Essential Question(s):

How can you tell if a quadratic functiona)opens up or downb)has a minimum or maximum, andc)how many x-intercepts it has?

“Wait… didn’t we do this already?”• I tried to warn you…• The notes that follow in yellow, I will expect you to

memorize (meaning: they won’t be given to you on a quiz)

Quadratic Functions are parabolas (‘U’ shaped) anda)Can open either upward or downwardb)Always have a vertex which is either the maximum or

minimum Opening up == minimum, opening down == maximum

c)Always have exactly one y-interceptd)Can have 0, 1, or 2 x-intercepts

The x-intercept(s) are the solution(s) [roots] of the equation

Quadratic Functions can be written in one of three forms• Transformation form: f (x) = a(x – h)2 + k

Most useful for finding the vertex of a parabola Vertex is at (h, k) (Set inside parenthesis = 0 & solve, number outside)

If a is positive, the graph opens up. If a is negative, graph opens down. The y-intercept is at ah2 + k

The x-intercepts are at

kh

a

Using Transformation Form• Find the vertex of the function and state

whether the graph opens upward or downward• g(x) = -6(x – 2)2 – 5

• h(x) = -x2 + 1

h = 2 and k = -5, so vertex is at (2, -5)Because a = -6, graph opens down

There is no h, and k = 1 so vertex is at (0, 1)Because a = -1, graph opens down

Polynomial form: f (x) = ax2 + bx + c• Yeah, we’ve seen this plenty already…• Most useful for finding the y-intercept

y-intercept is at (0, c)• If a is positive, the graph opens up.• If a is negative, graph opens down.

• The vertex is at

• The x-intercepts are at 2 4

2

b b ac

a

,22

bf

a

b

a

Using Polynomial Form• Determine the y-intercept and state whether

the graph opens upward or downward• g(x) = x2 + 8x – 1

• g(x) = 2x2 – x + 5

The y-intercept is at (0, -1)Because a = 1, graph opens up

The y-intercept is at (0, 5)Because a = 2, graph opens up

x-intercept form: f (x) = a(x – s)(x – t)• This is simply polynomial form factored out• Most useful for finding the x-intercepts (duh)

x-intercepts are at (s, 0) and (t, 0)• If a is positive, the graph opens up.• If a is negative, graph opens down.

• The vertex is at

• The y-intercepts is at (0, ast)

,2 2

s t s tf

Using x-intercept Form• Determine the x-intercepts and state whether

the graph opens upward or downward• h(x) = -2(x + 3)(x + 1)

• f(x) = -0.4(x + 2.1)(x – 0.7)

The x-intercept are at (-3, 0) and (-1, 0)Because a = -2, graph opens down

The x-intercepts are at (-2.1, 0) and (0.7, 0)Because a = -0.4, graph opens down

Assignment• Page 170• Problems 1-25, odd problems