**Precalculus 2.1 Quadratic Functions**

**2.1 Quadratic Functions**

You can find extra notes on this website:

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

**I. Quadratic Functions**

Definition:Let “n” be a non-negative integer and let a

Definition:

_{n}, a

_{n-1}, …, a

_{2}, a

_{1}, a

_{0}

be real numbers with a

_{n}≠ 0.

The Function

f (x) = a

_{n}x

^{n}+ a

_{n-1}x

^{n-1}+ … + a

_{2}x

^{2}+ a

_{1}x + a

_{0}

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*) = ax

^{2}+ 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*) = a

*x*

^{2}+ b

*x*+ 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) = ax

^{2}

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 x

^{2}

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) = x

^{2}+ 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) = 0

^{2}+ 9(0) + 8 = 8

(0, 8)

**Let y = 0**

0 = x

^{2}+ 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)

= x

^{2}- 10x + 2x - 20

= x

^{2}- 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)

= -(x

^{2}- 8x - 20)

= -x

^{2}+ 8x + 20