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NC Math 1 Unit 5

Quadratic Functions

Main Concepts

Study Guide & Vocabulary

Classifying, Adding, & Subtracting Polynomials

Multiplying Polynomials

Factoring Polynomials

Review of Multiplying and Factoring Polynomials

Graphing Quadratic Functions & Characteristics of Quadratic Functions

Application of Quadratic Functions

Quadratic Regression and Choosing a Model of Best Fit

Review & Practice Test

Common Core Standards

NC.M1.A-

APR.1

Build an understanding that operations with polynomials are comparable to operations with integers by adding and subtracting quadratic expressions and by adding, subtracting, and multiplying linear expressions.

NC.M1.A-

APR.3

Understand the relationships among the factors of a quadratic expression, the solutions of a quadratic equation, and the zeros of a quadratic function.

NC.M1.A-

CED.2

Create and graph equations in two variables to represent linear, exponential, and quadratic relationships between quantities.

NC.M1.A-

REI.1

Justify a chosen solution method and each step of the solving process for linear and quadratic equations using mathematical reasoning.

NC.M1.A-

REI.11

Build an understanding of why the x-coordinates of the points where the graphs of two linear, exponential, and/or quadratic equations 𝑦 = f(𝑥) and 𝑦 = g(𝑥) intersect are the solutions of the equation f(𝑥) = 𝑔(𝑥) and approximate solutions using graphing technology or successive approximations with a table of values.

NC.M1.A-

REI.4

Solve for the real solutions of quadratic equations in one variable by taking square roots and factoring.

NC.M1.A-

SSE.1a

Interpret expressions that represent a quantity in terms of its context. a) Identify and interpret parts of a linear, exponential, or quadratic expression, including terms, factors, coefficients, and exponents.

NC.M1.A-

SSE.1b

Interpret expressions that represent a quantity in terms of its context. b) Interpret a linear, exponential, or quadratic expression made of multiple parts as a combination of entities to give meaning to an expression.

NC.M1.A-

SSE.3

Write an equivalent form of a quadratic expression, 𝑎𝑥2 + 𝑏𝑥+ 𝑐, where a is an integer, by factoring to reveal the solutions of the equation or the zeros of the function the expression defines.

NC.M1.F-

BF.1b

Write a function that describes a relationship between two quantities. b) Build a function that models a relationship between two quantities by combining linear, exponential, or quadratic functions with addition and subtraction or two linear functions with multiplication.

NC.M1.F-

IF.2

Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

NC.M1.F-

IF.4

Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums.

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I understand…

the effect of gravity controls the a value of a quadratic in standard form when modeling projectile motion.

initial height controls the c value of a quadratic in standard form when modeling projectile motion.

initial velocity controls the b value of a quadratic in standard form when modeling projectile motion.

that the x-intercepts of quadratic functions are the solutions of quadratic equations.

that the rate of change of a quadratic function does not remain constant.

that the x-intercepts/solutions/zeros/roots can be determined by factoring for some quadratic functions. I can…

use function notation to evaluate a quadratic function given a value in the domain.

interpret the contextual meaning of a given point from a quadratic function in function notation.

interpret the meaning of the independent and dependent variables in context of a quadratic function.

interpret and analyze key features of a quadratic function in context including positive/negative,

increasing/decreasing, intercepts, maximum/minimum and domain/range when given the function as a table,

graph, and/or verbal description.

identify the terms, factors and coefficients of a quadratic expression.

interpret the terms, factors and coefficients of a quadratic expression in terms of the context.

create an equation in two variables to represent a quadratic relationship between two quantities.

graph a quadratic equation that represents a relationship between two quantities.

choose an appropriate domain and range for a quadratic function.

identify the maximum and minimum of quadratic functions.

identify where a quadratic function is increasing and decreasing.

compare linear and quadratic functions symbolically, graphically, verbally, and using tables.

build a quadratic function by multiplying linear equations and combining two quadratic equations with

addition and subtraction.

Essential Questions:

How can projectile motion be modeled using a quadratic function? How does knowing the definition of a maximum or minimum help you visualize the graph of a

quadratic function? How do you determine which solution to use for a quadratic equation? How is factoring connected to the distributive property? How can I compare operations with integers to operations with quadratic expressions? What types of information are contained in various forms of a quadratic function?

Vocabulary:

axis of symmetry

binomial

constant

degree of a monomial

degree of a polynomial

difference of squares

extreme values

factoring

intercepts

linear expression

monomial

polynomial

relative minimum or maximum

solutions or roots of a quadratic equation

standard form of a polynomial

symmetry

trinomial

vertex

x-intercepts of a quadratic function

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Algebra Tiles

x2 = –x2 = x = –x = +1 = –1 =

You simplify variable expressions by combining like terms. Use zero pairs, where needed, and

perform as many operations as possible within the expression.

Example. 3a + (–a) = + = = 2a

0

2y + 7 + y – 3 = + + = 3y + 4

–3

Use models to simplify these expressions.

1. 4x + 7x = 2. 5b – 2b =

3. 3a + (-2a) = 4. 5y – (–2y) =

5. 3c2 – 8 + 2c2 = 6. 8 – 5x + 3x2 – 2 + 3x =

Remember 2(x + 3) means + = 2x + 6

Use models to simplify these expressions.

7. 2(y2 – 4) = 8. 3(a + 3) =

9. 2(z – 4) + 9 = 10. 2(2b2 – 1) + 7 =

11. 10c2 + 4(c2 + 3c) – 5c = 12. 8y + 2(3 – y) – 4 =

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What are polynomials?

A polynomial is a monomial or the sum of monomials.

A monomial is a number, a variable, or a product of numbers and variables.

After being simplified, a polynomial can be named based on its degree and its number of terms.

The degree of a monomial is the sum of the exponents of the monomial’s variables.

The degree of a polynomial is equal to the degree of the monomial term with the largest degree.

Polynomial Degree Number of Terms

Name Graph

3x + 2 1 2 linear binomial

a2 – 3a + 5 2 3 quadratic trinomial

6z3 3 1 cubic monomial

2x4 + 2x - 4 4 3 4th degree trinomial

7 0 1 constant monomial

(or just “constant”)

Polynomials are usually written in standard form. The standard form of a polynomial is when the

polynomial is written such that the degrees of the monomial terms decrease from left to right.

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What is the degree of each monomial?

Write each polynomial in standard form and then name it.

Modeling Polynomials with Algebra tiles

Add and subtract by combining like terms and by using “adding zero” where needed.

Ex. (2x2 + x + 3) + (x2 – 3x + 1)

+

Form a zero pair with +x and –x.

= 3x2 – 2x + 4

Try these:

1. (2x2 + 3x + 3) + (x2 + 2x – 2)

2. (3x2 + 4x – 3) + (4x2 – 2x – 4)

3. (x2 + 3x – 4) – (2x2 + 2x – 3)

4. (5x2 – 2x + 1) – (x2 +2x – 4)

5. 3x – 2x2 + 3x4 6. 9x3 7. 8 – 4x

8. 5b2 + b3 – 2b +

7

9. 5x – 3x2 + 2x2 – 3 10. 3 – 3x2

1. 3ab 2. 814 3. 7x4 4. 7a2b

0

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EXPLORING MULTIPLYING POLYNOMIALS

x2 = –x2 = x = –x = +1 = –1 =

ex. 1. x(x + 1) 2. 2x(–x + 3)

x + 1

–x2 x x x

x x2 x = x2 + x = –2x2 + 6x

–x2 x x x

Try these examples:

1. 2(3x + 1)

2. x(x + 2)

3. –2x(x + 2)

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Homework

Simplify each sum or difference. Write your answer in standard form. Name each polynomial. (To avoid

sign errors, carefully use the definition of subtraction to rewrite all subtraction as addition, and carefully

use the distributive property when a polynomial with more than one term is being subtracted.)

1. (3x2 – 2x – 7) + (5x2 – 7x + 3)

2. (8x2– 10) (3x2 – 5x 9)

3. (6t5 – 4t2 + 7) (9t4 – 2t5 2t)

4. (11 – 2a + 3a3) + (7 – a4 – 5a + 7a2)

5. (3n3 – 2n4 – n) (5n5 – n2 + 7)

6. x(2x – 1)

7. 3x(2x + 1)

8. 2x(–x + 4)

9. -x ( -3x + 2)

10. challenge 4x2 (x + 4)

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ex. 1. (x + 1)(x + 1) 2. (x + 1)(–x + 3)

-x + 3

x + 1

x –x2 x x x

x x2 x = x2 + 2x + 1 = –x2 + 2x + 3

+

+ 1 –x 1 1 1

1 x 1

Draw out the algebra tiles

1. (x + 2)(x + 1)

2. (x + 2)(x + 2)

3. (–2x – 1)(x + 2)

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Multiplying Binomials: The Distributive Property

*Without really thinking about it, how would you simplify (a +

b)2 ?

To calculate the product of (2x + 4) and (3x + 5), you can use a

rectangle to represent the problem.

You can then calculate the area of each part and then calculate

the sum of all of the parts to find

the area of the whole rectangle.

(2x + 4)(3x + 5) = 6x2 + 12x + 10x + 20 = 6x2 + 22x + 20

You can also use the distributive property without the graphic

organizer.

(2x + 4)(3x + 5) = (2x)(3x) + (2x)(5) + (4)(3x) + 4(5) = 6x2 + 22x + 20

A mnemonic often used when multiplying two binomials is FOIL. FOIL is an acronym for

First, Outer (or outside), Inner (or inside), and Last. In the example (2x + 4)(3x + 5)

F: (2x)(3x) = 6x2 ; O: (2x)(5) = 10x ; I: (4)(3x)=12x ; L: (4)(5) = 20

If the two polynomial factors are not both binomials, you cannot use the FOIL method. When

you use the distributive property, a box, or area, diagram can help you organize your work.

When multiplying polynomials, think about the distributive property very carefully. Pay

attention to the sign of each term.

1. (2a – 1)(3a + 7)

F:

O:

I:

L:

2. (3a – 7)(a 2)

F:

O:

I:

L:

3. (x + 1)(x + 4)

F:

O:

I:

L:

4. (7n – 5)(5n + 4)

F:

O:

I:

L:

2x + 4

3x

+5

2x + 4

3x

6x2

12x

+5

10x

20

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5. (x – 3)(x2 – 2x + 5) 6. (y + 6)(y2 + 4y 3)

7. (5x2 – 3)(5x2 + 3) 8. (x2 – 2)(x + 3)

9. (2a2 + 4)(a + 7) 10. (2b2 – 7)(b + 3b2)

11. (9w2 + 5)(2w3 + 3w + 1) 12. (y 2)(3y2 + 6y 7)

13. Mr. Wilson has two congruent, rectangular gardens 3 feet apart from

each other. The length of each garden is 4 times its width. There is a 3-ft wide

walkway around each garden.

a. Write an expression for the combined areas of the gardens and walkway.

b. Write an expression for the area of the walkway only.

c. The walkway’s area is 150 square feet. How big is each garden?

3ft

3ft 3ft

3ft 3ft

3ft 3ft

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14. Calculate the surface area of the space figure.

15. Calculate the volume of the space figure.

Complete each area diagram. Write down the problem that it illustrates and simplify the product.

16.

17.

18.

These 3 products deal with special binomial patterns.

19.

20.

21.

(4x – 3) cm

(x + 1) cm

(2x + 4) cm

(3x 2) cm

5

5x

15x2

3

3x –7

6x2

–12x

2x 5

8x2

3

6x

a b

a

b

a –b

a

–b

a b

a

–b

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Binomials: Multiplying Special Cases

Use the algebra tiles to model (x + 3)2 Use the algebra tiles to model x2 + 9

Draw below Draw below

Write the algebraic equation for (x+3)2 Write the algebraic equation for x2 + 9

Are the results the same

Why???

A) The Square of a Binomial:

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

The square of a binomial is the square of the first term plus twice the product of the two terms

plus the square of the last term. Its product is called a perfect square trinomial.

B) The Difference of Squares

(a + b)(a – b) = a2 – b2

The product of the sum and difference of the same two terms is the difference of their squares.

(a + b) and (a – b) are conjugates of one another.

You can always use the distributive property to multiply these two special cases, but you will be

using them a lot, so you should have them memorized and mastered.

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Simplify each product.

1. (x – 2)(x + 2)

2. (x – 2)2

3. (x + 2)2

4. (x – 7)(x + 7)

5. (x – 7)(x – 7)

6. (x + 7)2

7. (3x – 9)(3x + 9)

8. (3x – 9)2

9. (3x + 9)2

10. Identify each polynomial as a difference of squares, a perfect square trinomial, or neither.

a. 8x2 – 9 b. 25x2 – 20x – 4 c. x2 – 1 d. 4x2 + 25 e. x2 + 6x + 9

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Practice Multiplying Polynomials

Simplify each product. Circle the ones that are one of the two patterns. Time yourself.

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Factoring Quadratic Polynomials

In order to factor polynomials, put your algebra tiles into a rectangular array. You may need to add zero

pairs to complete your model.

Ex. Factor x2 + 4x + 3

What should be on the

Rearrange left hand side?

above?

x2 + 4x + 3 = (x + 1)(x + 3)

Factor 2x2 + 3x

Rearrange

2x2 + 3x = x(2x + 3)

Factor –x2 + 3x + 4

Rearrange Notice I had to

add a zero

-pair (x & –x).

–x2 + 3x + 4 = (x + 1)(–x + 4)

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Factor each problem. Draw the algebra tiles

1. –3x2 + 2x

2. 5x2 –15x + 10

3. x2 + 6x + 8

4. a2 + 4a + 4

5. x2 – 6x + 9

6. y2 – 2y – 15

7. m2 + m – 12

8. 6x2 + 7x + 2

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Factoring without algebra tiles

In the first example, we’ll just look at the first and last steps of factoring polynomials.

Step 1: Factoring out the GCF

The greatest common factor, or GCF, is the largest factor the terms have in common. Factoring

a polynomial reverses the multiplication process. To find the GCF, list all the prime factors of

each term and then circle or underline the factors common to ALL terms. If the coefficient of the

first term of a polynomial written in standard form is negative, factor out –1.

Example 1: What is the GCF of 12x3 + 18x2 – 15x?

Calculate the GCF of the 3 terms

12x3: 2 * 2 * 3 * x * x * x

18x2: 2 * 3 * 3 * x * x

15x: 3 * 5 * x

So, the GCF is 3x

Use the distributive property to write the product. 3x (4x2 + 6x – 5)

To finish the problem, check each factor to make sure that it cannot be factored any

further. Multiply the factors to make sure it was factored correctly. Sign errors and

arithmetic errors are a fairly common errors, so pay close attention to your signs and

double check your multiplication to make sure you didn’t add.

Note: When finding the GCF of the variables, find the least power of each variable.

For example: x5y − x3y2 + x2y4

x2 is the least power of x

y is the least power of y

GCF is x2y

Final factored answer is x2y (x3 – xy + y3)

Factor the following polynomials by factoring out the GCF.

1) 5𝑥5 + 10𝑥3 2) 4𝑏3 + 12𝑏2 − 8𝑏 3) −3𝑑3 − 12𝑑2 + 15𝑑

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Step 2: Check for the two patterns

If the polynomial is a binomial, the only way you have learned to factor it so far is to factor out

the GCF and then check to see if it is a difference of squares. In a later math course, you will be

taught how to factor a binomial if it is a sum or a difference of cubes.

If the polynomial is a trinomial, after factoring out the GCF, check to see if it is a perfect square

trinomial. If it is, you can factor it quickly as the square of a binomial.

a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2

a2 – b2 = (a + b)(a – b)

4) 𝑥2 − 25 5) 4𝑥2 − 16 6) 𝑥2 − 6𝑥 + 9 7) 12𝑥3 + 36𝑥2 + 27𝑥

Step 3: Factoring Trinomials ax2 + bx + c

Factoring polynomials can help you solve many different types of problems. In Math I, you will

use the factors of quadratic polynomials to help you graph quadratic functions and solve

problems modeled by quadratic functions. Not all quadratic polynomials are factorable, but most

of the ones you deal with this year will be.

After factoring out the GCF, if you are left with a quadratic trinomial that is a not perfect square

trinomial, you may have to do a little bit of guessing and checking to calculate its factors.

You’ve already learned that the product of two linear binomials is a quadratic trinomial. Use that

knowledge to learn how to factor quadratic trinomials.

Using variables to represent coefficients and constants. Starting with the quadratic trinomial Ax2

+ Bx + C, we’ll assume that the GCF has already been factored out, so A is a counting number,

and B & C are integers. We can declare its linear binomial factors as (Dx + E) and (Gx + H). D

and G must be counting numbers, since A is a counting

number. Remember to factor out a –1 if the quadratic

term has a negative coefficient. E and H are the constant

terms will be integers.

We can use an area model to look for patterns that will

help us factor quadratic trinomials.

The factors of A will always be G and D. The factors of

C will always be E and H. DHx + GEx = (DH + GE)x

= Bx

Because the GCF was already factored out of Ax2 + Bx + C, G and H share no common factors

other than 1, and D and E share no common factors other than 1.

Dx + E

Gx

Ax2 = GDx2

GEx

+H

DHx

EH = C

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A key characteristic that will help you factor quadratic trinomials more quickly is that the factors

of AC have a sum of B. AC = (GD)(EH). The factors could also be written as (DH)(GE). If you

add DH and GE, you get the value of B.

Another key characteristic deals with the signs of the constants. If C is positive, E and H must be

the same sign. If C is negative, E and H must have different signs.

Before you start factoring a quadratic trinomial, remember to do the following:

1. Factor out the GCF. If the coefficient of the quadratic term is negative, factor –1 out of

the quadratic trinomial.

2. Check to see if the trinomial is a perfect square trinomial. If it is, write its factors as the

square of a binomial.

Sometimes, the coefficient of the quadratic term will be 1. These quadratic trinomials can be

factored fairly quickly. Since A = 1, we have a quadratic trinomial in the form of x2 + Bx + C.

From the area model, we know that the factors of A and C must have a sum of B. When A = 1,

the factors of C must equal B. G and D must both be equal to 1 since A is 1 and we only use

integers when we factor.

Example:

Factor completely. x2 – 7x -30

Draw a 2x2 box:

Place the highest degree term in the top left box and the constant in the bottom right box.

Use the “diamond” to find the other two boxes.

Find 2 terms that multiply to get ac and add/subtract to get b.

Find the GCF of the 1st row to find the 1st part of the factor on the left.

Find the top “factor”

Find the 2nd factor on the left

ac

b

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8) 𝑥2 + 7𝑥 + 12 9) 𝑔2 + 7𝑔 + 10 10) 𝑑2 − 17𝑑 + 42

Some polynomials have two variables. If they are factorable, each binomial factor will have two

linear terms instead of one linear term and one constant term.

11) 𝑥2 + 11𝑥𝑦 + 24𝑦2 12) 2𝑚2 − 34𝑚𝑛 − 120𝑛2

If the coefficient of the quadratic term isn’t 1, you can use the pattern we looked at earlier to help

you factor it. If graphic organizers help you, draw the area model.

13) 10𝑥2 + 17𝑥 + 3 14) 10𝑥2 + 11𝑥 − 6

Step 4: Check each factor again to make sure the polynomial is fully factored. Multiply

your factors to make sure you factored correctly.

Even though we haven’t yet covered steps 2 and 3, we’ll look at the final step. Every time you

fully factor a polynomial, you should check each factor when you’re done to make sure you

didn’t miss any factors. You should also multiply your factors to make sure the product is

correct.

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Fully factor each quadratic polynomial. Practice following the 4 steps.

1. x2 – 3x + 2

2. x2 + 17x + 16

3. x2 10x + 16

4. x2 – 8x + 16

6. y2 21y + 38

5. w2 + w 6

7. x2 – 4x – 5

8. x2 + 27x + 26

9. x2 15x + 26

10. x2 – 11x – 26

12. y2 3y 54

11. w2 + 25w – 26

13. m2 – 13mn + 42n2

14. x2 + 2xy 15y2

15. a2 3ab 10b2

16. x2 + 17xy + 72y2

18. x2 xy 72y2

17. x2 + 27xy + 72y2

19. x2 21xy 72y2

20. x2 12x + 32

21. x2 12x + 27

22. x2 – 21x – 100

24. y2 2y 3

23. w2 – 29w + 100

25. x2 – 6x + 9

27. x2 12x 36

26. x2 – 9

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Fully factor each polynomial. Practice following the 4 steps.

1. 6a2 + 11a – 7 2. 3a2 13a + 14 3. 2x2 + 5x + 2

4. 35n2 + 3n – 20 5. 3n2 7n + 2 6. 6n2 + 19n + 14

7. 4n2 11n – 3 8. 4n2 8n + 3 9. 4x2 4x 3

10. 6y2 25y + 14 11. 4x2 + 13x + 3 12. 5x2 + 21x + 4

13. 5y2 12y + 4 14. 5x2 + 9x + 4 15. 2x2 + 17x + 8

16. 4y2 16y + 15 17. 4x2 + 4x 15 18. 4x2 4x 15

19. 3y2 y 4 20. 3x2 + 11x 4 21. 3x2 + 13x + 4

22. 4x2 + 16 23. 3x2 + 6x 24. 12x2 – 3x

25. 4x2 + 8x + 4

26. 4x2 – 4 27. 9x2 – 49

28. 25x2 – 20x + 4

29. 12x2 – 243 30. 8x2 – 56x + 98

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More Practice Factoring Polynomials

Fully factor each polynomial. Circle the ones that can be factored by using one of the two patterns. Time yourself.

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Properties of Parabola

A ____________________ is a function that can be written in the Standard Form of 2y ax bx c where a, b, and c are real numbers and a 0. Ex:

25y x 22 7y x

2 3y x x

The domain of a quadratic function is ____________________.

The graph of a quadratic function is a U-shaped curve called a ________________.

All parabolas have a _______________, the lowest or highest point on the graph (depending upon

whether it opens up or down).

The ______________ _______ _____________________ is an imaginary line which goes through

the vertex and about which the parabola is symmetric.

Characteristics of the Graph of a Quadratic Function: 2y ax bx c

Direction of Opening: When 0a , the parabola opens ________:

When 0a , the parabola opens ________:

Stretch: When 1a , the parabola is vertically ________________.

When 1a , the parabola is vertically ________________.

Axis of symmetry: This is a vertical line passing through the vertex. Its equation is

_____________.

Vertex: The highest or lowest point of the parabola is called the vertex, which is on the axis of

symmetry. To find the vertex, plug in 2

bx

a

and solve for y. This yields a point (________,

__________) x-intercepts: are the 0, 1, or 2 points where the parabola crosses the x-axis. Plug in y = 0 and

solve for x.

y-intercept: is the point where the parabola crosses the y-axis. Plug in x = 0 and solve for y: y = c.

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Without graphing the quadratic functions, complete the requested information:

1.) 2( ) 3 7 1f x x x

What is the direction of opening? _______

Is the vertex a max or min? _______

Compare to y = x2? __________

2.)

25( ) 3

4g x x x

What is the direction of opening? _______

Is the vertex a max or min? _______

Compared to y = x2? __________

3.) The parabola y = x2 is graphed to the right.

Note its vertex (___, ___) and its width.

You will be asked to compare other parabolas to

this graph.

Graphing in STANDARD FORM (2y ax bx c ): we need to find the vertex first.

Vertex

- list a = ____, b = ____, c = ____

- find x = 2

b

a

- plug this x-value into the function (table)

- this point (___, ___) is the vertex of the parabola

Graphing

- put the vertex you found in the center of

your x-y chart.

- choose 2 x-values less than and 2 x-values more

than your vertex.

- plug in these x values to get 4 more points.

- graph all 5 points

Find the vertex of each parabola. Graph the function and find the requested information

4.) f(x)= -x2 + 2x + 3

Vertex: _______

Max or min? _______

Direction of opening? _______

Axis of symmetry: ________

5.) h(x) = 2x2 + 4x + 1

Vertex: _______

Max or min? _______

Direction of opening? _______

Axis of symmetry: ________

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

y

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Compare to the graph of y = x2?_______________ Compare to the graph of y = x2?________

Unit 6 Skills (Review & Preview)

naming polynomials

simplifying polynomials: standard form, adding, subtracting and multiplying including the 2 special cases

factoring polynomials including the 2 special cases

calculating zeros of quadratic functions

calculating the axis of symmetry using the zeroes(roots or x-intercepts) of a quadratic function

calculating the vertex using the axis of symmetry of a quadratic function

identifying quadratic functions by looking at the characteristics of a graph

without graphing, using the coefficient of the quadratic term to determine whether the parabola has a maximum vertex

and opens down or a minimum vertex and opens up as well as determining how wide or narrow it is compared to other

quadratic functions

graphing a quadratic function by using the characteristics as well as graphing it by using substitution

calculating the y-intercepts of a quadratic function

1. What is the standard form of a quadratic function?

2. What is the parabola’s vertex and where is it located?

State whether the parabola has a minimum or maximum point and draw its axis of symmetry (dashed line).

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3.

What is the vertex?

Compare the value of “a” to zero.

4.

What is the vertex?

Compare the value of “a” to zero.

5.

What is the vertex?

Compare the value of “a” to zero.

12. Order the quadratic functions from widest to narrowest graphs.

𝑓(𝑥) = 3𝑥2 − 2𝑥 + 3 ; 𝑔(𝑥) = 5𝑥2 ; ℎ(𝑥) = 6𝑥2 + 2𝑥 − 1 ; 𝑗(𝑥) =1

2𝑥2 ; 𝑘(𝑥) = 𝑥2

13: Use the characteristics of a quadratic function to calculate the vertex & 2 points on each side of the vertex.

Sketch the parabola.

𝑦 = −𝑥2 + 4

a. Factor the polynomial. Solve for the zeroes/roots of the

function.

b. What is the axis of symmetry? Graph it.

c. What is the vertex? Graph it

d. What is the y-intercept? Graph it and its reflection over the

axis of symmetry.

e. Calculate one other point on the graph. Graph it and its

reflection over the axis of symmetry.

14. 𝑦 = −𝑥2 Vertex: minimum or maximum

15. 𝑦 = 2𝑥2 − 5𝑥 − 1 Vertex: minimum or maximum

y

x

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Width compared to y = x2: wider, narrower or equal

Parabola opens: up or down

y-intercept:

Width compared to y = x2: wider, narrower or equal

Parabola opens: up or down

y-intercept:

1. 𝑦 = 𝑥2 − 4𝑥 + 3

a) What is the axis of symmetry?

Graph it.

b) What is the vertex?

Graph it

c) What is the y-intercept?

Graph it and its reflection over the

axis of symmetry.

d) Find one other point on the graph.

Graph it and its reflection over the

axis of symmetry.

e) Sketch the parabola.

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2. 𝑦 = −𝑥2 + 2𝑥 − 1

a) What is the axis of symmetry?

Graph it.

b) What is the vertex?

Graph it

c) What is the y-intercept?

Graph it and its reflection over the

axis of symmetry.

d) Find one other point on the graph.

Graph it and its reflection over the

axis of symmetry.

e) Sketch the parabola.

3. 𝑦 = 𝑥2 + 2𝑥

a) What is the axis of symmetry?

Graph it.

b) What is the vertex?

Graph it

c) What is the y-intercept?

Graph it and its reflection over the

axis of symmetry.

d) Find one other point on the graph.

Graph it and its reflection over the

axis of symmetry.

e) Sketch the parabola.

4. 𝑦 = −𝑥2 − 2𝑥 + 3

a) What is the axis of symmetry?

Graph it.

b) What is the vertex?

Graph it

c) What is the y-intercept?

Graph it and its reflection over the

axis of symmetry.

d) Find one other point on the graph.

Graph it and its reflection over the

axis of symmetry.

e) Sketch the parabola.

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Applications of Quadratics Examples

Only ask 4 questions:

a) zeros

b) max/min height (y value of the vertex)

c) when it hits the max/min (x value of the vertex)

d) height when x = 0 (y intercept)

1. Marta throws a baseball with an initial upward velocity of 70 feet per second. This equation

ℎ(𝑡) = −16𝑡2 + 70𝑡 models the situation.

a. Ignoring Marta’s height, how long after she releases the ball, will it hit the ground?

b. What is the maximum height of the baseball?

c. When does the ball reach the maximum height?

2. A rectangular lot is 50 feet wide and 60 feet long. If both the width and the length are increased by the same

amount, the area is increased by 1200 square feet. Find the amount by which both the width and the length are

increased.

3. A rectangular lawn has dimensions of 24 feet by 32 feet. A sidewalk will be constructed along the inside edges

of all four sides. The remaining lawn will have an area of 425 square feet. How wide is the walk?

4. The floor of a rectangular cage has a length 4 feet greater than its width, w. Bob will make a new cage and

increase each dimension of the cage’s floor by 2 feet.

a. Write an expression to represent the area of the new cage’s floor in terms of the width of the original cage’s

floor.

b. Compare the volume of the new cage to the volume of the original cage.

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Applications HW

1. A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. This is modeled by

the equation ℎ(𝑡) = −16𝑡2 + 240𝑡.

a. How long will it take the boulder to hit the ground?

b. How high was the boulder after 5 seconds?

2. A rectangular lawn is 60 feet by 80 feet. How wide of a uniform strip must be cut around the edge when mowing

the grass in order for half of the lawn to be cut?

3. A picture frame measures 12 cm by 20 cm (uniform width). The picture without the frame measures 84 2cm .

Find the width of the frame.

4. A rectangular picture is 12 inches by 16 inches. If a frame of uniform width contains an area of 165 square

inches (frame only), what is the width of the frame?

5. A farmer has 400 feet of fencing. He wants to fence off a rectangular field that borders a straight river (no fence

along the river). What are the dimensions that will give him the largest area?

6. A rectangular pen is to be built along a wall from 72 yards of fencing. Find the maximum area that can be

enclosed and the dimensions of the pen.

7. A rectangular piece of ground is to be enclosed on three sides by 160 ft of fencing. The fourth side is the barn.

Find the dimensions of the enclosure and the maximum area that can be enclosed.

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8. The length of a tropical garden at a local conservatory is 5 feet more that its width. A walkway 2 feet wide

surrounds the outside of the garden. If the total area of the walkway and garden is 594 square feet, find the

dimensions of the garden.

9. Helen has a rectangular garden 25 feet by 50 feet. She wants to increase the garden on all sides by an equal

amount. If the area of the garden will be increased by 400 square feet, by how much will each dimension be

increased?

10. A rectangle is 5 centimeters longer than it is wide. Find the possible dimensions if the area of the rectangle is

104 square centimeters.

11. The YMCA has a 40 foot by 60 foot area in which to build a swimming pool. The pool will be surrounded by a

concrete sidewalk of uniform width. What could the width of the sidewalk be if organizers want the pool to be

1500 square feet?

12. The function f(t) = –5t2 + 20t + 60 models the approximate height of an object in feet t seconds after it is

launched. (Challenge)

a. How many seconds does it take the object to hit the ground?

b. How high off the ground will the object get?

c. How long after the launch will the object be at its maximum height?

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Using Regression to Calculate a Function Rule for Data (Linear, Exponential, and Quadratic)

For each of the following, type in the data into a graphing calculator. Study the data and decide which

regression is the most appropriate. Use the appropriate regression program on the calculator to find the

function rule for the data.

When the x-values increase by 1, there are some quick calculations that can be done to see whether the data is linear,

exponential, or quadratic. As x-values increase by 1, if the y-values have a common difference, the data is linear. As

the x-values increase by 1, if the y-values have a common ratio, the data is exponential. As the x-values increase by

1, if the y-values have a common second difference (the difference of the differences), the data is quadratic.

1.

x y

2 13

1 10

0 7

1 4

2 1

Type: ______________

Function rule:

__________________

2.

x y

2 20

1 5

0 0

1 5

2 20

Type: ______________

Function rule:

__________________

3.

x y

1 2

0 4

1 8

2 16

3 32

Type: ______________

Function rule:

__________________

4.

x y

2 3125

1 625

0 125

1 25

2 5

Type: ______________

Function rule:

__________________

5.

x y

2 12

1 3

0 0

1 3

2 12

Type: ______________

Function rule:

__________________

6.

x y

2 7

1 5

0 3

1 1

2 1

Type: ______________

Function rule:

__________________

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Unit 6 Practice Assessment

1. Name each polynomial based on its degree and number of terms.

7x3 – 1 ________ 2x2 _______ –x – 2 ______

9 _______ 7x8 + 5x3 – 4x – 3 ________________

2. Write each polynomial in standard form.

5x – 3 + 5 – 6x2 – 2x + 4x = __________ 3 – 11x – 9x = _________

2 – 3x3 2x3 + 3 – 12x + 8x = ______________

6x2 – 6x + 2x – x2 – 2x + x – 3 = _______________

Calculate the sum or difference. Write the answer in standard form.

3. (3x2 – 3x + 5) – (6x2 + 6x – 9) =

4. (2x4 + 5x3 + 5) + (3x4 – 7x3 – 5x2 – 6x) =

Simplify each product. Write your answer in standard form. Show your work.

5. 4x(3 – 7x) =

6. (2x + 7)(2x – 7) =

7. (5x + 2)(5x + 2) =

8. (4x + 3)(2x – 5) =

Fully factor each polynomial.

9. 4x2 – 9 =

10. 16x2 + 72x + 81 =

11. x2 – 2x – 15 =

12. 4x2 + 4x – 3 =

13. Calculate the zeroes of the quadratic functions. Use your work from #11 & #12. Graph the zeroes and list

their coordinates.

y = x2 – 2x – 15 y = 4x2 + 4x – 3

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14. A quadratic function has zeroes at (2, 0) and (4, 0). What is the function’s axis of symmetry?

A quadratic function has zeroes at (7, 0) and (–3, 0). What is the function’s axis of symmetry?

Sketch the function. Your sketch must include the vertex and two points on each side of it.

15. y = x2 + 4x 16. y = x2 – 4x + 3

17. What is the axis of symmetry for #15?

What is the axis of symmetry for # 16?

18. What is the vertex for #15?

What is the vertex for # 16?

Use the function rules for #19 & #20.

f(x) = x2 – 3x – 4 ; g(x) = 6x2 ; h(x) = –x2 + 2x + 5 ; j(x) = 5x2 – 2x + 1

19. Which of the functions {f(x), g(x), h(x), and j(x)} open up?

Which of the functions {f(x), g(x), h(x), and j(x)} open down?

Which of the functions {f(x), g(x), h(x), and j(x)} have a maximum vertex?

Which of the functions {f(x), g(x), h(x), and j(x)} have a minimum vertex?

Which of the functions {f(x), g(x), h(x), and j(x)} is the narrowest?

20. What are the y-intercepts for each of the functions?

f(x): _______ g(x): _______ h(x): _______ j(x): _________