November 13, 2013

Pre-Calculus 3rd period

Introduction

Using six parents functions learned in Pre-Calculus class, I was able to create a logo for

my favorite social organization. The Junior State of America and the Junior Statesmen

Foundation is the organization that I will be creating a logo for using functions. I chose this

organization because it connected to my social justice topic; teenage apathy. JSA is a national

organization that exists in over 500 schools. The Junior Statesmen Foundation is a non-profit

educational corporation which provides administrative and educational support for the Junior

State of America. “The mission of the Junior State of America and Junior Statesmen Foundation

is to strengthen American democracy by educating and preparing high school students for life-

long involvement and responsible leadership in a democratic society” (JSA.org). These two

foundations work together to create a place for students who are interested in politics a place to

go and express themselves. Junior State chapters are school-sponsored extracurricular clubs that

serve as a center of political awareness at school. Lively debates, stimulating discussions and

engaging activities let students express their opinions while learning about issues. The student

leaders of the chapter gain valuable skills as they build and run the club. “The JSA conventions

bring thousands of students together to share opinions and learn from each other. JSA holds

college level summer schools on campuses of Georgetown, Princeton and Stanford, students

attend rigorous advanced courses in government, politics, history and public speaking” (Jsa.org).

It also conducts institutes that offer advanced study in national issues and state and local

government. I support this organization because it is making students more aware and helping to

create a work of informed, engaged and responsible teenagers; fighting apathy. Students who

participate in JSA are committing to becoming informed teenagers in our society and community

by becoming aware of what is happening nationally and locally.

Functions

In math there are many terms that need to be mastered, for Pre-Calculus class one will

need to have an excellent grasp of functions. Before learning what a function is one needs to first

define a relation. A relation is two quantities that are related to each other by some rule of

correspondence. A relation can be displayed in many ways; ordered pairs, a table, an equation, a

sentence and mapping. Now that we know what a relation is we can now conquer a function. A

function is a special relation such that for all x values there exist one and only one y value. In

simpler terms if the x value repeats it is not a function. Every function is a relation but not ever

relation is a function. We use the Vertical line test to determine whether a relation is a function

or not. The Vertical line test states that if you draw a vertical line that touches the graph more

than once then the graph is not a function. One way that we classify functions is the one to one

function. In a one to one function every element of the range of the function corresponds to

exactly one element of the domain. We use the horizontal line test to test if a function is one to

one or not. In a function, the set of all input values (x values) for which the value the function is

defined is the domain. The Domain is also known as the independent variable. The Range is the

set of all possible output (y values) for which the function is defined or dependent variable.

When explaining the domain and range of functions we use a set of symbols.

{}−set brackets ˂ - less than

│- such that ˃ - greater than

≤ - less than or equal to ∈−element of

≥ - greater than or equal to R−all realnumbers

Now that we know the terms and their definitions let’s look at some examples:

Example 1: Relation as a Function

(2, 7), (15, 23), (4, 8), (11, 16)

This is an example of a Function because as you can see none of the x values repeat. Each x

value has one y value.

Example 2: Relation that is not a function

(3, 1), (7, 13), (7, 22), (5, 14)

This example is not a function because the x value 7 is distributed to two y values.

Example 3: Domain and Range

From example one and two the domain and ranges are as followed

Example one domains –{x∨x=2,15,4,11 }

Example two domains- {x∨x=3,7,5}

This is the domain of these relations because they are the x values of the ordered pairs.

Example one ranges- {x∨x=7,23,8,16}

Example two ranges- {x∨x=1,13,22,14 }

This is the ranges of these relations because they are the y values of the ordered pairs

Example 4: Graphs that is a Function

Figure 1

This graph is a function because it passes the vertical line test. If we were to take a ruler and

place it along this graph at any point the graph will only touch the ruler once.

Example 5: Graphs that are not Functions

Figure 2

This graph is not a function because it fails the vertical line test. If we were to take a ruler and

place it at any point of this graph, the graph will touch the ruler at multiple points. Let’s use the y

axis as our vertical line. The graph touches the y axis at 3 different points.

Example 6: Domain and Range of Graph one

The domain - { x∨x∈ R }

The domain of figure one reads x such that x is an element of the set of all real numbers. This is

the domain because the graph has no end points and will go on forever, so it will include all real

numbers.

The Range - { y∨ y>0 }

The range of figure one reads y such that y is greater than or equal to 0. This is the range because

the graph will never cross the x axis and have negative y values.

Example 7: Domain and Range of Graph two

The domain- { x∨−1≤ x ≤1 }\

The domain of figure two reads x such that x is greater than or equal to -1 and less than or equal

to 1. This is the domain is because the lowest x value is -1 and the highest is 1. So all the x

values on this graph would fall between -1 and 1.

The Range-{ y∨−1≤ y≤ 1 }

The range of figure two reads y such that y is greater than or equal to -1 and less than or equal to

1.

Example 8: Function as One to One

(1, 2), (8, 5), (9, 4)

This is a one to one because every y has only one x.

Example 9: Function as not One to One

(4, 5), (3, 5), (7, 6)

This is not a one to one because not every y has one x.

This is the end of our introduction to functions.

Parent Functions

A parent function is the simplest function with the defining characteristics of the family.

Functions in the same family are transformations of their parent function. The parent functions I

will be using to create my logo are linear function, parabola function, absolute value function,

cubic function, semi-circle function and sine and cosine functions. Parabola and absolute value

functions have

Functions

Linear Function:

The parent function

equation for the linear

function is f(x) = x.

Figure 3

The domain and range of this function is all real numbers because the graph is not restricted and

will continue on in both directions. The transformation equation for this parent function is f(x) =

mx+b, (m) is slope and changes the steepness of the line and (b) is the y intercept, the point

where the graph crosses the y axis. Two linear equations from my image are f(x)1 =1.29x+ 1.39

and f(x)2 =1.29x+9.5. In my first linear equation f(x) =1.29x+ 1.39 compared to the parent

function has a slope of 1.29 and y intercept of 1.95. The last linear equation has a slope of 1.29

and a y intercept of 9.5. The domain of my first linear function is domain {xǀ-4≤x≤0} and the

range is {yǀ3.77≤y≤1.39}/. The domain and range of my second linear function is {xǀ-5≤x≤-2}

and the range is {xǀ3.05≤x≤6.92}.The two linear equations in my image represent the arms in my

image I choose to use the linear function to create the arms because it would give me the best

shape of a slanted line. To be able to manipulate the slope of the line was important and vital to

give the arm a tilted look.

Linear Function 1 f(x)=1.29x+ 1.39 {-4≤x≤0}

Linear Function 2 f(x)= 1.29x+9.5 {-5≤x≤-2}

Figure 4

Let’s find out what makes Absolute Value functions unique.

Absolute Value Function

The parent function

equation for the absolute

value function is f(x) =│x│

Figure 5

The domain is all real

numbers because the graph is not restricted, and will go on forever making every point a possible

value. The range of this function is all numbers greater than 0 because the lines of the graph will

continue to go on forever upward, on the positive side of the y-axis. The operator of this parent

function is the absolute value sign, this is the special attribute in this graph that tells it apart from

others. The transformation equation is f (x) = a ǀx-hǀ +k, the (a) value in the equation determines

the direction and stretch factor of the graph. If the (a) is negative then it will point downward, if

it is positive it will point upward, the smaller the number the wider it will spread. The (h) and (k)

values are the respectively the x and y values of the vertex of the function. The (h) value shifts

the graph horizontally along the x axis and the (k) value shifts it horizontally along the y axis.

Two absolute value functions from my image are f(x)1= -2ǀx-25ǀ+22 and f(x)2= -0.68ǀx-25ǀ+20.

The first absolute value function has an (a) of -2, it has a stretch of 2 and is turned down, the

points of the vertex is (25, 22) it shifted 25 units to the right on the x axis and 22 units up the y

axis. The second absolute value function has an (a) of -0.68, its stretch is 0.68 and is turned

down, the points of the vertex is (25, 20), it is shifted 25 units to the right on the x axis and 20

units up the y axis. The domain of absolute value one is {xǀ24.5≤x≤ 25.5} and the range is

{yǀ21≤y≤22}. The domain of absolute value function two is {xǀ23.5≤x≤26.47} and the range is

{yǀ19≤y≤20}. These two absolute value function are very similar but they differ manly in the

stretch, the second function has a wider stretch, the closest the number is to 0 the wider the

stretch. These two absolute value functions help to make a star in my image. I choose to use

absolute value functions because of the sharp point at the vertex and straight legs that are perfect

for the sides of a star.

Absolute Value Function 1 f(x)={24.5 ≤ x ≤ 25.5:-2ǀx-25ǀ+22 }

Absolute Value Function 2 f(x)={23.5 ≤ x ≤ 26.47: -0.68ǀx-25ǀ+20. }

Absolute Value Function 3 f(x)={25 ≤ x ≤ 27:0.76 ǀx-25.5ǀ +10.3}

Absolute Value Function 4 f(x)={14.374 ≤ x ≤ 15.625:-0.8 ǀx-15ǀ +13.5}

Absolute Value Function 5 f(x)={23.07 ≤ x ≤ 23.93.51 ǀx-23.5ǀ +8}

Absolute Value Function 6 f(x)={-19.8 ≤ x ≤ -10.2:-2.1 ǀx+15ǀ +20}

Figure 6

Now it’s time to learn about Parabola functions.

Parabola Function

The parent function

equation for the

parabola function is

f(x)=x2

Figure 7

The domain for this function is all real numbers because the graph is not restricted and will go on

forever touching all the values on the x axis. The range of this function is all numbers greater

than 0 because the lines of the graph will continue to go on forever upward, on the positive side

of the y-axis. The operator of this parent function is the squared, it gives the graph its curve at

the vertex. The transformation equation is f (x) = a (x-h)2 +k, the (a) value in the equation

determines the direction and stretch factor of the graph. If the (a) is negative then it will point

downward, if it is positive it will point upward, the smaller the number the wider it will spread.

The (h) and (k) values are the respectively the x and y values of the vertex of the function. The

(h) value shifts the graph horizontally along the x axis and the (k) value shifts it horizontally

along the y axis. Two parabola functions in my image are f(x)1 = -0.9(x-32)2 +14.5 and f(x)2 = -

7(x-31)2 +18.79. The first parabola function has an (a) of -0.9, it has a stretch of 0.9 and is turned

down, the points of the vertex is (32, 14.5) it shifted 32 units to the right on the x axis and 14.5

units up the y axis. The second parabola function has an (a) of -7, its stretch is 7 and is turned

down, the points of the vertex is (31, 18.79), it is shifted 31 units to the right on the x axis and

18.79 units up the y axis. The domain of absolute value one is {xǀ31.3≤x≤ 32.7} and the range is

{yǀ14.059≤y≤14.5}. The first parabola has a very wide stretch while the other is narrow. The

domain of absolute value function two is {xǀ30.46≤x≤31.44} and the range is

{yǀ16.781≤y≤18.79}. The domain was determined by looking at my image at finding what part

of my parabola I wanted to be shown and then choose the two points I was going to restrict by

graph by to get that image. The first parabola represents the upper lip of lady liberty in my

image; the second parabola is part of a set of parabola that makes up the crown on the head of

lady liberty.

Parabola Function 1 f(x)= -0.9(x-32)2 +14.5 {xǀ31.3 ≤ x ≤ 32.7}

Parabola Function 2 f(x)= -7(x-31)2 +18.79 {xǀ30.46 ≤ x ≤ 31.44}

Parabola Function 3 f(x)= -6.5(x-32)2 +19 {xǀ31.51 ≤ x ≤ 32.49}

Parabola Function 4 f(x)= -10(x-30.26)2 +17.9 {xǀ29.89 ≤ x ≤ 30.5}

Parabola Function 5 f(x)= -7(x-33)2 +18.79 {xǀ32.564 ≤ x ≤ 33.549}

Parabola Function 6 f(x)= -7(x-33.8)2 +17.9 {xǀ33.548 ≤ x ≤ 34.23}

Parabola Function 7 f(x)= -0.8(x-31)2 +16.5 {xǀ32.3 ≤ x ≤ 33.3}

Parabola Function 8 f(x)= -0.8(x-32)2 +16.5 {xǀ30.73 ≤ x ≤ 31.76}

Parabola Function 9 f(x)= 2(x)2 +3 {xǀ-2≤ x ≤ 2}

Parabola Function 10 f(x)= -7(x-15)2 +17 {xǀ14.002 ≤ x ≤ 15.978}

Parabola Function 11 f(x)= 0.3(x-22.5)2 +16.16 {xǀ21.38 ≤ x ≤ 23.663}

Parabola Function 12 f(x)= 0.3(x-20.26)2 +15.76 {xǀ19.288 ≤ x ≤ 21.41: }

Parabola Function 13 f(x)= :0.3(x-20.26)2 +16.16 {xǀ19.306 ≤ x ≤ 21.41}

Parabola Function 14 f(x)= 0.6(x-22.5)2 +15.76 {xǀ21.38 ≤ x ≤ 23.663}

Figure 8

It’s cubic

functions time to shine.

Cubic Function

The parent function equation for the cubic function is f(x) = x3

Figure 9

Both domain and range for this function is all real numbers. The operator of this parent function

is the squared. The transformation equation is f (x) = a (x-h)3 +k, the (a) value in the equation

determines the direction and stretch factor of the graph. If the (a) is negative then it will point

downward, if it is positive it will point upward, the smaller the number the wider it will spread

from the axes. The (h) and (k) values are the respectively the x and y values of the inflection

point of the function. The inflection point is where the graph changed direction. The (h) value

shifts the graph horizontally along the x axis and the (k) value shifts it horizontally along the y

axis. Two absolute value functions in my image are f(x)1=2.6(x-31.5)3 + 17 and f(x)2= -2.6(x-

32.5)3 + 17. The first cubic function has a stretch of 2.6 and is positive the inflection point is at

coordinates 31.5,17. The second cubic function has a stretch of 2.6 and is negative and the

inflection point is 32.5, 17. The domain of the first cubic function graph {30.395≤x≤32} and the

range is {13.5≤x≤17.32}. The second cubic function has a domain of {32≤x≤33.6} and a range

of {13.5≤x≤17.32}. }. The domain was determined by looking at my image at finding what part

of my parabola I wanted to be shown and then choose the two points I was going to restrict by

graph by to get that image. These two cubic function graphs make up the face of lady liberty.

Cubic Function 1 f(x)= 2.6(x-31.5)3 + 17 {30.395 ≤ x ≤ 32}

Cubic Function 2 f(x)= -2.6(x-32.5)3 + 17 {32 ≤ x ≤ 33.6}

Cubic Function 3 f(x)= -2.3(x-20.5)3 + 17 {19.882 ≤ x ≤ 20.739}

Cubic Function 4 f(x)= 2.4(x-20)3 + 16 {20.58 ≤ x ≤ 20.87}

Cubic Function 5 f(x)= 0.15(x-39)3 + 9 {-40 ≤ x ≤ -34.81}

Cubic Function 6 f(x)= 100(x-14.6)3 + 16.2 {14.477 ≤ x ≤ 14.7442}

Cubic Function 7 f(x)= 100(x-15.2)3 + 16.2 {15.074 ≤ x ≤ 15.344}

Cubic Function 8 f(x)= 100(x-15.4)3 + 16.2 {15.274 ≤ x ≤ 15.4609}

Cubic Function 9 f(x)= 100(x-15)3 + 16.2 {14.874 ≤ x ≤ 15.144}

Cubic Function 10 f(x)= 100(x-4.8)3 + 16.2 {14.674 ≤ x ≤ 14.9442}

Cubic Function 11 f(x)= 0.2(x-0.2)3 + 9.11 {-2 ≤ x ≤ 0}

Cubic Function 12 f(x)= 0.3(x-1.6)3 + 2.77 {1 ≤ x ≤ 2.3}

Figure 10

We are moving on up to Semi-Circle functions.

Semi-Circle Function

The parent function for

the semi-circle function

is √1−(x) ²

Figure 11

The domain for the semi -circle function is numbers between -1 and 1 and the range is numbers

between 0 and 1. This operator for this function is the squared and the square root. This function

has a radius and a center. The transformation equation is f (x) = a√r−(x−h) ² + k, the (a) value

in the equation determines the direction and stretch factor of the graph. If the (a) is negative then

it will point downward, if it is positive it will point upward, the smaller the number the wider it

will spread from the axes. The (h) and (k) values are the respectively the x and y values of the

center of the function. The (h) value shifts the graph horizontally along the x -axis and the (k)

value shifts it horizontally along the y axis. Two semi-circle function graphs are 1.19

√7−(x−32) ² +16 and 1√5−(x−21.5) ² +16. The first semi-circle function graph has a radius

of 7, a stretch of 1.19 and the center point at 32 and 16. The semi-circle cubic function graph has

a radius 5, a stretch of 1 and the center point of 21.5 and 16. For semi-circle function graphs you

do not have to restrict the domain, since there is a radius and the graph has to end a certain about

of units from the center point. I used the semi-circle functions to make circles and some letters.

Semi -Circle Functions 1 1.19√7−(x−32) ² +16

Semi -Circle Functions 2 -1.19√7−(x−32)² +16

Semi -Circle Functions 3 1√6−(x−15) ² +15

Semi -Circle Functions 4 -1√6−(x−15) ² +15

Semi -Circle Functions 5 1√6−(x−25) ² +20.5

Semi -Circle Functions 6 -1√6−(x−25) ² +20.5

Semi -Circle Function 7 1√6−(x−25) ² +10

Semi -Circle Function 8 -1√6−(x−25) ² +10

Semi -Circle Function 9 1.8√0.2−(x−24.5)² +8.8

Semi -Circle Functions 10 -1.8√0.2−(x−24.5)² +8.8

Semi -Circle Function 11 -1√5−(x−21.5) ² +16

Semi -Circle Function 12 1√5−(x−21.5) ² +16

Semi -Circle Function 13 1√4−(x−32)² +15.5

Semi-Circle Function 14 -1√0.1−(x−31.5) ² +16

Semi-Circle Function 15 -1√0.1−(x−32.5) ² +16

Figure 12

Let’s jump on the train and

continue on to Sine and Cosine functions

Sine and Cosine Functions

The parent function equation for the sine and cosine function is sin x or cos x

Figure 13

The domain of the sine, cosine function is all real numbers and the range is numbers between 1

and -1. The special attribute in this function is that it has a (b). The transformation equation for

this parent function is a sin/cos b (x-h) +k, (a) is the amplitude or the distance the curve gets

from the line y=0; it changes the height and the (b) is the period or the distance it takes to

complete one cycle/ pattern; it changes the number of springs. The (h) is the horizontal shift and

the (k) is the vertical shift. Since the graph runs along the x axis the horizontal shift is not that

important because no matter what part of the graph you look at the sequence will be the same at

another part. The difference between sine and cosine is sin crosses y axis at 0 and cos crosses y

axis at 1. Two sine functions from my image are f(x)1 0.06sin (9.56(x-15)) +14 and f(x)2 0.34sin

(-7.38(x-10)) + 15.3. The first sine function has an amplitude of 0.06, a period of 9.56, a (h) of

15 and a (k) of 14. The second sine function has an amplitude of 0.34, a period of -7.38, a (h) of

10 and a (k) of 15.3. The domain of the first sine function is {14.089 ≤ x ≤ 15.899} and the

range is {14.34 ≤ x ≤ 14.46}. The domain of the second sine function is {-6≤x≤-4} and the range

is {14.96≤x≤15.64}.

Sine Function 1 {-6 ≤ x ≤ -4: 0.34sin (-7.38(x-10)) + 15.3}

Sine Function 2 {-6 ≤ x ≤ -4: 0.34sin (-7.38(x-10)) + 14.5}

Sine Function 3 {-6 ≤ x ≤ -4: 0.34sin (-7.38(x-10)) + 13.8}

Sine Function 4 {-6 ≤ x ≤ -4: 0.34sin (-7.38(x-10)) + 12.8}

Sine Function 5 {14.089 ≤ x ≤ 15.899:0.06sin (9.56(x-15)) +

14}

Sine Function 6 {14.14 ≤ x ≤ 15.833:0.34sin (9.56(x-15)) +

14.4}

Sine Function 7 {14.118 ≤ x ≤ 15.866: 0.34sin (9.56(x-10)) +

14.2}

Sine Function 8 {14.061≤ x ≤ 15.931: 0.34sin (9.56(x-10)) +

13.8}

Sine Function 9 {14.214≤ x ≤ 15.776: 0.34sin (9.56(x-10)) +

14.8}

Sine Function 10 {14.197≤ x ≤ 15.783: 0.34sin (9.56(x-10)) +

14.7}

Cosine Function 11 {31.5 ≤ x ≤ 32.5: -0.1cos (-6.7(x-32)) + 15}

Figure 14

Non-

Functions and Shading

Figure 15

Non Function Equation Range

Non Functions 1 f(y)= -0.9sin(5(y-6.34)) +2 {3 ≤ y ≤ 9.2}

Non Functions 2 f(y)=2.18 ǀy-14ǀ -2 {5 ≤ y ≤ 22.7}

Non Functions 3 f(y)=1(y-14)2 -9 {10 ≤ y ≤ 17.95}

Non Functions 4 f(y)= -3 {11.55 ≤ y ≤

16.45}

Non Functions 5 f(y)= 2 {10.79 ≤ y ≤ 17.3}

Non Functions 6 f(y)= 0.08sin(30(y-15))

+14.8

{15 ≤ y ≤ 16}

Non Functions 7 f(y)= 0.08sin(30(y-15))

+15

{15 ≤ y ≤ 16}

Non Functions 8 f(y)= 0.08sin(30(y-15))

+15.2

{15 ≤ y ≤ 16}

Non Functions 9 f(y)= 0.08sin(30(y-15.5))

+15.4

{15 ≤ y ≤ 16}

Non Functions 10 f(y)= 0.08sin(30(y-15))

+14.6

{15 ≤ y ≤ 16}

Non Functions 11 f(y)= -0.7 ǀy-20ǀ +24.2 {19 ≤ y ≤ 21}

Non Functions 12 f(y)= -0.7 ǀy-20ǀ +25.8 {19 ≤ y ≤ 21}

Non Functions 13 f(y)= 24 {10.5 ≤ y ≤ 11.5}

Non Functions 14 f(y)= 25 {10 ≤ y ≤ 11.5}

Non Functions 15 f(y)= 26.75 {10 ≤ y ≤ 11.5}

Non Functions 16 f(y)= 25.5 {8 ≤ y ≤ 9.5}

Non Functions 17 f(y)= 26.1 {8 ≤ y ≤ 9.5}

Non Functions 18 f(y)=20 {14.32 ≤ y ≤ 15.5}

Non Functions 19 f(y)=20.76 {13.89 ≤ y ≤ 15.5}

Non Functions 20 f(y)=21 {13.778 ≤ y ≤

15.76}

Non Functions 21 f(y)=23 {14.333 ≤ y ≤

15.5}

Non Functions 22 f(y)=22 {13.778 ≤ y ≤

15.76}

Non Functions 23 f(y)=26 {13.09 ≤ y ≤ 15.5}

Non Functions 24 f(y)=-0.4(y-18)2 -20 {14 ≤ y ≤ 22.4}

Non Functions 25 f(y)= 0.4(y-11)2 -20 {7.5 ≤ y ≤ 14.8}

These functions are not functions because they fail the vertical line test. If you were to take a

ruler the graph would hit the ruler at more than one points. The vertical line test if you draw a

vertical line that touches the graph more than once then the graph is not a function.

And the moment we have all been waiting for.

Image

Figure 16

Conclusion

This work relates to math because we used mathematic equations, functions and non-

functions to create a logo for my organization. Junior Statesmen of America makes high school

studens more aware and helping to create a work of informed, engaged and responsible

teenagers; fighting apathy. Students who participate in JSA are committing to becoming

informed teenagers in our society and community by becoming aware of what is happening

nationally and locally. My image will help my organization because it looks interesting and will

catch people’s attention, making them look further into the organization. I recommend that

every school has JSA chapter where students could learn to be more active and informed in the

world we live in, in a exciting and interactive way.