Graphs%20of%20Quadratic%20Function

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Graphs of Quadratic Function

• Introducing the concept Transformation of the
  Graph of y x2

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Graph of f(x) ax2 and a(x-h)2

• Objective Graph a function f(x)a(x-h)2, and
  determine its characteristics.

Definition A QUADRATIC FUNCTION is a function
that can be described as f(x) ax2 bc c
0.
Graphs of QUADRATIC FUNCTIONS are called
PARABOLAS.
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Now let us see the graphs of quadratic functions
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Graph of QUADRATIC FUNCTION
LINE , OR AXIS OF SYMMETRY
VERTEX
VERTEX
LINE , OR AXIS OF SYMMETRY
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• Thus the y-axis is the LINE SYMMETRY. The point
  (0,0) where the graph crosses the line of
  symmetry, is called VERTEX OF THE PARABOLA

• Next consider f(x) ax2, we know the following
  about its graph. Compared with the graph of f(x)
  x2.
• If gt 1, the graph is stretched vertically.
• If lt 1, the graph is shrunk vertically.
• If a lt 0, the graph is reflected across the
  x-axis.

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EXAMPLEa. Graph f(x) 3x2b. Line of Symmetry?
Vertex?
LINE OF SYMMETRY The y-axis
VERTEX (0,0)
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Exercisea. Graph f(x) -1/4 x2b. Line of
symmetry and Vertex?

• Your answer should be like this

LINE OF SYMMETRY Y-AXIS
VERTEX (0,0)
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In f(x) ax2, let us replace x by x h. if h is
positive, the graph will be translated to the
right. If h is negative the translation will be
to the left. The line, or axis of symmetry and
the vertex will also be translated the same way.
Thus f(x) a(x-h)2, the axis of symmetry is x
h and the vertex is (h, 0).
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Compare the Graph of f(x) 2(x3)2 to the graph
of f(x) 2x2.
LINE OF SYMMETRY, X -3
VERTEX (0,0), SYMMETRY, Y-AXIS
VERTEX (0,3)
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EXAMPLEa. Graph f(x) - 2(x-1)2b. Line of
Symmetry and Vertex?
VERTEX (h, 0) (1,0)
LINE OF SYMMETRY, X1
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EXERCISESa. Graph f(x) 3(x-2)2b. Line of
Symmetry and Vertex?
VERTEX (2,0)
LINE OF SYMMETRY, X2
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Graph of f(x) a(x-h)2k

• Objective Graph a function f(x) a(x-h)2 k,
  and determine its characteristics.

In f(x) a(x-h)2, let us replace f(x) by f(x)
k f(x) k a(x-h)2 Adding k on both sides
gives f(x) a(x-h)2 k. The Graph will be
translated UPWARD if k is Positive and DOWNWARD
if k is NEGATIVE. The Vertex will be translated
the same way. The Line of Symmetry will NOT be
AFFECTED
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Guidelines for Graphing Quadratic Functions,
f(x)a(x-h)2 k

• When graphing quadratic function in the form
  f(x)a(x-h)2k,
• The line of symmetry is x-h0, or x h.
• The vertex is (h,k).
• If a gt 0, then (h,k) is the lowest point of the
  graph, and k is the MINIMUM VALUE of the
  function.
• If a lt 0, then (h,k) is the highest point of the
  graph, and k is the MAXIMUM VALUE of the
  function.

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Examplea. Graph f(x) 2(x3)2 2b. Line of
Symmetry, Vertex?c. is there a min/max value? If
so, what is it?
LINE OF SYMMETRY, X-3
VERTEX ( -3,-2) MINIMUM -2
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Exercisesfor each of the following, graph the
function, find the vertex, find the line of
symmetry, and find the min/ max value.

• 1. f(x) 3(x-2)2 4
• 2. f(x) -3(x2)2 - 4

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Answer 1
VERTEX (2,4) MIN 4 LINE OF SYMMETRYX 2
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Answer 2
VERTEX (-2,-1) MAX -1 LINE OF SYMMETRYX -2
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ANALYZING f(x) a(x-h)2k

• Objective Determine the characteristics of a
  function f(x) a(x-h)2k

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EXAMPLEWithout graphing, find the vertex,line of
symmetry, min/max value.Given1. f(x)
3(x-1/4)242. g(x) -4x5)27
a. What is the Vertex?
b. Line of Symmetry?
c. Is there a Min / Max Value?
d. What is the min / max value?
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Answer in 1 and 2
a. What is the Vertex? 1. (1/4, -2) 2. ( -5, 7)
b. Line of Symmetry? X ¼ X -5
c. Is there a Min / Max Value? Minimum. The graph extends upward since 3gt0 Maximum. The graph extends downward since 4lt0.
d. What is the min / max value? Min.Value is 2 Max.Value is 7