Completing the Square
Section 5.4
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Ex.
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Solve 2x  7x  5  0 by completing the
square
2
Solve 3x 2  2x  1  0 by completing the
square
Day 2 Vertex form
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The coordinates of the vertex of the graph
2
y  ax  bx  c where a  0 are (h,k)
then the parabola can be written as
y  a(x  h)  k
2
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which is called the vertex form of a quadratic
function.
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The axis of symmetry is a line that divides the
parabola into two parts that are mirror images
of each other.
The axis of symmetry passes through the
vertex of the parabola. x = ?
Review transformations from chapter 2
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If y = f(x), then
y = af(x) is a vertical stretch or compression
of f
y = f(bx) is a horizontal stretch or
compression of f
y = f(x) + k is a vertical translation of f
y = (x – h) is a horizontal translation of f.
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Given h(x) = 2x 2  16x  23 write the
function in vertex form and give the
coordinates of the vertex and the equation of
the axis of symmetry. Then describe the
transformation from f(x) = x 2 to h.
Step 1 Write in vertex form.
2
h(x) = 2x  16x  23
- factor GCF from
the first 2 terms
2
h(x) = 2(x  8x)  23
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Complete the square. h(x) = 2(x 2  8x)  23
h(x) = 2(x2 + 8x + 16) + 23 – 2(16)
h(x) = 2(x + 4)2 – 9
Coordinates of vertex (-4, -9)
The axis of symmetry: x = -4
Translations: vertical stretch of 2, horizontal
translation 4 units to the left, and vertical
translation 9 units down.
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A softball is thrown upward at an initial
velocity of 32 ft per second from 5 feet above
the ground. The ball’s height in feet above
the ground is modeled by h(t) = -16t2 + 32t +
5, where t is the time in seconds after the ball
is released. Complete the square and find
the maximum height of the ball.
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Each side of a square is increased by 2 cm,
producing a new square whose area is 30
cm2. Find the length of the sides of the
original square.