Graphs, Lines, and Functions

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Graphs, Lines, and Functions

• Sections 2.5, 3.1, 3.2

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The Cartesian Coordinate System

• The Cartesian coordinate system provides a method
  to represent geometrically functions and
  equations in two variables.
• We can thus incorporate a large amount of
  information into a format that is easily
  accessible.

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The Cartesian Coordinate System, cont.
The coordinate system consists of two
perpendicular coordinate axes,
y
I
II
xgt0, ygt0
xlt0, ygt0
which divide the plane into 4 regions, called
quadrants.
x
III
IV
xlt0, ylt0
xgt0, ylt0
The axes are two number lines which allow us to
assign to any point in the plane a unique ordered
pair of coordinates.
The first number in the ordered pair is the
coordinate directly above or below on the x-axis,
the second number is the coordinate directly to
the left or right on the y-axis.
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Graphs of equations and functions

• The graph of an equation in two variables
  consists of all ordered pairs which make the
  equation true.
• Ex. Sketch the graph of the equation x2 y2
  25.

We can identify several ordered pairs which
satisfy the equation, for example (0, 5), (0,
-5), (5, 0), (-5, 0), (4, 3), (4, -3), (3, 4),
and (3, -4).
After plotting these and other points we can
estimate the shape of the graph.
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Intercepts

• Definition An x-intercept of a graph is a point
  where the graph crosses the x-axis. A y-intercept
  of a graph is a point where the graph crosses the
  y-axis.
• The y coordinate of an x-intercept is always 0,
  the x coordinate of an y-intercept is always 0.
• So to find an x-intercept, we can set y 0 and
  solve the equation for x. To find a y-intercept,
  we can set x 0 and solve for y.
• Definition A zero or root of a function is any
  value of x for which f(x) 0.

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Graphs of linear functions

• The graph of a linear function (a function in the
  form f(x) ax b) is a line.
• A linear function has the property that the rate
  of change (the slope) is constant.
• The slope m of a linear function is defined as

where (x1, y1) and (x2, y2) are any two points on
the line.
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Equations of a line

• y - y1 m(x - x1) is the point-slope form of a
  line with slope m passing through the point (x1,
  y1).
• y mx b is the slope-intercept form of a line
  with slope m and y-intercept (0, b).
• We need only the slope and one point or the
  coordinates of two points on the line to find the
  equation of a line.

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Using the Point-Slope Form

• Find the equation of the line passing through the
  points (-3, 20) and (5, 44)
• First we need to find the slope of the line

3

• Then using the point-slope form we get

y - 20 3(x - (-3))
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Linear Functions in Practice

• A truck rental company rents trucks for 30 per
  day plus 0.40 per mile driven. Express the
  daily cost of renting a truck as a function of
  the number of miles driven.
• Let x be the number of miles driven.
• Let C be the total daily cost.
• Then C(x) 30 0.4x

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Linear Functions and Isocost Lines

• A company manufactures two products, lawn chairs
  and hammocks. The lawn chairs cost 5.50 each to
  produce, the hammocks 9. Find an equation for
  the total number of each that can be manufactured
  if the total cost is to be 9900.
• Let x be the number of lawn chairs produced.
• Let y be the number of hammocks produced.

• Then 5.5x 9y 9900
• This line is called an isocost line.
• We can quickly graph this line by finding the
  x-intercept and y-intercept and drawing the line
  between the two.

(0, 1100)
(1800, 0)
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Equations and FunctionsTwo tests

• We can graph any equation in two variables on the
  Cartesian coordinate system. But we can easily
  tell from the graph if it is the graph of a
  function.
• Vertical line test If every vertical line
  crosses a graph at most once, then it is a
  function.

Function
Not a function
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Equations and FunctionsTwo tests

• Some equations can be represented as a function
  in either variable.
• Horizontal line test If every horizontal line
  crosses a graph at most once, then it is monotone
  or one-to-one.
• A monotone function is always increasing or
  always decreasing.
• A monotone function has an inverse.

Not one-to-one
One-to-one
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Demand and Supply Equations

• The graphs of demand and supply equations (called
  demand and supply curves) are customarily graphed
  with the quantity q on the horizontal axis and
  the price p on the vertical axis.
• Demand and supply equations are generally
  monotone functions, so we can express p as a
  function of q or q as a function of p .

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Demand and Supply Equations
Graph the supply equation p .2q 2.
Graph the demand equation
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Key Suggested Problems

• Sec. 2.5 9, 11, 15, 19, 23, 25, 29, 35, 41, 42
• Sec. 3.1 3, 7, 11, 15, 23, 25, 33, 51, 55, 63,
  65
• Sec. 3.2 5, 9, 11, 15, 17, 19, 20, 23, 29, 30