Quadratic Functions (3.1)

12102function theofMin/Max

2at vertex 2

122

27222

272242

2742

782

2

22

222

2

2

x

x

x

xx

xx

xx

Identifying the vertex (e2, p243) Complete the square

110 function ofmin/max ;3at vertex

13

833

8336

86

86

2

22

222

2

2

x

x

x

xx

xx

xx

Identify both the vertex and the x intercept (e3, p244)

intercepts axisx 4,2)4)(2(862 xxxxxFactor it

Given vertex and a point find the equation of a parabola (e4) (p244) plug in all 4 values given.

212

22136

),(

2

2

2

xy

aa

khvertexkhxay

Higher Degree Polynomials (3.2)

negative coefficient reflects the graph in the x-axi

Degree is odd

upward shift, by one unit  left  shift,  by  one  unit

the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right

the degree is even and the leading coefficient is positive, the graph rises to the left and right

the  degree  is  odd  and  the  leading  coefficient  is  positive,  the  graph falls to the left and rises to the right

The Leading Coefficient Test only tells you whether the graph eventually  rises  or  falls  to  the  right  or  left.   Other  characteristics  of  the graph, such as intercepts and minimum and maximum points, must be determined by other tests.

Page 255

Page 255

Apply Leading Coefficient Test. Because the leading coefficient is positive and the degree is even, you know that the graph eventually rises to the left and to the right

2346 23 xxxxfExample 11 (P274): Find the zeros of

Step1: plot the graph – let the calculator/computer do the work

There is a zero here, between 0.6 and .07

Step2: Rational Zero Test (P270)

6

1,

3

2,

3

1,

2

1,2,1

6,3,2,1

2,1

6 of factors

2 of factors

Setp3: Test

023

233243

26

234623

23

xxxxf Plug it into the calculator, don’t try to evaluate it by hand

Yes 2/3 is a zero

x3 x2 x c

2/3 6 -4 3 -2

4 0 2

6 0 3 0

Setp4: Synthetic Division

363

2 2

xxxf

Remainder is 0 Another proof that 2/3 is a zero

6(2/3)

Remainder and factor theorems on page 268; e5,6

We will skip the upper and lower bound rule on page 258

6x2 3

x-2/3 6x3 -4x2 3x -2

6x3 -4x2

3x -2

0

Now the long division is much easier (p264, e1,2,3)

Example 3 (p28): 8122 235 xxxxxf

Possible zeros, repeated (touches)?

A zero

8,4,2,11

8,4,2,1

1 of factors

8 of factors

Step1: Plot it

Step2: Rational Zero Test (p 256)

x5 x4 x3 x2 x c

-2 1 0 1 2 -12 8

-2 4 -10 16 -8

1 -2 5 -8 4 0

48522 234 xxxxx

x4 x3 x2 x c

1 1 -2 5 -8 4

1 -1 4 -4

1 -1 4 -4 0

4412 23 xxxxx

x3 x2 x c

1 1 -1 4 -4

1 0 4

1 0 4 0

4)1(12 2 xxxx

Setp3: Test: Plug in 1 and 2

Setp4: Synthetic Division

)2)(2)(1(12

2)1(124)1(12 222

ixixxxx

ixxxxxxxx

Example 1, p286

100001.

01.100

1100

1)(

f

f

xxf

As x (input) gets bigger y (output) gets smaller

As x (input) gets smaller y (output) gets larger

0

domain

asymptotea straight line associated with a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero and the slope of the curve at the point approaches the slope of the line

Rational Functions and Asymptotes (3.5)

Degree  of  the  numerator  is  equal to the  degree  of  the denominator Horizontal asymptote: y= ratio of leading coefficients

Degree  of  the  numerator  is  less  than  the  degree  of  the denominator Horizontal asymptote: y = 0

Vertical asymptotes: set the denominator equal to zero and solve the resulting equation for x

Degree  of  the  numerator  is  greater than the  degree  of  the denominator No horizontal asymptote

Example 4 (p289)

Example 5 (p290)

Degree of numerator < denominator Horizontal asymptote y = 0

0 intercept y 00

0 set x to

f

0

02

0 y to

2

xxx

x

set

Example 3 (p298)

x2 x c

-1 1 -1 0

-1 2

1 -2 2

1

22

1

221

1

2

x

xx

xx

x

xxxf

Synthetic Division

Slant Asymptotes – Page 299

Slant asmyptote

1

2

1

21

1

22

xx

x

xx

x

xxxf

x2 x c

1 1 -1 -2

1 0

1 0 -2

Example 5 (p299)

Slant asmyptote