3 polynomial and rational functions © 2008 pearson addison-wesley. all rights reserved sections...

3Polynomial and Rational Functions

© 2008 Pearson Addison-Wesley.All rights reserved

Sections 3.1–3.2

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-2

3.1 Quadratic Functions and Models

3.2 Synthetic Division

Polynomial and Rational Functions3


Quadratic Functions and Models3.1Quadratic Functions ▪ Graphing Techniques ▪ Completing the Square ▪ The Vertex Formula ▪ Quadratic Models and Curve Fitting

Polynomial Functions

A polynomial function of degree n, where n is a nonnegative integer, is a function defined by an expression of the form

where are real numbers where .


,)( 011

1 axaxaxaxf nn

nn

01,, aaa nn 0na

Quadratic Function

A function f is a quadratic function if

where a, b, and c are real numbers with a ≠ 0.

The graph of a quadratic function is a parabola. Parabolas have a line of symmetry thru the vertex.


,)( 2 cbxaxxf


3.1 Example 1(b) Graphing Quadratic Functions (page 304)

Graph g(x) = –2x2 and compare to y = x2 and y = 2x2. Give the domain and range.

The graph of g(x) = –2x2 is a narrower version of the graph of y = x2 and is a reflection of the graph of y = 2x2 across the x-axis.

Domain: Range:

Graphing Techniques for the Quadratic Equation

Standard form of the quadratic function:

1. The vertex is the point (h,k). The axis of symmetry is x = h.

2. The orientation is given by a:– if a > 0 the graph opens up– if a < 0 the graph opens down

4. The y-intercept is f(0) = c.

5. The x-intercept(s) are the solutions to f(x) = 0.


khxaxf 2)(

Graphing Techniques for the Quadratic Equation

Non-Standard form of the quadratic function:

1. The vertex is the point . The axis of symmetry is x = -b/2a.

2. The orientation is given by a:– if a > 0 the graph opens up– if a < 0 the graph opens down

4. The y-intercept is (0,c).

5. The x-intercept(s) are the solutions to f(x) = 0.


cbxaxxf 2)(

a

bf

a

b

2,

2


Graphing Quadratic Function in standard form

Graph F(x) = –2(x + 3)2 + 5 using transformations. Give the domain and range.

The graph of F(x) is the graph of g(x) translated 3 units to the left and 5 units up. It is stretched vertically by a factor of 2. it is reflected across the x-axis.

Domain: Range:


Graphing a Parabola with the non-standard form

Graph f(x) = x2 + 2x – 5 by locating the vertex and the x-and y-intercepts.•The graph opens up because a > 0.•The vertex is found by :

•The y-intercept is (0,-5)•The x-intercepts are the solutions to . This calls for the quadratic formula.

Vertex (–1, –6) Axis: x = –1

65)1(2)1()1( ,1)1(2

2

22

fk

a

bh

a

bf

a

b

2,

2

520 2 xx

Graphing a Parabola with the non-standard form


0,61 and 0,61

61

2

622

2

242

12

51422

520

2

2

x

x

x

xx

The two x-intercepts are:


Example 2 – Non-standard form for Quadratic Function

Find the axis and vertex of the parabola f(x) = –3x2 + 12x – 8 using the vertex formula.

Axis: x = 2

a = –3, b = 12, c = –8

Vertex: (2, f(2))

Vertex: (2, 4)


3.1 Example 5(a) Solving a Problem Involving Projectile Motion (page 308)

A ball is thrown directly upward from an initial height of 75 ft with an initial velocity of 112 ft per sec. Give the function that describes the height of the ball in terms of time t.

The projectile height function is


3.1 Example 5(b) Solving a Problem Involving Projectile Motion (page 308)

After how many seconds does the ball reach its maximum height? What is the maximum height?

The maximum height occurs at the vertex.

The ball reaches its maximum height, 271 ft, after 3.5 seconds.

Verify with a graphing calculator.


3.1 Example 5(c) Solving a Problem Involving Projectile Motion (page 308)

For what interval of time is the height of the ball greater than 200 ft?Solve the quadratic inequality .

Use the quadratic formula to find the values of x that satisfy .

a = –16, b = 112, c = –125

or


3.1 Example 5(c) Solving a Problem Involving Projectile Motion (cont.)

The two numbers divide a number line into three regions, Choose test valuesto see which interval satisfies the inequality.

The ball will be greater than 200 ft above ground level between 1.39 and 5.61 seconds after it is thrown.


3.1 Example 5(c) Solving a Problem Involving Projectile Motion (cont.)



3.1 Example 5(d) Solving a Problem Involving Projectile Motion (page 308)

After how many seconds will the ball hit the ground?

Use the quadratic formula to find the positive solution of

Reject

The ball hits the ground after about 7.62 sec.



Synthetic Division3.2Synthetic Division ▪ Evaluating Polynomial Functions Using the Remainder Theorem ▪ Testing Potential Zeros

Properties for Using Synthetic Division

Division Algorithm:

Let f(x) and g(x) be polynomials with g(x) of lower degree than f(x). There exists a unique quotient, q(x) and remainder, r(x) where

Special Case:

For this case you can use synthetic division.


).()()()( xrxqxgxf

).()()()( xrxqkxxf


3.2 Example 1 Using Synthetic Division (page 323)

Use synthetic division to divide .

x – 3 is in the form x – k.

Bring down the 4, and multiply 3(4) = 12.


3.2 Example 1 Using Synthetic Division (cont.)

Add –15 and 12 to obtain –3. Multiply (–3)(3) = –9.

Add 11 and –9 to obtain 2. Multiply (2)(3) = 6.


3.2 Example 1 Using Synthetic Division (cont.)

Add –10 and 6 to obtain 4.

Since the divisor x – k has degree 1, the degree of the quotient will always be one less than the degree of the original polynomial.

The Remainder Theorem

If the polynomial f(x) is divided by x-k, then the remainder is equal to f(k).

Zeros, Roots and x-intercepts:

A zero of a polynomial function f is a number such that f(k) = 0 (the remainder). The real number zeros are the x-intercepts of the graph of the function. The Remainder Thm is a quick way to decide if a number k is a zero of f(x). A zero of f(x) is a root or solution of the equation f(x)=0.



3.2 Example 2 Applying the Remainder Theorem (page 324)

Let . Use the remainder theorem to find f(4).

Use synthetic division with k = 4.

f(4) = 17

Insert 0 as the coefficient for the missing x2-term.


3.2 Example 3(a) Deciding Whether a Number is a Zero (page 325)

Let . Is k = 2 a zero?

Use synthetic division with k = 2.

2 is not a zero.

Insert 0 as the coefficient for the missing x-term.


3.2 Example 3(b) Deciding Whether a Number is a Zero (page 325)

Let . Is k = –3 a zero?

Use synthetic division with k = –3.

–3 is a zero.


3.2 Example 3(c) Deciding Whether a Number is a Zero (page 325)

Let . Is k = 1 + 3i a zero?

1 + 3i is a zero.

Use synthetic division with k = 1 + 3i.

3 polynomial and rational functions © 2008 pearson addison-wesley. all rights reserved sections...

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