Polinomials Functions

8/8/2019 Polinomials Functions

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3.1

Quadratic Functions and Models


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A quadratic function is a function of the form:


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a > 0Opens up

Vertex is lowest point

Axis of symmetry

Graphs of a quadratic function f ( x) = ax 2 + bx + c

a < 0Opens down

Vertex is highest point

Axis of symmetry


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Steps for Graphing a Quadratic Function

by Hand Determine the vertex. Determine the axis of symmetry.

Determine the y-intercept, f (0) . Determine how many x-intercepts the graph has. If there are no x-intercepts determine another

point from the y-intercept using the axis of

symmetry. Graph.


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Without graphing, locate the vertex and find the axis of symmetryof the following parabola. Does it open up or down?

Vertex:

Since -3 < 0 the parabola opens down.


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Finding the vertex by completing the square:


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10 0 10

15

15

(0,0)

(2,4)

y x2


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10 0 10

15

15

(0,0)

(2, -12)


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10 0 10

15

15

(2, 0)

(4, -12)


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10 0 10

15

15

(2, 13)

Vertex


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Determine whether the graph opens up or down.Find its vertex, axis of symmetry, y-intercept, x-intercept.

x-coordinate of vertex:

Axis of symmetry:

y-coordinate of vertex:


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There are two x-intercepts:


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5 0

10

10

Vertex: (-3, -13)

(-5.55, 0) (-0.45, 0)

(0, 5)


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3.3Polynomial Functions and

Models


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A polynomial function is a function of the form


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Polynomial. Degree 2.

Not a polynomial.

Not a polynomial.

Determine which of the following arepolynomials. For those that are, state the degree.

(a)

(b)

(c)


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If f is a polynomial function and r is a real number

for which f (r )=0, then r is called a (real) zero of f , or root of f . If r is a (real) zero of f , then

(a) r is an x-intercept of the graph of f .

(b) ( x - r ) is a factor of f .


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Use the above to conclude that x = -1 and x = 4 arethe real roots (zeroes) of f .


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1 is a zero of multiplicity 2.

-3 is a zero of multiplicity 1.-5 is a zero of multiplicity 5.


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.

If r is a Zero or Odd Multiplicity

If r is a Zero or Even Multiplicity


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Theorem

If f is a polynomial function of degreen, then f has at most n-1 turning points.


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TheoremFor large values of x, either positiveor negative, the graph of thepolynomial

resembles the graph of the power function.


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For the polynomial

(a) Find the x- and y-intercepts of the graph of f .

(b) Determine whether the graph crosses ortouches the x-axis at each x-intercept.

(c) Find the power function that the graph of f resembles for large values of x.

(d) Determine the maximum number of turningpoints on the graph of f .


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For the polynomial

(e) Use the x-intercepts and test numbers to findthe intervals on which the graph of f is above the x-axis and the intervals on which the graph is

below the x-axis.(f) Put all the information together, and connectthe points with a smooth, continuous curve toobtain the graph of f .


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(b) -4 is a zero of multiplicity 1. (crosses)-1 is a zero of multiplicity 2. (touches)5 is a zero of multiplicity 1. (crosses)

(d) At most 3 turning points.

(a) The x-intercepts are -4, -1, and 5.y-intercept:


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Test number: -5

f (-5) 160Graph of f : Above x-axis

Point on graph: (-5, 160)


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Test number: -2

f (-2) -14

Graph of f : Below x-axis

Point on graph: (-2, -14)

-4 < x


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Test number: 0 f (0) -20

Graph of f : Below x-axis

Point on graph: (0, -20)

-1 < x < 5


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Test number: 6

f (6) 490

Graph of f : Above x-axis

Point on graph: (6, 490)


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8 6 4 2 0 2 4 6 8

300

100

100

300

500(6, 490)

(5, 0)(0, -20)

(-1, 0)

(-2, -14)(-4, 0)

(-5, 160)


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3.7The Real Zeros of a Polynomial

Function


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Theorem: Division Algorithm for Polynomials

If f ( x) and g( x) denote polynomial functions and if g( x) isnot the zero polynomial, then there are uniquepolynomial functions q( x) and r ( x) such that

where r ( x) is either the zero polynomial or apolynomial of degree less than that of g( x).


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Remainder Theorem

Let f be a polynomial function. If f ( x) is

divided by x - c, then the remainder is f (c).


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Find the remainder if

x + 3 = x - (-3)

is divided by x + 3.

30


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Factor Theorem

1. If f (c)= 0, then x - c is a factor of f ( x).

2. If x - c is a factor of f ( x) , then f (c)=0 .


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Use the Factor Theorem to determine whether the

function has the factor

(a) x + 3

(b) x + 4

x +3 is not a factor of f ( x).

x + 4 is a factor of f ( x).(b) f (-4) = 0


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Theorem Number of Zeros

A polynomial function cannot have

more zeros than its degree.


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Theorem Rational Zeros Theorem

Let f be a polynomial function of degree 1 orhigher of the form

where each coefficient is an integer. If p/q in the

lowest terms, is a rational zero of f , then p must bea factor of a 0 and q must be a factor of a n.


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List the potential rational zeros of

p:

q:


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Find the real zeros of

Factor f over the reals.

There are at most five zeros.

Write factors of -12 and 1 to obtain the potentialrational zeros.


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Thus, -3 is a zero of f and x + 3 is a factor of f .

Thus, -2 is a zero of f and x + 2 is a factor of f .


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Thus f(x) factors as :


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Theorem Bounds on Zeros


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Let f denote a continuous function. If a


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Use the Intermediate Value Theorem to showthat the graph of function

has an x-intercept in the interval [-3, -2].

f (-3) = -11.2 < 0

f (-2) = 1.8 > 0

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