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.

Parabolas y = ax2 + bx + c

Vertex (axis of symmetry, max/min)

x-intercept x-intercept

y-intercept

Properties of a Parabola

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 )

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.

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

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

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

Step 3: Simplify

Step 4: Solve 2

1352

135 and

2144255

21695

428

9218

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 = _____

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

Properties of Parabolas

1) x2 + 4x – 5

2) -x2 − 2x + 1