unit a student success sheet (sss)

Unit A Student Success Sheet (SSS) Introduction to Functions (sections 1.1-1.2)

Standards: Alg 1 5.0, Alg 1 7.0, Alg 1 8.0, Alg 1 16.0, Alg 1 18.0

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

Concept # What we will be learning… Mandatory Homework

Optional Extra practice from textbook

1 Finding the slope of a line given two points Worksheet 1 1 #7-10

2 Writing the equation of a line given slope and a point or two points Worksheet 1 Page 12 #25-42

3 Writing the equations of parallel and perpendicular lines given a line and a point Worksheet 1 Page 12 #45-46

4 Identifying functions while looking at ordered pairs or a graph (vertical line test) Worksheet 1 Page 14 #85, page 27 #78

5 Evaluating functions with numbers or variable expressions Worksheet 1 Page 12 #57-66

6 Writing linear models and evaluating for word problems Worksheet 2 Page 24 #5-10, page 25 #13-24

7 Writing and solving Cost, Profit, and Revenue word problems Worksheet 2 Page 25 #25-36; page 26 #43

8 Evaluating piecewise functions Worksheet 3 Page 25 #37-42; page 26 #44-46

9 Finding the domain of a function (rational, even/odd radical, polynomials) in interval notation

Worksheet 3 Page 26 #53-62

10 Evaluating the difference quotient (linear and quadratic) Worksheet 4 Page 29 #87-92

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 A Concept 1 – Worksheet 1

INTEGRITY Tim Werdel

Need Help? Support is available! Mr. Werdel: Monday afterschool 3 – 4 pm

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

Wednesday – Friday afterschool from 3-4pm

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

Edmodo.com. Math Analysis group code to join is

xd1wmo

Success comes from knowing that you did your best to

become the best that you are capable of becoming.

John Wooden

IN THIS UNIT…

We will be reviewing key concepts from Algebra 1 and Algebra 2 that are foundational to success in Math Analysis. More specifically, we will begin exploring

functions and different representations of functions, mostly linear in Unit A. Functions can be represented in four different ways: Verbally, Algebraically,

Numerically, and Graphically (remember this with VANG). Functions are represented verbally when we use word problems to model functions. In

this chapter, we will be looking at two linear models through word problems. Algebraically represented functions show up when we see equations, such as f(x)

or g(x). We will be evaluating functions (including regular one-piece functions and newer piece-wise functions) as well as finding and interpreting domain for

several key functions: polynomials, rationals, and radicals. Functions are represented numerically with a table or with ordered pairs. We will be defining what a

function is by looking at ordered pairs and their relationships. Lastly, functions can be represented graphically. We will be looking at graphs in this chapter, but

our graphical analysis of functions will begin its journey in Unit B, and continue for most of the first semester.

http://www.brainyquote.com/quotes/quotes/j/johnwooden386888.html

---Unit A Student Success Sheet---Introduction to Functions (sections 1.1-1.2) ---Math Analysis 2011-2012---

#1 Finding the slope of a line given two points

, given Points: (x1, y1) and (x2,y2) you can find the slope of the line by plugging into this formula!

Sometimes it is tricky to remember the equations of horizontal and vertical lines.

Here are some phrases to help you!

Horizontal - Y_____H! (this is because all of the ordered pairs on

the line have the same y-value!)

Vertical - X____ V_____ (this is because all of the ordered pairs on

the line have the same x-value!)

1.

2.

3.

4.


#2 Writing the equation of a line given slope and a point or two points

1 1y y m x x is called point-slope form of a line

m represents the slope

1 1,x y represents any point on the line

Begin all equations with point-slope form!!

y= # is a horizontal line (slope of 0) because… the y-value (height) stays the same for every ordered pair

x= # is a vertical line (undefined slope) because… the x-value stays the same the whole time for every ordered pair

Write the equation of the following lines with the given information. Leave in point-slope form

1. through: 1, 5 , slope 2 2. 3

through: 2,1 , slope2

3.

4.

Write the equation of the following lines with the given information. Write your final answer in slope-intercept form.

5. 6.

7.

8.


#3 Writing the equations of parallel and perpendicular lines given a line and a point

The opposite reciprocal of “undefined” is… 0!

The opposite reciprocal of “0” is… undefined!

Using that… solve these tricky ones:

1.

2.

3.

4.

Use Unit A Concept 2 to write the equations of the following lines:

5. 6.

7. 8.


#4 Identifying functions while looking at ordered pairs or a graph (vertical line test)

Many relationships exist in math. There is a very special relationship, however, that we focus on greatly. This relationship is called a FUNCTION relationship.

To be a FUNCTION, the following specifications must be met:

(1) When describing the relationship using ordered pairs, a. Every x-value can only be associated with one y-value b. It is okay if two different x-values are matched up with the same y-value.

(2) When showing the relationship on a graph, a. The graph must pass the vertical line test, meaning that if you drew a vertical line anywhere on the

graph, it would only touch the graph at most ONE time.

This relationship is a function. -2 matches up with 5 -1 matches up with 3 3 matches up with 7 4 matches up with 12

This relationship is NOT a function. 1 matches up with 3 2 matches up with 3 0 matches up with BOTH -2 and 7!!

This relationship is a function. Wherever I draw a vertical line, the most it would touch the graph is once.

This relationship is NOT a function. There are many places where if I draw a vertical line, it would touch the graph more than once.

Directions: Label the following as “FUNCTION” (F) OR “NOT A FUNCTION” (NAF)


#5 Evaluating functions with numbers or variable expressions

So, you have a function…like f(x) = 2x2 + 3x – 4

You are told to find f(3), so you must change the function to

f(3) = 2 2 + 3 - 4 Plug in “3” (in parentheses of course!) anywhere there is a box, and using PEMDAS, simplify!

2

3 2 3 3 3 4f Plug in “3” wherever you see “x”

3f = 2(9) + 3(3) – 4 Use the order of operations to simplify exponents

3f = 18 + 9 – 4 Use the order of operations to simplify multiplication

3f = 23 Add like terms together

So, you have a function…like f(x) = 2x2 + 7x + 4

You are told to find f(x+4), so you must change the function to

f(x+4) = 2 2 + 7 + 4 Plug in “x+4” (in parentheses of course!) anywhere there is a box, and using PEMDAS, simplify!

2

4 2 4 7 4 4f x x x Plug in (x+4) wherever you see “x”

=2(x2 + 8x + 16)+ 7(x + 4) + 4 Use FOIL with anything squared

=2x2 + 16x + 32 + 7x + 28 + 4 Distribute everything that is needed

4f x =2x2 + 23x + 64 Combine like terms

1. 2.

3. 4.

5. 3 22 3 1;Find g x x x x g 6.

7. 8. 22 4; Find f x x f x h f x


#6 Writing linear models and evaluating for word problems

From the problem, you can set up two sets of ordered pairs. (Then, use concept #2 to solve the problem) The x-value will always represent time The y-value will always represent an amount

1. Mary just self-published her first book and is selling it on amazon.com. In her first week, she sold 6 copies of the book. By the 5th

week,

she had sold a total of 46 copies of the book. Assuming her sales follow a linear model, (a) Write the linear equation to model her total

book sales; (b) calculate how many books Mary will have sold by the 4th

week; and (c) predict how many total books Mary will have sold

by the 52nd

week, one year after publishing, if this pattern continues.

translates into (week #, books sold) (1,6) and (5,46)

Solve for b to get b = -4

a) f(x) = 10x – 4

b) plug in 4 for x; 10(4) – 4 = 36 books by week 4

c) plug in 52 for x; 10(52) – 4 = 516 books by week 52

2. Roberto loves doing Math Analysis problems for fun. He keeps a secret notebook of the extra problems he has done and hopes to give it

to Mr. Werdel as a present at the end of the year. During the first week of school, he had done 11 extra problems. By the sixth week of

school, he had done a total of 111 extra problems. Assuming his completion of problems follows a linear model, (a) write the linear

equation to model the number of extra problems he has done; (b) calculate how many extra problems Roberto had completed by week 3;

and (c) predict how many total extra problems Roberto will have completed by week 18, the end of the first semester, assuming this

pattern continues.

( , ) and ( , )

3. Sally sells sea shells in a shack on the sea shore. During week 1, she sells 4 shells. During week 6, because her popularity had increased,

she sold 34 shells. Assuming her sales follow a linear model, (a) Write the linear equation to model her sales; (b) calculate how many

shells Sally sold during week 3; and (c) predict how many shells she will sell during week 21 if this pattern continues.

( , ) and ( , )

4. Juan collects starfish from the shores of Huntington Beach on the weekends. During week 1, he finds 3 starfish to start his collection.

During week 13, he finds enough starfish to make his total collection 87 starfish. Assuming his findings follow a linear model, (a) Write the

linear equation to model his starfish collection; (b) calculate how many starfish Juan had in his collection during week 7; and (c) predict

how many starfish he will have during week 32 if this pattern continues.

( , ) and ( , )


#7 Writing and solving Cost, Profit, and Revenue word problems

1. You are starting a picture frame

business. It costs you $3200 for monthly

equipment and rental fees, and $2.50 for

supplies for each picture frame. You sell

each picture frame for $11. Write your

(a) Cost Function; (b) Revenue Function;

(c) Profit Function; and (d) estimate the

number of picture frames you will have

to sell in order to break even (round up

to the nearest picture frame if necessary

and find the amount of profit if you do

round up).

Costs: Fixed - $3200 Variable - $2.50 C(x) = 3200 + 2.50x Revenue: $11 per frame R(x) = 11x Profit: =(11x) – (3200+2.50x) P(x) = 8.50x – 3200 Break-Even Point: 0 = 8.50x – 3200 +3200 +3200 3200 = 8.50x 8.50 8.50 X = 376.47 frames; BEP must be 377 frames at a very small profit of P(377) = 8.50(377 – 3200) = $4.50!

2. You are starting a pencil selling business. It costs you $1400 for monthly equipment and rental fees, and $.25 for

supplies for each pencil. You sell each pencil for $.75. Write your (a) Cost Function; (b) Revenue Function; (c) Profit

Function; and (d) estimate the number of pencils you will have to sell in order to break even (round up to the

nearest pencil if necessary).

3. You are starting a beach towel business. It costs you $1500 for monthly equipment and rental fees, and $3.25 for supplies for each beach towel. You sell each

beach towel for $14. Write your (a) Cost Function; (b) Revenue Function; (c) Profit Function; and (d) estimate the number of beach towels you will have to sell in

order to break even (round up to the nearest beach towel if necessary).

4. You are starting a used book business. It costs you $2200 for monthly equipment and rental fees, and $.50 for each used book. You sell each used book for $4.

Write your (a) Cost Function; (b) Revenue Function; (c) Profit Function; and (d) estimate the number of used book you will have to sell in order to break even

(round up to the nearest used book if necessary).


#8 Evaluating piecewise functions

A piecewise function is just that… a function broken into PIECES. The pieces are divided out by certain x-values that “work” for each piece of the function, given by inequality expressions.

When evaluating piecewise functions, first decide which piece it fits with. Then, plug it in to that piece and that piece only! (Remember, these are piecewise FUNCTIONS, meaning every x-value can only match up to one y-value! You can’t plug the x-value in to two

different places!)

A piecewise function can have anywhere from two to infinite number of pieces. In this unit, we will just be working with 2 and 3 pieces.

1.

x Top or

bottom? f(x)

-1

0

1

2

5.

x Top, middle, or bottom? f(x)

-3

-2

0

4

2. x

Top or

bottom? f(x)

-1

0

1

2

6.

x Top, middle,

or bottom? f(x)

-3

-2

1

2

3.

x Top or

bottom? f(x)

-1

0

1

2

7.

x Top, middle,

or bottom? f(x)

-4

-3

0

1

4. x

Top or

bottom? f(x)

-4

-3

-2

-1

8.

x Top, middle,

or bottom? f(x)

-4

-3

0

1


#9 Finding the domain of a function (rational, even/odd radical, polynomials) in interval notation

DOMAIN represents any x-values that “work” with the function, meaning you can plug it in and get a real answer.

If you get an imaginary answer (even root of a negative number) or an undefined answer (divide by zero), those x-values are restrictions on the domain.

We write our Domain in interval notation, which is a notation that shows all of the UNrestricted values (i.e. all the values that “work”)

Interval Notation

Consists of:

Parentheses ( ) Brackets [ ] “Union” symbols (U)

Used whenever the number is NOT included in the domain

Used whenever the number IS included in the domain

Used whenever the domain has multiple parts to it and you want to connect the answers together as one

Used when you would also use < or > symbols Used when you would also use < or > symbols

and are always noted with parentheses… remember INFINITY… PARENTHESES!

Write the following

statements in interval notation

1.

2.

3.

4.

5. 6.

7. 8.

9. 10.

11.

12.


Domain of functions OLYNOMIAL FUNCTIONS ATIONAL FUNCTIONS

In this chapter, we are looking at the domain of

Any function with whole-number powers; is not split into a numerator and denominator. Examples are

linear functions, quadratics, cubics, quartics.

Any function that has a numerator and denominator; both numerator and denominator themselves must be

polynomials.

The domain of a polynomial is ALL REAL NUMBERS, meaning that ANY x-value that is plugged in will give

you a real y-value as its pair

The domain of a rational function is restricted whenever the DENOMINATOR EQUAL ZERO (can’t

divide by zero!). So, set the denominator equal to zero and solve, to find those restricted values!

(-∞,∞) Restricted values: p=-1, p=4

(-∞-1)U(-1,4)U(4,∞)

VEN-ROOTED RADICAL

FUNCTIONS

DD-ROOTED RADICAL

FUNCTIONS

Any function with a radical whose index is an even number, such as a square root, fourth root, etc.

Any function with a radical whose index is an odd number, such as a cube root, fifth root, etc.

The domain of an even-rooted radical is restricted whenever the radicand (under the radical) is negative, because that would give you an imaginary answer. So, set the radicand (just the stuff under the radical!) > 0

and solve

The domain of a odd-rooted is ALL REAL NUMBERS, meaning that ANY x-value that is plugged in will give

you a real y-value as its pair.

When you solve, you get x > 4

*4,∞) (-∞,∞)

1. 2.

3. 4.

5. 6.

7. 8.


#10 Evaluating the difference quotient (linear and quadratic)

f x h f x

h

The difference quotient is an expression that is used for a fundamental skill in Calculus. We will be seeing it several times this year before we

utilize it in Unit U. Your goal for this chapter is to be able to plug into it and simplify correctly.

#1 Take your equation f(x+h) -f(x) Combine f(x+h)- f(x) Combine

#2 Take your equation f(x+h) -f(x) Combine f(x+h)- f(x) Combine


#3. Work Simplified

f(x+h)

-f(x)

Combine f(x+h)- f(x)

and Simplify, put it all over “h”

Factor out “h” from top if

needed

Not needed (monomial)

Cancel “h” on top with the

“h” on the bottom

#4. Work Simplified

f(x+h)

-f(x)




needed



#5. Work Simplified

f(x+h)

=

-f(x)

=




needed

Yes, because it’s not a monomial



=

#6. Work Simplified

f(x+h)

-f(x)




needed




#7. (*Bonus) Work Simplified

f(x+h)

-f(x)



*Because this is a RATIONAL function, we must do some more simplification before moving on to the next step!

Add fractions together by

finding an LCD

Distribute numerators

Combine like terms

--dividing by ‘h” is the same as

multiplying

by

=

Factor out “h”

from top if needed

Not needed; numerator is a monomial



#8. (*Bonus) Work Simplified

f(x+h)

-f(x)


and Simplify

Add fractions together by

finding an LCD

Distribute

numerators

Combine like

terms

--dividing by ‘h” is the same as

multiplying

by


needed



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