Maggie Cole Final Project

0005: Understand the principles and properties of patterns and

algebraic operations and relations

Determining Algebraic Expressions that best represent patterns among data presented in tables,

graphs and diagrams

Algebraic Expression: a mathematical expression that consists of variables, numbers and operations

such as addition, subtraction, multiplication and division

Types of Algebraic Expressions:

- Linear: An algebraic expression that has degree of 1

o Table: as the independent variable is changing

at a constant rate, the dependent variable is

also changing at a constant rate

o Graph: Straight line

o Diagram:

- Quadratic: An algebraic expression that has degree of 2

o Table: as independent variable is changing at a

constant rate, and the second differences of the

dependent variables are constant

o Graph: form a parabola, a U shape right side up

or upside down

o Diagram:

- Exponential: variable is in the exponent

o Table: as the independent variable is changing

at a constant rate, there is a common ratio

between the dependent variables

o Graph: rises or falls quickly in one direction,

level off on one side of the function

o Diagram:

More Information on this: https://www.ck12.org/algebra/identifying-function-models/lesson/Linear-

Exponential-and-Quadratic-Models-BSC-ALG/

x y

0 12

1 18

2 24

3 30

4 36

x y

0 0

1 1

2 4

3 9

4 16

+6

+6

+6

+6

+1

+1

+1

+1

+1

+1

+1

+1

+1

+3

+5

+7 V

V

V

+2

+2

+2

𝑦 = 6𝑥 + 12

𝑦 = 𝑥2

𝑦 = 2𝑥

V

x y

0 1

1 2

2 4

3 8

4 16

+1

+1

+1

+1

V

V

V

x2

x2

x2

x2

Number of squares in n steps 6𝑛 + 12

Number of circles in n steps 𝑛2

Number of squares in n steps 2𝑛

https://www.ck12.org/algebra/identifying-function-models/lesson/Linear-Exponential-and-Quadratic-Models-BSC-ALG/https://www.ck12.org/algebra/identifying-function-models/lesson/Linear-Exponential-and-Quadratic-Models-BSC-ALG/

Generalizing patterns using explicitly defined and recursively defined functions

Explicitly defined function – A function in which you can write the dependent variable in terms of the

independent variable. This type of function allows you to find any dependent variable given the

independent variable.

Recursively defined function – A function in which you need the value of a previous point of the

function. This type of function allows you to find the 𝑛𝑡ℎ value of a function as long as you know the

value of the (𝑛 − 1)𝑡ℎ term.

When looking at sequences of numbers and trying to find the functions that represent these sequences,

there may be arithmetic or geometric sequences.

- Arithmetic: A sequence in which each term is a fixed number larger than the term before it

- Geometric: A sequence of numbers in which the ratio between consecutive terms is constant

The following sequence is arithmetic, and the explicit and recursive functions are given

{-5, -3, -1, 1,…} Explicit: 𝑎𝑛 = −5 + 2(𝑛 − 1)

Recursive: 𝑎1 = −5

𝑎𝑛 = 𝑎𝑛−1 + 2

The following sequence is geometric, and the explicit and recursive functions are given

{2, 6, 18, 54,…} Explicit: 𝑎𝑛 =2

3(3)𝑛

Recursive: 𝑎1 = 2

𝑎𝑛 = 3(𝑎𝑛−1)

For more on explicit and recursive functions:

https://www.ck12.org/book/CK-12-Math-Analysis/section/7.1/

Performing and analyzing basic operations on numbers and algebraic expressions

Basic operations on numbers:

- Addition: (x+n) increasing a given number x, by another number, n

- Subtraction: (x-n) decreasing a given number x, by another number, n

- Multiplying: (𝑥 × 𝑛) adding the number n, x times

- Dividing: 𝑥

𝑛 splitting up x into n equal parts

- Exponents: 𝑥𝑛 n copies of x multiplied together

Basic Operations on algebraic expressions:

- Addition/subtraction: combining like terms

o (2𝑥 + 4) + (6𝑥 − 1) = 8𝑥 + 3

- Multiplication: distributive property

o 2(3𝑥 + 9) = 6𝑥 + 18

https://www.ck12.org/book/CK-12-Math-Analysis/section/7.1/

o 𝑥(𝑥 + 4) = 𝑥2 + 4𝑥

- Multiplication and Addition/Subtraction

o (2𝑥 + 3)(𝑥 − 7) = 2𝑥2 − 14𝑥 + 3𝑥 − 21 = 2𝑥2 − 11𝑥 − 21

Deriving an algebraic model that best represents a given situation and evaluating the strengths and

weaknesses of that model

Given data (a table of values, a scatterplot, a set of ordered pairs, or a mapping) it is sometimes useful

to find an algebraic model (or equation) that you can use to represent the given situation. You may want

to represent the data using a linear function, quadratic function or an exponential function.

One way you could go about doing this if you are given a table of values or a set of ordered pairs is to

input the given information into your calculator and create a scatterplot and imagine what kind of

function best fits the points.

Once you decide which type of relation you think will fit the data best, you can use your calculator to

find either a linear regression, a quadratic regression or an exponential regression. Then your calculator

will give you an estimated equation that you can use to model your data. It is important to note that this

regression model you are given is not exact and will not give you an exact value. This function will just

allow you to model data in an algebraic way and allow you to predict what would happen given the

data.

Regression: finding the best fit equation for a relationship of data

Linear Regression: If the scatter plot looks as if it is a straight line, you would guess that a linear

regression would fit this data the best. When you find a linear regression on your calculator, you will get

a linear correlation coefficient ‘r’ which tells you the strength of the linear relationship between the two

variables is. The closer the ‘r’ is to ±1 , the stronger the model is and the closer to 0, the weaker the

model is.

Quadratic Regression: If the scatter plots looks as if it is the shape a quadratic function, then you would

use the quadratic regression setting in your calculator

Exponential Regression: If the scatter plots looks as if it is the shape an exponential function, then you

would use the exponential regression setting in your calculator

Quadratic relation Linear relation Exponential relation

Coefficient of Determination, ‘𝑟2′ or ′𝑅2′:

- In a linear regression, this will be represented by ‘𝑟2′ while in a quadratic or exponential

regression, this will be represented by ′𝑅2′

- This is a measure of how well a regression equation represents the data

More on linear correlation coefficient and coefficient of determination:

https://mathbits.com/MathBits/TISection/Statistics2/correlation.htm

https://mathbitsnotebook.com/Algebra1/StatisticsReg/ST2FittingFunctions.html

Applying algebraic concepts of relation and function (e.g., range, domain, inverse) to analyze

mathematical relationships

Relation: the relationship between two sets of information

Function: a type of relation, but every point in the domain, goes to one and only one point in the range

Domain: All of the inputs of a function

Range: All of the outputs that the given inputs give

Inverse: Given an equation, y=f(x), the inverse is a rule that associates the x with the given y, or in other

words, when you solve for x in terms of y.

Ex. Equation: 𝑦 = 𝑥3

Inverse: 𝑥 = √𝑦3

Important facts about inverses:

- y becomes the independent variable and x becomes the dependent variable

- the domain of the inverse is the range of the original equation

- the range of the inverse is the domain of the original equation

- since the x and y variables change from one being independent to the other, you have to change

the labeling of your graph when you graph an inverse function

- When graphing inverse you can do this in two ways:

o Take each coordinate (x,y) from the original function, and change the coordinates to

(y,x)

o Reflect the function over the line y=x

- Not all inverses of functions are functions, need to careful for when you solve for x and get ‘𝑥 =

±’

o For example:

𝑦 = 𝑥2

𝑥 = ±√𝑦 , and this inverse is not a function because each x value can have

more than one output

More information on inverses:

https://www.purplemath.com/modules/invrsfcn3.htm

https://mathbits.com/MathBits/TISection/Statistics2/correlation.htmhttps://mathbitsnotebook.com/Algebra1/StatisticsReg/ST2FittingFunctions.htmlhttps://www.purplemath.com/modules/invrsfcn3.htm

Analyzing the results of transformations (e.g., translations, dilations, reflections, rotations) on the

graphs of functions

Given a function, f(x):

- Translations:

o f(x)+b

if b>1, shift function up b units

if b1, shift function to the left b units

if b1, compresses the function horizontally by b

when 0

Examples:

1. Use differences or ratios to tell whether the table of values represents a linear function, a

quadratic function or an exponential function. Then state the function

a)

2. Determine an explicit and recursive formula for the following

(http://www.visualpatterns.org/)

3. At the state fair, there is a game where you throw a ball at a pyramid of cans. If you knock over all of

the cans, you win a prize. The cost is 3 throws for $1, but if you have an armband you get 6 throws for

$1. The armband costs $10.

a) If Bob wants to throw the ball 15 times should he buy the armband?

b) If Bob plans on throwing the ball 65 times should he buy the armband?

c) After how many throws does it make sense for Bob to buy the armband?

4. The sales of a company (given in millions) is shown in the table below. Determine which type of

regression would best represent this data, and state the function and the coefficient of determination.

x y

-2 1/9

-1 1/3

0 1

1 3

x y

-2 -2

-1 1

0 4

1 7

2 10

x y

-2 11

-1 5

0 3

1 5

2 11

b) c)

x (Year) 2005 2006 2007 2008 2009 2010 2011 2012 2013

y (Sales) 12 19 29 37 45 55 62 68 76

http://www.visualpatterns.org/

5. The temperature (in degrees Fahrenheit) was measured at various altitudes (in thousands of feet)

above Los Angeles. The graph and table shown below represent the data. Decide which type of

regression would give you an equation that best fits this data and then give the equation.

6. One of your students claims that unless a function is one-to-one, it does not have an inverse. Is she

correct? Why or why not?

7. At Genesee High School, the sophomore class has 60 more students than the freshman class. The

junior class has 50 fewer students than twice the students in the freshman class. The senior class is three

times as large as the freshman class. If there are a total of 1,424 students at Genesee High School, how

many students are in each class?

8. Let 𝑓(𝑥) = |𝑥 − 3| for every real number 𝑥. Transformations of the graph of 𝑦 = 𝑓(𝑥) are described

below in b-d. After each description, write the equation, in vertex form, for the transformed graph.

Then, sketch the graph of the equation you write for part (e).

a. Graph 𝑦 = 𝑓(𝑥).

b. Translate the graph left 5 units and down 2 units.

c. Reflect the resulting graph from part (a) across the 𝑥-axis.

d. Scale the resulting graph from part (b) vertically by a scale factor of 1

2.

e. Translate the resulting graph from part (c) right 3 units and up 3 units. Graph the resulting

equation.

Altitude 0 0.5 3 4 7 9.5 13 20 28 35

Temperature 63 62 55 49 38 27 12 -13 -26.5 -62

Solutions:

1. Solution: Part (a): the independent variable (x-value) is changing at a constant rate of one, now if we

look at the dependent variables, we notice that the differences are not constant, but they are

following a pattern of being multiplied by 3 each time. Therefore this function represents an

exponential function and the function is 𝑓(𝑥) = 3𝑥. Part (b): the independent variable is changing at

a constant rate of +1 and the dependent variable is also changing at a constant rate of +3. Therefore

this is a linear function and since the rate of change is 3, and from the table we also know the y-

intercept is 4, the function is 𝑓(𝑥) = 3𝑥 + 4. Part (c): the independent variable is changing at a

constant rate of +1 and the dependent variables first differences are: -6, -2, +2, +6. So the second

differences are a constant +4 and therefore this function is quadratic and the function is 𝑓(𝑥) =

2(𝑥)2 + 3.

2. Explicit formula: 𝑓(𝑛) =(𝑛+1)(𝑛+2)

2

- To find this formula, I thought of each pattern as half of a rectangle, so use the first one as an

example:

Recursive formula: 𝑎1 = 3

𝑎𝑛 = 𝑎𝑛−1 + 𝑛 + 1, for 𝑛 ≥ 2

- To find this formula, you notice that the first term, 𝑎1 = 3 and then each of the following

patterns has (n+1) more blocks

3. Part (a) let the equation for cost with no armband be: 𝑛 =1

3𝑥, where x is the number of throws and

n is the cost, in dollars. Now let the equation for cost with an armband be: 𝑎 =1

6𝑥 + 10, where x is

the number of throws and a is the cost, in dollars. Now for part (a) we will need to evaluate these

equations at x=15. So we get 𝑛 =1

3(15) = 5, so with no armband he’d pay $5. When he does have

the armband, 𝑎 =1

6(15) + 10 = 12.5, so with an armband he’d pay $12.50. Hence for 15 throws,

he should not buy the armband. Part (b) Now we need to do the same thing but with x=65. For no

armband, we get 𝑛 =1

3(65) = 21.66667, so he would have to pay $21.67 for 65 throws without

the armband. With the armband, 𝑎 =1

6(65) + 10 = 20.833333, so with the armband, for 65

throws he’d pay $20.83. Therefore if he wants 65 throws, he should buy the armband. Part (c) for

this part I will set the two equations equal to each other and solve for the number of throws where

the cost will be the same so, 1

3𝑥 =

1

6𝑥 + 10,

1

6𝑥 = 10, and 𝑥 = 60. Therefore this tells us that for 60

throws, he would pay the same amount whether he had the armband or not. Therefore it would

make sense for him to get the armband if he is planning on getting 60 or more throws.

So if we consider this rectangle with the first 3 squares and

then I added in the dotted lined squares, the area of this entire

rectangle would be 2x3. Now if we think of this figure to be the

first element of the sequence, our n-value would be one, so we

can label the sides as (n+1) and (n+2). Therefore if we just want

to find the area of half of this rectangle (the purple part) we

would get (𝑛+1)(𝑛+2)

2

4. The following is a scatter plot that shows these data points:

a. This graph looks to be linear, so I will use a linear regression

b. Add this data into lists in my calculator, click on menu, click on

statistics, stat calculations, linear regression (mx+b)

𝑚 = 8.11666667

𝑏 = −16261.60555556

𝑟2 = 0.99729059

𝑟 = 0.998644377

c. 𝑦 = 8.1167𝑥 − 16261.6056

d. The coefficient of determination is, 𝑟2 = 0.9986 and therefore

this linear regression represents the data very strongly

5. When you first look at the given graph, it looks as though this scatter plot would be best

represented using a linear function. But when you plug the given data into your calculator, you will

find that the coefficient of determination for the linear regression is 𝑟2 = .990282820 and the

coefficient of determination for the quadratic regression is 𝑅2 = .991365280 and therefore

although a linear model would work fairly well, the quadratic regression is a better regression to

model the given data. So when I do a quadratic regression of this model, I obtained:

𝑎𝑥2 + 𝑏𝑥 + 𝑐

𝑎 = 0.01224691

𝑏 = −3.9056300

𝑐 = 63.95884619

𝑅2 = 0.991365280

𝑦 = 0.0122𝑥2 − 3.9056𝑥 + 63.9588

6. Solution: Yes your student is correct because if a function is one-to-one that means that every

output in the range has exactly one input that gives that output. Therefore when you find the

inverse, since the domain and range switch, every input would still go to one output so the inverse

would be a function. If a function is not one-to-one, then there would be at least one output in the

range that maps to more than one input. Then when you find the inverse of this function, since the

domain and range change, you will have an input map to more than one output, and therefore this

would not be a function.

7. Let f=# of students in the freshman class, s=# of students in the sophomore class, j=# of students in

the junior class, and x=# of students in the senior class.

From the problem, we can create the following equations:

𝑠 = 𝑓 + 60

𝑗 = 2𝑓 − 50

𝑥 = 3𝑓

𝑓 + 𝑠 + 𝑗 + 𝑥 = 1424

Now since we have s, j, and x in terms of f, we can plug our equations for those variables into the

last equation so that we have an equation with one variable.

𝑓 + (𝑓 + 60) + (2𝑓 − 50) + 3𝑓 = 1424

Combining like terms, we get:

7𝑓 + 10 = 1424

7𝑓 = 1414

𝑓 = 202

Now that we know f we can find the rest of the variables:

𝑠 = 202 + 60 = 262

𝑗 = 2(202) − 50 = 354

𝑥 = 3(202) = 606

Hence we now know, there are 202 students in the freshman class, 262 students in the sophomore

class, 354 students in the junior class and 606 students in the senior class.

8.

a)

b) 𝑓(𝑥 + 5) − 2 = |𝑥 + 5 − 3| − 2 = |𝑥 + 2| − 2

c) −𝑓(𝑥) = −(|𝑥 + 2| − 2) = −|𝑥 + 2| + 2

d) 𝑓 (1

2𝑥) = − |(

1

2𝑥) + 2| − 2

e)

𝑓(𝑥) = |𝑥 − 3|

𝑦 = − |(1

2) 𝑥 + 2| − 2

References:

Artzt, A.F., Sultan, A. (2018) The mathematics that every secondary school math teacher needs to know.

New York, NY: Routeledge.

All other websites that were used and can be used for more information can be found at the end of each

exemplar.