precalculus i

1

QuadraticFunctions

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)(

abx

2

A quadratic function of the form

is in standard form.axis of symmetry: x = h

vertex: (h, k)

Standard Form

0,)()( 2 akhxaxf

Characteristics of Parabola

symmetryofaxis

symmetryofaxisa > 0

a < 0

vertex: minimum

vertex: maximum

7

Higher DegreePolynomial Functions


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

.

xasxf )(

xasxf )(

an < 0

xasxf )(

xasxf )(

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

xasxf )(

xasxf )(

xasxf )(

xasxf )(

Leading Coefficient Test: n even

The following statements are equivalent forreal 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 numbersand 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 andSynthetic Division


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


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 Signs0

11)( axaxaxf n

nn

n

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 -37c = -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

precalculus i

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