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1 Quadrati c Function Dr. Claude S. Moore Danville Community College PRECALCULUS I
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1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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Page 1: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

1

Quadratic

Functions

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

Page 2: 1 Quadratic Functions Dr. Claude S. Moore Danville 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

Page 3: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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)(

Page 4: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

For a quadratic function of the form

gives the axis of

symmetry.

Axis of Symmetry

cbxaxxf 2)(

a

bx

2

Page 5: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

A quadratic function of the form

is in standard form.

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

Standard Form

0,)()( 2 akhxaxf

Page 6: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

Characteristics of Parabola

symmetryofaxis

symmetryofaxisa > 0

a < 0

vertex: minimum

vertex: maximum

Page 7: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

7

Higher DegreePolynomial Functions

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

Page 8: 1 Quadratic Functions Dr. Claude S. Moore Danville 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

Page 9: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

.

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

Page 10: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

.

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

Page 11: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 12: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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)

Page 13: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 14: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 15: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

15

Polynomial and

Synthetic Division

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

Page 16: 1 Quadratic Functions Dr. Claude S. Moore Danville 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

Page 17: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 18: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 19: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

If a polynomial f (x)

is divided by x - k,

the remainder is r = f (k).

Remainder Theorem

Page 20: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

A polynomial f (x)

has a factor (x - k)

if and only if f (k) = 0.

Factor Theorem

Page 21: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

21

Real Zeros of Polynomial Functions

Dr. Claude S. MooreDanville Community

College

PRECALCULUS I

Page 22: 1 Quadratic Functions Dr. Claude S. Moore Danville 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

Page 23: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 24: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 25: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 26: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 27: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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

Page 28: 1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I.

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