Transcript 3.2 Quadratic Functions & Graphs

3.2 Quadratic Functions & Graphs
Quiz
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Write out the general form of a quadratic equation.
f(x) = _____________
Quadratic Function
Quadratic Function
General Form
f(x) = ax2 + bx + c
(a ≠ 0)
Standard( tranformation)
Form
f(x) = a(x - h)2 + k
(a ≠ 0)
Parabola
Complete the Square
Complete the Square
General Form
f(x) = ax2 + bx + c
Standard Form
f(x) = a(x – h)2 +
k
x2 + 2px + p2 = (x + p)2
x2 - 2px + p2 = (x - p)2
Complete the Square
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Example: Given f(x) = 2x2 - 8x + 1, complete the square
to put it into the form f(x) = a(x – h)2 + k.
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How about f(x) = x2 - 3x + 1?
The Graph of a Quadratic
a>0
a<0
Maximum point y
y
x
x
Minimum point
Vertex
Axis of symmetry
x=h
Vertex
Axis of symmetry
x=h
Find vertex of a parabola
Transformation form: f(x) = a(x – h)2 + k
Vertex : ( h, k )
Axis of symmetry: x = h
Transformation form: f(x) = ax2 + bx + c
Vertex : ( - b/2a, f(- b/2a) ) Axis of symmetry: x =
-b/2a
Graphing parabolas
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Determine if the graph opens up or down
Determine the vertex ( h, k )
Find the y – intercept
Plot the vertex and at least 2 additional points on one
side of the vertex
Use symmetry finish the other half
Example: f(x) = 2x2 + x - 3
Application
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Write the equation of the parabola with vertex at (8, 3)
passing through (10, 5).
f(x) = a ( x – h )2 + k
5
10 8
3
Height of a Projected Object
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If air resistance is neglected, the height s ( in feet ) of an
object projected directly upward from an initial height s0
feet with initial velocity v0 feet per second is
s (t) = -16t2 + v0t + s0,
where t is the number of seconds after the object is
projected.
Application
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6.
A ball is thrown directly upward from an initial height of 100 feet
with an initial velocity of 80 feet per second.
Give the function that describes the height of the ball in terms of
time t.
Graph this function so that the y-intercept, the positive xintercept, and the vertex are visible.
If the point (4.8, 115.36) lies on the graph of the function. What
does this mean for this particular situation?
After how many seconds does the projectile reach its maximum
height? What is the maximum height? Solve analytically and
graphically.
For what interval of time is the height of the ball greater than 160
feet? Determine the answer graphically.
After how many seconds will the ball fall to the ground?
Determine the answer graphically.
Homework
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PG. 174: 3-48(M3), 17 instead of 18
PG. 175: 57 – 72 (M3)
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Key: 6, 27, 36, 57, 63
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Reading: 3.3 Quadratic Equation & Ineq.