Geometry Integrated with Algebra

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Geometry Integrated with Algebra

• Section 4-1
• Graphing Quadratic Functions

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Focus

• Understand how the coefficients of a quadratic
  function influence its graph the direction it
  opens, its vertex, its line of symmetry, and its
  y-intercept

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On the north bank of the Chicago River, the Water
Arc sprays recirculated water across the river
toward a terrace along the south bank. The curve
of water is big enough for boats to sail under.
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Parabola

• The path of the water is an example of a
  parabola, a curve that can be modeled with a
  quadratic function.
• A quadratic function is a function that can be
  written in the standard form
• y ax2 bx c, where a ? 0

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Water Arc

• The path of the Water Arc can be modeled using
  the function
• y -0.006x2 1.2x 10

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Talk it Over 1

• Use a graphics calculator to graph the function.
  The water cannon is about 10 ft above the river
  surface.
• A. What point on your graph represents the water
  cannon?
• B. What is the greatest height the water reaches?
• C. How far across the river does the water reach?

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Solution Talk it Over 1
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Solution Talk it Over 1A

• Use a graphics calculator to graph the function.
  The water cannon is about 10 ft above the river
  surface.
• A. What point on your graph represents the water
  cannon?
• (0,10)

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Solution Talk it Over 1B

• Use a graphics calculator to graph the function.
  The water cannon is about 10 ft above the river
  surface.
• B. What is the greatest height the water reaches?
• 70 feet

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Solution Talk it Over 1C

• Use a graphics calculator to graph the function.
  The water cannon is about 10 ft above the river
  surface.
• C. How far across the river does the water reach?
• 208 feet

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Talk it Over 2

• What happens to the graph if you change the
  coefficient of x2 in the equation from -0.006 to
  0.006. Describe the new graph.

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Solution Talk it Over 2

• What happens to the graph if you change the
  coefficient of x2 in the equation from -0.006 to
  0.006. Describe the new graph.

The graph still passes through (0,10) but opens
up instead of down and is shifted to the left
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Solution Talk it Over 2
The graph still passes through (0,10) but opens
up instead of down and is shifted to the left
0.006
-0.006
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Orientation of Parabolas
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Sample 1

• Use the function y -3x2 2x 1
• A. Tell whether the the graph opens up or down.
• B. Tell whether the vertex is a maximum or a
  minimum.
• C. Find an equation for the line of symmetry.
• D. Find the coordinates of the vertex.

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Solution Sample 1A

• Use the function y -3x2 2x 1
• A. Tell whether the the graph opens up or down.

The value of a is -3. It is negative, so the
graph opens down.
y ax2 bx c y -3x2 2x 1
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Solution Sample 1B

• Use the function y -3x2 2x 1
• B. Tell whether the vertex is a maximum or a
  minimum.
• Because the graph opens down the vertex is a
  maximum.

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Solution Sample 1C

• Use the function y -3x2 2x 1
• C. Find an equation for the line of symmetry.

y ax2 bx c y -3x2 2x 1
The line of symmetry is the equation x ?
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Solution Sample 1D

• Use the function y -3x2 2x 1
• D. Find the coordinates of the vertex.
• The vertex lies on the line of symmetry, so x ?

y -3x2 2x 1
(?, 1?)
y -3(?)2 2(?) 1
y -3( ) ? 1
y -? ? 1
y 1?
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How to Graph

• The vertex is one point that helps you sketch a
  parabola
• The intercepts, where the parabola crosses the
  x-axis and y-axis, also help you sketch a graph
• Sketch a graph of the Water Arcy -0.006x2
  1.2x 10

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• y -0.006x2 1.2x 10

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• y -0.006x2 1.2x 10

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Sample 2

• Use the function y x2 0.5x 3.74
• A. Find the y-intercept of the graph.
• B. Use a graph to estimate thex-intercepts.
  Check one x-intercept by substitution.

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Solution Sample 2A

• Use the function y x2 0.5x 3.74
• A. Find the y-intercept of the graph.
• y x2 0.5x 3.74
• y (0)2 0.5(0) 3.74
• y 0 0 3.74
• y -3.74

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Solution Sample 2B

• Use the function y x2 0.5x 3.74
• B. Use a graph to estimate thex-intercepts.
  Check one x-intercept by substitution.
• Use a graphics calculator
• The x-intercepts are -2.2 and 1.7

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Solution Sample 2B

• Use the function y x2 0.5x 3.74
• B. Use a graph to estimate thex-intercepts.
  Check one x-intercept by substitution.
• Use a graphics calculator
• The x-intercepts are -2.2 and 1.7
• Check Substitute 1.7 for x and 0 for y
• 0 (1.7)2 0.5(1.7) 3.74
• 0 2.89 .85 3.74
• 0 3.74 3.74
• 0 0 ?

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Look Back

• Explain how you would find the maximum or minimum
  value of a quadratic function.
• First find the equation of the line of symmetry
  for the function in the form y ax2 bx c
• It will have the form __________
• Since _____ is the x-coordinate of the vertex,
  substitute _______ in the original equation to
  find the y-coordinate of the vertex.
• If a is__________, the vertex is a _________.
• If a is __________ the vertex is a ___________.

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Look Back

• Explain how you would find the maximum or minimum
  value of a quadratic function.
• First find the equation of the line of symmetry
  for the function in the form y ax2 bx c
• It will have the form __________
• Since _____ is the x-coordinate of the vertex,
  substitute _______ in the original equation to
  find the y-coordinate of the vertex.
• If a is__________, the vertex is a _________.
• If a is __________ the vertex is a ___________.

positive
minimum
negative
maximum
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THE END