Quadratic Functions p' 264269

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Quadratic Functions p. 264-269

• OBJECTIVES
• Analyze graphs of quadratic functions
• Write quadratic functions in standard form and
  use the results to sketch graphs of functions
• Use quadratic functions to model and solve
  real-life problems

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Graphing the parabola y f (x) ax2 , p. 265

• Consider the equation y x2

Axis of symmetry x 0 ( yx2 is
symmetric with respect to the y-axis )
0
4
1
1
4
(1, 1)
(0, 0)
(1, 1)
(2, 4)
(2, 4)
y
When a gt 0, the parabola opens upwards and is
called concave up. The vertex is called a
minimum point.
Vertex(0, 0)
x
3

• For the function equation y x2 , what is a ?

a 1 . What if a does not equal 1?
Consider the equation y 4x2 .
What is a ? a 4
y
0
4
4
16
16
(0, 0)
(2, 16)
(1, 4)
(1, 4)
(4, 16)
x
When a lt 0, the parabola opens downward and is
called concave down. The vertex is a maximum
point.
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Properties of the Parabola f (x) ax2 p. 266

• The graph of f (x) ax2 is a parabola with the
  vertex at the origin and the y axis as the line
  of symmetry.
• If a is positive, the parabola opens upward, if
  a is negative, the parabola opens downward.
• If ?a? is greater than 1 (?a? gt 1), the parabola
  is narrower then the parabola f (x) x2.
• If ?a? is between 0 and 1 (0 lt ?a? lt 1), the
  parabola is wider than the parabola f (x) x2.

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Graphing the parabola y f (x) ax2 k
Consider the equation y 4x2 3 . What is a
? a 4
Graphical Approach
Numerical Approach
x
Vertex(0, -3)
Algebraic Approach y 4x2 3
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The graph y 4x2 is shifted down 3 units.
x
y 4x2
Vertex(0, -3)
y 4x2 3 .
In general the graph of y ax2 k is the
graph of y ax2 shifted vertically k units. If
k gt 0, the graph is shifted up. If k lt 0, the
graph is shifted down. (P. 267)
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Consider the equation y 4(x 3)2 . What is
a ?
a 4. What effect does the 3 have on the
function?
The axis of symmetry is x 3.
Numerical Approach
y
x
Axis of symmetry is shifted 3 units to the right
and becomes x 3
y 4x2
y 4(x 3)2
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Standard Form of a Quadratic Function p.267

• The quadratic function
• f(x) a(x h)2 k, a 0
• is in standard form.
• The graph of f is a parabola .
• Axis is the vertical line x h.
• Vertex is the point (h, k).
• If a gt 0, the parabola opens upward.
• If a lt 0, the parabola opens downward.

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p. 269 EZ way to find the vertex

• f (x) 2x2 x 1
• a 2 b -1 c 1

Vertex
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The position equation- s -16t2 v0t s0 , p.
117. is the distance the object is above
ground. Objects dropped from rest have an initial
velocity of 0 feet per second, or v0 0. s
-16t2 s0 How far has the object traveled from
the point of release? s 16t2
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• Direct Variation as an nth Power (n 2), p. 316
• The following statements are equivalent.
• 1. y varies directly as the 2nd power of x.
• 2. y is directly proportional to the 2nd power
  of x.
• 3. y kx2 for some constant k.
• s 16t2
• s varies directly as the 2nd power of t, k 16
• How can we measure the distance an object travels
  as it falls?
• How can we slow the falling object?

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Real Experiment
FULL LENGTH 8 feet
6 feet
4 feet
2 feet
1.432 sec required for 2 ft.
2.912 sec required for 8 ft.
2.038 sec required for 4 ft.
2.49 sec required for 6 ft.
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HOMEWORK

• Work p. 270-274 1-98 alternate odd
• p. 321-323 13-16, 39, 55
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