since all points on the x-axis have 0 as
  a y-coordinate.

The y intercept is 6 and the x intercept is 10.