5-Minute Check on Activity 4-2

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5-Minute Check on Activity 4-2

• In the general equation, y ax2 bx c,
  identify what a, b, and c tell us about the
  graph.
• In the following equations, identify the
  direction it opens and what the y-intercept is
  for each
• y -2x2 6x - 2
• y 2x2 12x 4
• y 3x2 6x 9
• What is the domain of problem 2?

a if a gt 0 then it opens up otherwise it
opens down b helps determine the vertex and
line of symmetry c is the y-intercept of the
parabola
a lt 0, so it opens down y-intercept -2
a gt 0, so it opens up y-intercept 4
a gt 0, so it opens up y-intercept -9
Domain x x? Real s
Click the mouse button or press the Space Bar to
display the answers.
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Activity 4 - 3

• The Shot Put

3
Objectives

• Determine the vertex or turning point of a
  parabola
• Identify the vertex as a maximum or a minimum
• Determine the axis of symmetry of a parabola
• Identify the domain and range
• Determine the y-intercept of a parabola
• Determine the x-intercept(s) of a parabola using
  technology
• Interpret the practical meaning of the vertex and
  intercepts in a given problem
• Identify the vertex from the standard form y
  a(x h)² k of the equation of a parabola

4
Vocabulary

• None new

5
Activity

• Parabolas are good models for a variety of
  situations that you encounter in everyday life.
  Examples include the path of a golf ball after it
  is struck, the arch (cable system) of a bridge,
  the path of a baseball thrown from the outfield
  to home plate, the stream of water from a
  drinking fountain, and the path of a cliff diver.
• Consider the 2000 mens Olympic shot put event,
  which was won by Finlands Arsi Harju with a
  throw of 69 feet 10¼ inches. The path of his
  winning throw can be approximately modeled by the
  quadratic function defined by
• Y -0.015545x²
  x 6
• Where x is the horizontal distance in feet from
  the point of the throw and y is the vertical
  height in feet of the shot above the ground.

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Activity

• Y -0.015545x² x 6
• Which way will the graph of the parabola open?
• What is the y-intercept of the graph of the
  parabola?
• What is the practical meaning of this value?
• What is the practical domain of the graph?
• What is the practical range of the graph?

Since a (-0.015545)lt 0, it opens down
c ? y-intercept so c 6
The shot is 6 feet above the ground when its
released
0 x 69 10 ¼
0 y 22.03
X 10 20 30 40 50 60 70
Y
14.4 19.8 22.0 21.1 17.1 10.0 0
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Vertex of a Parabola

• A Parabola, y ax2 bx c has a vertex at the
  coordinateswhere a is the coefficient of the
  x2 term, b is the coefficient of the
  x-term and c is the y-intercept
• If parabola opens up, then vertex is a
  minimum down, then vertex is a maximum

8
Example 1

• Determine the vertex of y -3x2 12x 5
• By Formula
• By Calculator

(2, 17)
Graph function (Y1) 2nd Trace, select
minimum Move so Left Bound on left side and
enter Move so Right bound on right side and
enter Move toward vertex and enter
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Axis of Symmetry of a Parabola

• A parabola, y ax2 bx c has an axis of
  symmetry defined by the vertical linewhere a
  is the coefficient of the x2 term, b
  is the coefficient of the x-term and c is
  the y-intercept
• The axis of symmetry
• A vertical line that passes through the x-value
  of the vertex coordinates
• Divides the parabola into right and left halves

-b x ------- 2a
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Example 2

• Determine the axis of symmetry of y -3x2
  12x 5
• By Formula

x 2
b 12 x - -------
- ------- 2a 2(-3)
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X-intercepts of a Parabola

• A parabola, y ax2 bx c has x-intercepts if
  it crosses the x-axis (y 0). A special
  equation will give us these if they exist.
• Graphical Conditions (nonexistence)
• If a parabola opens up and the vertex is above
  the x-axis, then there are no x-intercepts
• If a parabola opens down and the vertex is below
  the x-axis, then there are no x-intercepts
• X-intercepts are also known as the zeros of a
  function

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Example 3

• Determine if y -3x2 12x 5 has any
  x-intercepts?
• Determine if y x2 12x 6 has any
  x-intercepts?

By its graph it crosses the x-axis between -1 and
0 and again between 4 and 5
By its graph it opens upward and its vertex is
above the x-axis ? no x-intercepts
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Standard Form of a Parabola

• A parabola, y ax2 bx c is in standard form
  when the vertex can be read from the equation.
• Parabola Standard Form y
  f(x) a (x h)2 kwhere (h, k) are the
  coordinates of the vertex

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Example 4

• Sketch y 3x2
• Sketch y 3(x 2)2 ? horizontal shift to
  the right
• Sketch y 3(x -2)2 5 ? vertical shift up
• Where is the vertex for each?

vertex (0, 0)
vertex (2, 0)
vertex (2, 5)
Remember Horizontal shifts are inside and
vertical shifts are outside the function
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Domain and Range of a Parabola

• Domain all permissible x-values
• Range all possible y-values
• A parabola, y ax2 bx c has the following
  domain and rangeDomain x all real
  numbersRange parabola opens up y
  y-value of vertexparabola opens down y
  y-value of vertex

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Example 5

• Determine the domain and range ofy -3x2 12x
  5
• Determine the domain and range of y x2 12x
  6

Its vertex is (2, 17) and it opens
downward Domain x all real numbers Range y
17
Its vertex is (-3, 3) and it opens upward Domain
x all real numbers Range y 3
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Summary and Homework

• Parabola Summary So Far
• Quadratic Form y f(x) ax2 bx c
• Standard Form y f(x) a(x h)2 k
• Vertex (-b / 2a, -b2
  4ac / 4a)
• Axis of Symmetry x -b/(2a) (Vertical
  Line)
• Domain x all real
  numbers
• Range y min or y
  max
• a determines if it opens up (agt 0) or down (a lt
  0)
• Homework
• pg 430 434 problems 1 4, 9, 10