the set s consists of all multiples of 6. which of the following sets are contained within s ?
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The set S consists of all multiples of 6. Which of the following sets are contained within S ?. The set of all multiples of 3 The set of all multiples of 9 The set of all multiples of 12. Reminder: Citizenship points start today. Please work quietly so others can focus. Parabolas. - PowerPoint PPT PresentationTRANSCRIPT
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The set S consists of all multiples of 6. Which of the following sets are contained within S ?
I) The set of all multiples of 3II) The set of all multiples of 9III) The set of all multiples of 12
Reminder: Citizenship points start today.Please work quietly so others can focus.
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Parabolas y = ax2 + bx + c
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Vertex (axis of symmetry, max/min)
x-intercept x-intercept
y-intercept
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Properties of a Parabola
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To Find the y-intercept:
1) Write the equation in standard form, set
x equal to 0 and solve for y.
2) The y-intercept is always c.
y-int = ( 0, c )
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To find the x-intercepts
• Set y equal to 0 and solve for x.• The most common methods for solving
quadratic equations are by factoring or using the quadratic formula.
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The length of a rectangular window is 5 feet more than its width, w. The area of the window is 36 square feet. Write an equation that can be
used to find the dimensions of the window.
Width = w
Length = 5+w
Area = 36 = l x w
(5+w)(w)
5w + w2 = 36
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Solve for w: 5w + w2 = 36 1) Rearrange equation to set it equal to zero.
2) Factor to get: 3) In order for the product to equal 0, at least one
of the factors has to be 0.4) So you have 5) Therefore, 6) Which answer makes more sense and why?
(w + 9)(w - 4) = 0w2 + 5w - 36 = 0
w=-9 and w=4
4 because dimensions cannot be negative.
W + 9 = 0 and w – 4 = 0
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Solving Using the Quadratic Formula
Example: 5w + w2 = 36 → w2 + 5w - 36 = 0 For ax2 + bx + c = 0, the value of x is given by:
Step 1: Note that
Step 2: Substitute)1(2
)36)(1(455 2
a = 1, b = 5, & c = -36
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Step 3: Simplify
Step 4: Solve 2
1352
135 and
2144255
21695
428
9218
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Max/Min• If a is positive, the graph opens upward
(smile) and you will have a min.• If a is negative, the graph opens downward
(frown) and you will have a max.
• The max/min is the y-coordinate of the vertex. y = ____• The Axis of Symmetry is the x-coordinate of
the vertex. x = _____
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To find the Vertex…
1) Use to find the x- coordinate.
2) Plug the x-coordinate into the original equation
and solve to find the y-coordinate.
ab
2
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Properties of Parabolas
1) x2 + 4x – 5
2) -x2 − 2x + 1