unit e student success sheet (sss)

Unit E Student Success Sheet (SSS) Basics of Polynomials (sections 2.1-2.2)

Standards: Alg 2 8.0, Alg 2 9.0, Alg 2 10.0, Analysis 4.0

Mr. Werdel Segerstrom High School Math Analysis Honors 2011-2012

Concept # What we will be learning… Mandatory Homework

1 Identifying x-intercepts, y-intercepts, vertex (max/min), axis of quadratics and graphing them. Quadratics in standard form. Worksheet 15

Worksheet 16

2 Finding maximum and minimum values of quadratic applications using calculator, interpretation of solutions. [PATH OF FOOTBALL]

Worksheet 17

3 Finding maximum and minimum values of quadratic applications using calculator, interpretation of solutions. [MAXIMIZING AREA]

Worksheet 17

4 Using the leading coefficient test for polynomials to write end behavior (limit notation) Worksheet 18

5 Finding zeroes and multiplicities of polynomials using factoring Worksheet 19

6 Writing equations of polynomials given zeroes and multiplicities Worksheet 19

7 Graphing polynomials, including: x-int, y-int, zeroes (with multiplicities), end behavior. All polynomials will be factorable. Worksheet 20

Worksheet 21

Name: __________________________ Period: _____

Reminders:

Homework is completed in spiral bound notebook only.

Homework not done in homework notebook will not be

accepted.

All pages in homework notebook should be labeled

accordingly:

Unit ______ Concept ______ - (title of assignment) Examples:

Unit E Concept 1 – Practice Quiz

Unit E Concept 1 – Quiz Review

Unit E Concept 1-4 – Practice Test

Mr. Werdel: Monday after school, 2nd

lunch.

Mrs. Kirch: Monday – Wednesday Mornings 7-8am &

Wednesday – Friday afterschool from 3-4pm

Ms. Tamaoki: Tuesday & Thursday mornings 7:30-8am

Success in almost any field depends more on energy

and drive than it does on intelligence.

Sloan Wilson

This unit focuses on characteristics of polynomials. A polynomial is defined as a function with real-number coefficients and non-negative, whole-number

exponents. Its graph is continuous (don’t have to lift your pencil from the paper to draw it) and has smooth, rounded turns (no sharp points like an absolute value

graph) A polynomial can have one term (monomial), two terms (binomial), three terms (trinomial), or four+ terms (polynomial). In addition, a polynomial is

described by the largest exponent that it has in the equation. If the largest exponent is 0, it is a constant function. If the largest exponent is 1, it is linear; if it

is 2, it is quadratic; if it is 3, it is cubic; if it is 4, it is quartic; if it is 5, it is quintic; if it is anything 6 or larger, it is just “degree 6” or “degree 7” (and so on).

We will begin by looking at quadratic functions in more detail. We are familiar with the name parabola to describe the graphs of quadratics. However, we will be

examining those graphs and identifying key parts, such as the vertex (which is either a maximum or a minimum), intercepts, and axis.

Next, we will begin our journey through the world of general polynomials, focus on cubics, quartics, and quintics. We will look at their end behavior, their

zeroes, their intercepts, and other characteristics that define these graphs, including a review of extrema and intervals of increase and decrease.

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---Unit E Student Success Sheet---Basic of Polynomials (sections 2.1-2.2)--- Math Analysis 2011-2012---

Polynomials

How do you

classify

polynomials

by degree?

How do you

define a

polynomial?

How do you

classify

polynomials by

# of terms?

What are the

important

pieces/sections

of a polynomial

graph?

Real-number

coefficients

Continuous

(don’t lift pencil

from the paper

when drawing)

Smooth/

rounded turns

(no sharp

points)

Non-negative,

whole #

exponents


Identifying x-intercepts, y-intercepts, vertex (max/min), axis of quadratics and graphing them. Quadratics in standard form.

In Unit B, we sketched graphs of quadratics using the shifts of the parent function. In this Unit, we will be using that knowledge, but going a step further to make our sketches more accurate and detailed.

Our equations will start in standard form f(x) = ax2 + bx + c. In order to graph them more easily, we must complete the square to put it in the parent function form, like this:

( ) ( )

1.

Vertex Form equation:

_____________________

Vertex: _______ (max or min?)

y-intercept: _______

Axis: _______

x-intercept(s): (exact & approximate)

________ ________

________ ________

2.

Vertex Form equation:

_____________________

Vertex: _______ (max or min?)

y-intercept: _______

Axis: _______

x-intercept(s): (exact & approximate)

________ ________

________ ________


Finding maximum and minimum values of quadratic applications using calculator, interpretation of solutions. [PATH OF FOOTBALL]

Many real-life motions can be modeled by quadratic equations. There are generally three important pieces of the graph.

- Where the object begins, which is the y-intercept, (0,___)

- Where the object changes from increasing to decreasing (or vice versa), represented by the vertex of the graph as a maximum or minimum value

- Where the object reaches the ground, represented by the x-intercept of the graph.

SUGGESTIONS FOR FINDING THE CORRECT WINDOW:

1. Use “Zoom” – “Fit” to get an approximate view of the graph

2. Then, increase the x-max and y-max under the “Window” menu until you can see the whole graph.

3. Sometimes it takes a couple of changes to see what you want to see, don’t give up!

(a) How high is the football off the ground when it is punted?

(b) What is the highest height the football reaches in the air?

(c) How far did the football travel before it hit the ground again?


Finding maximum and minimum values of quadratic applications using calculator, interpretation of solutions. [MAXIMIZING AREA]

We can also use quadratics to find the maximum area of a region given certain boundary constraints.

Remember: Area of a rectangle = bh Perimeter of a rectangle = add up all the sides

A lifeguard has 500 feet of rope with buoys attached to layout a rectangular swimming area on a lake. If the beach forms one side of the rectangle, find the dimension of the rectangle that will have the greatest swimming area. What are the dimensions? What is the size of the swimming area?


Using the leading coefficient test for polynomials to write end behavior (limit notation)

End behavior describes how a graph acts at

the extremes – as we go really far to the left (get closer to negative infinity) or as we go really far to the right (get closer to positive infinity)

Polynomial end behavior is quite predictable. It is based on two things:

1) Degree

2) Leading Coefficient

DEGREE: ODD LEADING COEFFICIENT: POSITIVE

DEGREE: ODD LEADING COEFFICIENT: NEGATIVE

DEGREE: EVEN LEADING COEFFICIENT: POSITIVE

DEGREE: EVEN LEADING COEFFICIENT: NEGATIVE

Because end behavior is based solely on the degree and the LC, these are the only four possible representations.

Equation

LC Degree End behavior (limit statement)

1. +1 3 As x ∞, f(x) ∞ As x - ∞, f(x) - ∞

2.

3.

4.

5.


Finding zeroes of polynomials using factoring

Besides its end behavior, the next key trait of a polynomial is its zeroes, A.K.A. x-intercepts, roots, solutions One method of finding zeroes is by factoring. However, we will find that many polynomials will not factor, and we will have to use other methods (which we will explore in Unit F). All polynomials in this Unit will be factorable.

Equation Factored Form Zeroes

1.

2.

3.

4.

5.

6.

7.

8.

9.


The last five problems have something peculiar about them…a zero shows up more than once! This is called

multiplicity. We will see later what multiplicity of zeroes do to a graph.

10.

11.

12.

13.

14.

Writing equations of polynomials given zeroes and multiplicities

If we know a polynomial’s zero(es), then we have enough information to write the standard form equation of the function.

(take all the factors and multiply them together to get standard form!)

IN THIS UNIT, WE ARE ASSUMING THE LEADING COEFFICIENT IS 1

1.

(x+5)(x+5)(x+5)

2. 3.


4.

5. 6.

Graphing polynomials, including: x-int, y-int, zeroes (with multiplicities), end behavior. All polynomials will be factorable.

We are now ready to do a full evaluation of factorable polynomials and describe every aspect of them: how they behave at the extremes, how they behave in the middle, where their highest and lowest points are, where their intercepts are, and under what intervals they are increasing or decreasing. Gosh, that’s a lot of stuff we can use to describe these cool little graphs!

On a graph, multiplicity of zeroes is a big deal.

You must pay careful attention to a zero’s

multiplicity so you know how to act around the x-axis!

(In this Unit we will only be dealing with multiplicities of 1 and 2 on the graph – we’ll throw in multiplicity of

3 on graphs next Unit!


Equation 1.

Factored Equation

End behavior

x-intercepts (with multiplicities)

y-intercept

Extrema

Intervals of Increase & Decrease

Equation 2.

Factored Equation

End behavior


y-intercept

Extrema



Equation 3.

Factored Equation

End behavior


y-intercept

Extrema


Equation 4.

Factored Equation

End behavior


y-intercept

Extrema



Equation 5.

Factored Equation

End behavior


y-intercept

Extrema


Equation 6.

Factored Equation

End behavior


y-intercept

Extrema


unit e student success sheet (sss)

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