1

Quadratic

Functions

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

A polynomial function of degree n is

where the a’s are real numbers and the n’s are nonnegative integers

and an 0.

Polynomial Function

01

1)( axaxaxf nn

nn

A polynomial function of degree 2 is called a quadratic function.

It is of the form

a, b, and c are real numbers and a 0.

Quadratic Function

cbxaxxf 2)(

For a quadratic function of the form

gives the axis of

symmetry.

Axis of Symmetry

cbxaxxf 2)(

a

bx

2

A quadratic function of the form

is in standard form.

axis of symmetry: x = hvertex: (h, k)

Standard Form

0,)()( 2 akhxaxf

Characteristics of Parabola

symmetryofaxis

symmetryofaxisa > 0

a < 0

vertex: minimum

vertex: maximum

7

Higher DegreePolynomial Functions

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

The graph of a polynomial function…

1. Is continuous.

2. Has smooth, rounded turns.

3. For n even, both sides go same way.

4. For n odd, sides go opposite way.

5. For a > 0, right side goes up.

6. For a < 0, right side goes down.

Characteristics

.

xas

xf )(

xas

xf )(

an < 0

xas

xf )(

xas

xf )(

graphs of a polynomial function for n odd:0

11)( axaxaxf n

nn

n

Leading Coefficient Test: n odd

an > 0

.

an < 0

graphs of a polynomial function for n even:0

11)( axaxaxf n

nn

n

an > 0

xas

xf )(

xas

xf )(

xas

xf )(

xas

xf )(

Leading Coefficient Test: n even

The following statements are equivalent for

real number a and polynomial function f :

1. x = a is root or zero of f.

2. x = a is solution of f (x) = 0.

3. (x - a) is factor of f (x).

4. (a, 0) is x-intercept of graph of f (x).

Roots, Zeros, Solutions

1. If a polynomial function contains a factor (x - a)k, then x = a is a repeated root of multiplicity k.

2. If k is even, the graph touches (not crosses) the x-axis at x = a.

3. If k is odd, the graph crosses the x-axis at x = a.

Repeated Roots (Zeros)

If a < b are two real numbers

and f (x)is a polynomial function

with f (a) f (b),

then f (x) takes on every real

number value between

f (a) and f (b) for a x b.

Intermediate Value Theorem

Let f (x) be a polynomial function and a < b be two real numbers.

If f (a) and f (b)

have opposite signs

(one positive and one negative),

then f (x) = 0 for a < x < b.

NOTE to Intermediate Value

15

Polynomial and

Synthetic Division

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

If f (x) and d(x) are polynomialswith d(x) 0 and the degree of d(x) isless than or equal to the degree of f(x),

then q(x) and r (x) are uniquepolynomials such that

f (x) = d(x) ·q(x) + r (x)where r (x) = 0 or

has a degree less than d(x).

Full Division Algorithm

f (x) = d(x) ·q(x) + r (x)

dividend quotient divisor remainder

where r (x) = 0 orhas a degree less than d(x).

Short Division Algorithm

ax3 + bx2 + cx + d divided by x - k

k a b c d

ka

a r coefficients of quotient remainder

1. Copy leading coefficient.

2. Multiply diagonally. 3. Add vertically.

Synthetic Division

If a polynomial f (x)

is divided by x - k,

the remainder is r = f (k).

Remainder Theorem

A polynomial f (x)

has a factor (x - k)

if and only if f (k) = 0.

Factor Theorem

21

Real Zeros of Polynomial Functions

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

a’s are real numbers, an 0, and a0 0.

1. Number of positive real zeros of f equals number of variations in sign of f(x), or less than that number by an even integer.

2. Number of negative real zeros of f equals number of variations in sign of f(-x), or less than that number by an even integer.

Descartes’s Rule of Signs

01

1)( axaxaxf nn

nn

a’s are real numbers, an 0, and a0 0.

1. f(x) has two change-of-signs; thus, f(x) has two or zero positive real roots.

2. f(-x) = -4x3 5x2 + 6 has one change-of-signs; thus, f(x) has one negative real root.

Example 1: Descartes’s Rule of Signs

654)( 23 xxxf

Factor out x; f(x) = x(4x2 5x + 6) = xg(x)

1. g(x) has two change-of-signs; thus, g(x) has two or zero positive real roots.

2. g(-x) = 4x2 + 5x + 6 has zero change-of-signs; thus, g(x) has no negative real root.

Example 2: Descartes’s Rule of Signs

xxxxf 654)( 23

If a’s are integers, every rational zero of f has the form

rational zero = p/q,

in reduced form, and p and q are factors of a0 and an, respectively.

Rational Zero Test0

11)( axaxaxf n

nn

n

f(x) = 4x3 5x2 + 6

p {1, 2, 3, 6}

q {1, 2, 4}

p/q {1, 2, 3, 6, 1/2, 1/4, 3/2, 3/4}

represents all possible rational roots of f(x) = 4x3 5x2 + 6 .

Example 3: Rational Zero Test

f(x) is a polynomial with real coefficients and an > 0 with

f(x) (x - c), using synthetic division:

1. If c > 0 and each # in last row is either positive or zero, c is an upper bound.

2. If c < 0 and the #’s in the last row alternate positive and negative, c is an lower bound.

Upper and Lower Bound

2x3 3x2 12x + 8 divided by x + 3

-3 2 -3 -12 8 -6 27 -45

2 -9 15 -37

c = -3 < 0 and #’s in last row alternate positive/negative. Thus, x = -3 is a

lower bound to real roots.

Example 4: Upper and Lower Bound