1.2 Graphs of Equations

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1.2 Graphs of Equations
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Objective

• Sketch graphs of equations
• Find x and y intercepts of graphs of equations
• Use symmetry to sketch graphs of equations
• Use graphs of equations in solving real-life
  problems.

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The graph of an equation

• A relationship between two quantities can be
  expressed as an equation in two variables.
• The graph of an equation is the set of all points
  that are solutions of the equation.

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Determining Solutions

• Determine whether (1, 3) and (-2, 4) are
  solutions of the equation

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Sketching the graph of an equation

• Example 2 Using the point-plotting method, sketch
  the graph of

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• Using the point-plotting method, sketch the graph
  of

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• If you have two few points with point-plotting
  technique you could badly mispresent the graph of
  an equation.
• For example, using only the four points (-2, 2),
  (-1, 1), (1, -1) and (2, 2) any of these three
  graphs would be reasonable.

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Intercepts of a Graph

• Intercepts are points that have zero as either
  the x-coordinate or the y-coordinate.
• It is possible for the graph to have one or
  several intercepts.

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Finding intercepts

• To find x-intercepts, let y be zero and solve the
  equation for x.
• To find y-intercepts, let x be zero and solve the
  equation for y.

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Find the x and y intercepts
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Find the x and y intercepts
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Symmetry

• X-axis symmetry

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• Y-axis symmetry

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• Origin symmetry

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Graphical Tests for Symmetry

• 1. A graph is symmetric with respect to the
  x-axis if, whenever (x, y) is on the graph (x,
  -y) is also on the graph.
• 2. A graph is symmetric with respect to the
  y-axis, if, whenever (x, y) is on the graph (-x,
  y) is also on the graph.
• 3. A graph is symmetric with respect to the
  origin if, whenever (x, y) is on the graph (-x,
  -y) is also on the graph.

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Algebraic Tests for Symmetry

• 1. The graph of an equation is symmetric with
  respect to the x-axis if replacing y with y
  yields an equivalent equation.
• 2. The graph of an equation is symmetric with
  respect to the y-axis if replacing x with x
  yields an equivalent equation.
• 3. The graph of an equation is symmetric with
  respect to the origin if replacing x with x and
  y with -y yields an equivalent equation.

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Example 5 Testing for symmetry
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Application

• The net profits P (in millions of dollars) for a
  company from 2000 through 2005 can be
  approximated by the mathematical model P 48.3t
  100.04 where t is the calendar year, with t 0
  corresponding to 2000

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• Construct a table of values that shows the net
  profits for each year.

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• Use the table of values to sketch a graph of the
  model. Then use the graph to estimate
  graphically the net profit for the year 2012.

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• Use the model to confirm algebraically the
  estimate you found in part (b)

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• How do you identify intercepts and symmetry in
  order to sketch graphs of equations?
• Find intercepts by letting one variable be zero
  and solving for the other variable.
• Identify symmetry by choosing a point (x, y)
  on the graph and checking to see whether the
  points (x, -y), (-x, y) or (-x, -y) are also on
  the graph.