Sec 2.5 Quadratic Functions Maxima and Minima

Objectives:• Express a quadratic in vertex form.• Find coordinates of vertex by completing the square.• Find maximum and minimum values of a quadratic.

Standard Form

A quadratic function f(x) = ax2 + bx + c can be expressed in vertex form:

f(x) = a(x – h)2 + k by completing the square.

The graph of f is a parabola with vertex (h, k).

Ex 1. Let f(x) = x2 – 12x + 23.

a) Express f in vertex form.

b) Find the vertex.

Ex 2. Let f(x) = 3x2 – 6x + 10

a) Express f in vertex form.

b) Find the vertex.

Class Work

1. Let f(x) = 2x2 +8x -5.

a) Express f in vertex form

b) Find the vertex.

Maximums and MinimumsIf a quadratic function has vertex (h, k), then

the function has:

A ______________value at the vertex if it opens ________. (if a > 0)

A ______________ value at the vertex if it opens __________. (if a < 0)

minimumup

maximumdown

Let f be a quadratic function with vertex form f(x) = a(x – h)2 + k

The maximum or minimum value of f occurs at ________ and is __________.x = h f(x) = k

Ex 3. Consider the quadratic function f(x) = 5x2 – 30x + 49

a)Express f in vertex form.

b) Find the minimum value of f.

Class Work2. Consider the quadratic function

f(x) = 2x2 – 20x + 53

a)Express f in vertex form.

b) Find the minimum value of f.

Class Work3. Consider the quadratic function f(x) = –2x2 + 4x + 5

a)Express f in vertex form.

b) Find the maximum value of f.

Finding max and min without completing the square.The maximum or minimum value of a

quadratic function f(x) = ax2 + bx + c

occurs at: ________2bxa

Ex 4. Find the maximum or minimum of each quadratic function.

a) f(x) = x2 + 4x

b) g(x) = –2x2 + 4x – 5

Class Work

Find the maximum or minimum value of each quadratic function.

4. f(x) = x2 + 8x

5. g(x) = –4x2 + 8x – 2

HW #6 p200 1,4,5,8,10,13,14,19,22,25,26,

29,32,34,39,59,61,62