Wednesday, September 26, 2007

You can find extra notes on this website:

http://scidiv.bcc.ctc.edu/FL/MATH105/sso0201.pdf

Definition:
Let “n” be a non-negative integer and let an, an-1, …, a2, a1, a0
be real numbers with an ≠ 0.

The Function

f (x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0

is called a POLYNOMIAL FUNCTION of x with degree “n

Polynomial functions are classified by degree:

f (x) = a is a constant function
f (x) = mx + b is a linear function
f (x) = ax2 + bx + c is a quadratic function where a ≠ 0 and {a, b, c } is contained in the set of Real Numbers

Parabola – a graph of a quadratic function is a special type of u-shaped curve.

Since this graph opens upward, given f (x) = ax2 + bx + c, "a" is greater than 0.
if "a" is less than 0, then the parabola is opened downward.

Vertex is the turning point and the axis of symmetry is perpendicular through the x-value of the vertex. Can be found by x = -b/(2a)

Key: using f(x) = ax2
if "a" is greater than 1, the graph is a vertical stretch of the graph y= f(x)
if 0 is less than "a" which is less than 1, the graph is a vertical shrink of the graph y = f(x)

Standard Form of a Quadratic Function:

f(x) = a (x - h)2 + k, where a ≠ 0,
the axis of symmetry is the vertical line x = h, and the vertex of the function is (h, k)

Example 1: Given the vertex (4, -1) and the point (2, 3), what is the equation of the quadratic function.

y = a (x - h)2 + k
3 = a (2 - 4)2 + (-1)
4 = a (-2)2
4 = a (4)
1 = a

y = 1 (x - 4)2 - 1

Example 2: Given the vertex is (5/2, -3/4) and a point (-2, 4), what is the equation of the quadratic function.

y = a(x - h)2 + k
4 = a(-2 - 5/2)2 + (-3/4)
4.75 = a(-4.5)2
4.75 = 20.25 a
19/81 = a

y = (19/81)(x - 5/2)2 - 3/4

II. Maximum or Minimum values:
To find the maximum or minimum, find the vertex by using
x = -b/(2a)

Example: Given C = 800 - 10x + 0.25 x2
find the minimum cost and the number of fixtures:

x = -b/(2a) = 10/((2)(.25) = 10/.5 = 20

therefore there will be 20 fixtures

C = .25 (20)2 - 10(20) + 800
C = 700

So the minimum cost is \$700

You can check by graphing, and the minimum point is (20, 700)

III. Identify the vertex and the intercepts Algebraically.

f(x) = x2 + 9x + 8

Vertex is x=-b/2a = -9/2 = -4.5

f (-9) = (-4.5)2 + 9(-4.5) + 8
= 20.25 - 40.5 + 8
= -12.75

vertex is (-4.5, -12.25)

Intercepts:
Let x = 0
f (0) = 02 + 9(0) + 8 = 8
(0, 8)

Let y = 0
0 = x2 + 9x + 8
0 = (x + 8)(x + 1)
x = -8 and x = -1
so
(-8, 0) and (-1, 0)

IV. Find the equation of a quadratic

Given the two x-intercepts (-2, 0) and (10, 0), can you find the equation?

A. if the parabola opens upward, "a" is greater than 0,

y = a (x - p)(x - q)

because there can be lots of answers depending upon what the value of "a", so we will
Let a = 1

f (x) = (x - (-2))(x - 10)
= (x + 2)(x - 10)
= x2 - 10x + 2x - 20
= x2 - 8x - 20

B. if the parabola opens downward, "a" is less than 0,
y - a (x - p)(x - q)
because again, there can be lots of answers depending upon what the value of "a", so we will
Let a = -1

f(x) = -(x + 2)(x - 10)
= -(x2- 8x - 20)
= -x2 + 8x + 20