Lending A Hand When Times Are Tough:

Relations, Functions, Graphs

By: Jessica Jimenez

Viridiana Diaz

Iriana Barajas

Susana Ruiz

Table of Contents

Unit: Relations, Functions, Graphs Function Operations pg.2 Domain & Range, Function, and Relations pg.7 Using the Vertical Line Test pg.13 Evaluating Functions pg.18 Evaluating Compound Functions pg.22 Finding Domain Constraints pg.26 Transformations pg.31 Stretching, Shrinking, Reflecting pg.37 Family of Functions pg.41 Geometric Transformations pg.41 Writing Equations pg.47

Reference Sheet: Function Operations

State the term: Function operations

State the mathematical definition of the term.

Addition (f+g) (x)= f(x)+g(x)Multiplication (f*g) (x) = f(x)*g(x)Subtraction (f-g) (x) = f(x)-g(x)Division (f/g) (x) = f(x)/g(x), g(x) not equal to 0

Explain the meaning of the term in your own words.

When given a function just plug it to the operation to get a function operation

Show an example of the term and show something that looks like an example but is not. Explain what the difference between the two examples is.Domain Range Domain Range -1 -1 -2 -1 0 3 0 3 2 5 5 4 3 Function not a function

Difference: The difference is that the function of the domain is paired with exactly one element of the range, and the non function isnt because, the element -2 of the domain is paired with both -1 and 3 of the range.

How to Write a Skill Description

Function operations

This skill is good for finding successive discounts etc.

Examples (Use one or two challenging and relevant examples)

Suppose you are shopping in the store. You have a coupon worth $5 off any item.a. Use functions to model discounting an item by 20% and to model applying the coupon x=the original price . cost with 20% discount = f(x)= x-0.2x = 0.8x. cost with a coupon for $5 = g(x) = x-5

b. Model how much you would pay for an item if the clerk applies the discount first and then the coupon. (g*f) (x) = g (f(x)) = g(0.8x) = 0.8x-5 applying discount first

c. Coupon first and then the discount . (f*g) (x) = f(g(x)) = f(x-5) = 0.8(x-5) = 0.8x-4 applying coupon first

d. How much more is an item if the clerk applies the coupon first ? (f*g) (x) – (g*f) (x) = (0.8x-4)- (0.8x-5) = 1 any item will cost $1 more

Application (Explain how, if you were looking at a word problem, you would know that this skill could help you solve the problem)

If I was looking at a word problem, I would know that this skill could help me solve the problem by looking at what it’s asking me and that way I would know if it relates or not.

Description of the process (Describe the process in a list or paragraph form. Be sure to explain the purpose of each step)

First, look at the domain and the rangeSecond, match one element of the domain with one element of the range Third, determine weather it’s a function or not

Reference Sheet: Domain, Range, Function, and Relations State the term: Domain, range, function, and relations

State the mathematical definition of the term.

Domain – the set of all possible inputs of the functionRange – the set of all possible outputs of the functionFunction – a relation in which each element of the domain is paired with exactly one element in the rangeRelation – a set of pairs of input and output values

Explain the meaning of the term in your own words.

Show an example of the term and show something that looks like an example but is not. Explain what the difference between the two examples is.

Domain range domain range4 7 a 45 2 b 6 14 c 20Function neither

Difference:

The difference between these 2 examples is that one is a function and the other one isn't. Because the non function one doesn’t match completely.

Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea.

The domain and the range need to function together in order to be a function

How to Write a Skill DescriptionSkill Title Relation function or neither

Skill Description

The skill is determining if it is functions and if it doesn’t match then it’s not a function. Also remember that the domain does not repeat, if it does then it’s not a function.

Examples (Use one or two challenging and relevant examples)

Domain rangeA 74 7 11Function

Domain range X1 y1 x2 Y2 Relation

domain range A 4 B C 20 Neither

Application

When you have a function find the domain and the range then you will see if it’s a function or not

5. Description of the process First, find the domain and range.Second, match a value from the domain with a value from the rangeThird, if each value matches with exactly one pair then it is a function

There are more ways of determining if it is a function or not, besides mapping.

Reference Sheet: Using the Vertical Line Test

6. State the term:

Using the vertical line test7. State the mathematical definition of the term.

To determine weather it is a function or not

8. Explain the meaning of the term in your own words.

Determine whether or not a relation is a function

9. Show an example of the term and show something that looks like an example but is not. Explain what the difference between the two examples is.

Function not function

Difference: The difference between these 2 examples is that the vertical line in the non-function passes through at least 2 points on the graph, therefore its not a function

Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea.

Function

How to Write a Skill Description

Skill Title Determine if a graph is a function Skill Description Can be used on graph of a relation to tell whether the relation is a function if the vertical line passes through at least 2 points on the graph, then one element of the domain is paired with more than one element of the range and the relation is not a function.Examples

Application You could draw a vertical line through and find out if it would work or not. Description of the process First , look at the graph Second , draw thevertical line throughThird , count how many points it passes throughFourth, if it passes through more than two then it isn’t a function

You need to draw a vertical line to test if it’s a function.There could be cases where if you drew the line in one spot of the graph it would appear to be a function but everywhere else it doesn’t work.

Not function – crosses through at least 2 points

Reference Sheet: Relations, Functions & Graphs1. State the term: Evaluating FunctionsMathematical definition When evaluating a function, simply replace / substitute the input for “x” in the rule

Meaning in my personal words.When you are given a function like so: Find f(7) if f(x) = 5x2+56x+13, you will just plug in the 7 to every “x” in the function, so it would be , 5(7)2+56(7)+13=650. Example)Find f(3) if f(x) = 4x2-2x +5 Simplify replace “x” with the 3 in the rule and evaluate. F(3)=4(3) – 2(3) + 5 = 4(9) -6+5 =36 -6 + 5 =35

Find f (3x-5). If f(X) = 3x2-12x+10

Simplify replace “x” with (3x-5) in the rule and evaluate.

F (3x-5) =3(3x-5)2 – 12(3x-5)+10=27x-75-36x-70+10= -9x-145

Non-Example

Find f(5) if f(x)=5x-25-3ya.) Not possible

Difference: To evaluate a function you need to have a FULL function to solve. As, you can see in the non-example it simply askes of you to find f(5), that is not enough information to find your answer, because you will need y(x) as well.

Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea. Find f(5) if f(x)= 7x2-14x+28

5 5 5 5 ↓ ↓ → ↓ ↓ F(x)= 7x2-14x+28 f(x)= 7(5)2-14(5)+28

How to Write a Skill Description

6. Skill Title

Evaluating Functions

7. Skill Description

Shows you a different way to approach a function when eveluating it.

8. Examples

Given f(x)=2/(x-4) and g(x)=x2 -2,evaluate the following.

a.) -4g(2x) c.)g(-4)+f(5)-4(2x2-2) (-4)2-2+2/(5-4)=16-4(4x2-2) -4g(2x)=-16x2+8 9. Application

When you have a function and you wanted to find your varuiable you can evaluate it and find your answer.

10. Description of the process

To evaluate a function you have to do the following steps:• First check to see that you have a full function; Given f(x)=2/x-4 and g(x)=x2 -2• Then simply look at the evaluation that lays infront of you, and plug in the given function to whatever variable it corresponds to.*Remember if your denominator ends up being zero than the equation is considered “undefined.”• With your numbers set in place you can finally solve the equation and get your answer.

Reference Sheet: Relations, Functions, and Graphs State the term: Compound functions

Definition of the term.

A compound function is a function obtained from two given functions, where the range of one function is contained in the domain of the second function, by assigning to an element in the domain of the first function that element in the range of the second function whose inverse image is the image of the element.

Explain the meaning of the term in your own words.

A compound function is a function that uses a different rule depending on what the input value is.

Example:F(x)= 3x+4 if “x” > 05x if “x”= 0-2x-1 if “x” < 0

*Find f(0), f(-4) and f(2)

For f(0), x=0,therefore I would use the 2nd rule, f (0) = 5 (0) = 0For f(-4), x< 0, therefore I would use the 3rd rule, f(-4)= -2 (-4)-1=7For f(2), x > 0, therefore I would use the 1st rule, f(2)=3(2)+4=10

Non-Example:F(x)=6x+3 if “x” >104x+1 if “x” >20 2x if “x” >30

Find f(0), f (-2) and (4)

Difference: The differences is that unlike the real example the non-example cannot have a solution that pair up with the ”x.” The domain does not cover all numbers, nor do all the cases of “x” covered. They follow certain rules. Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea.

When evaluate a compound function, look at the input constraints to determine which rule to use.

6. Skill Title Evaluating Compound Functions

7.Skill Description Provides you with several rules to help you figure out the answer for your function.

8.Examples

If f(x) = -x+12 if x < 8 1/2(2x-2) if x≥ 8

FindF(-2)= -(-2)+12=14

F(10)= 1/2(20)-2=8

F(8)= ½(16)-2=6

Application

I could use this skill to find several functions at once by following rules tha t are given to me.

9.Description of the process

To evaluate a compound function do as follows: To evaluate a compound function, you must first figure out what ruole goes with yor function. After you have found the right rule to use for your function plug it in. Work it out. And you should get your answer.

Reference Sheet: Relations, Functions & Graphs Term: Domain and Range Constraints

The mathematical definition of the term.Domain constraints allow us to test the values inserted into the database and to test the queries to make sure comparisons made are appropriate. Range constraints a constraint are a rule that restricts the contents of a database relation. In relational algebra, every query result is in the form of a relation, so constraints may appear in the form of a statement involving one or more relational algebra expressions. Meaning of the term in your own words.

The domain are the x values. The range is the y values.

For example (1,2) (4,5) (7,8)... etc.

Domain = 1,4,7 etc. and range = 2,5,8

Example of the term and difference:

X2-5=0 X2=5 X=±√5 Therefore, the domain excludes +√5 and-√5 x<-1 No solution

Difference: If you have neither a radicand or rational expression then the domain has no constraint.

Another way that you can think about this term or something that will help you remember the concept. When you have functions like; X2=-144=0 X2=144 X=12

Hint: Whenever you see a function with this type of format; look for the values that make the denominator equal zero.Therefore, the answer to the example would be… the domain excludes +√5 and-√5. X+1< 0X<-1 Hint: Whereas. whenever you see a function with this type of format: look for the values that exclude all numbers less than you answser….so the answer for the example would be ….Domain excludes all numbers < -1.

How to Write a Skill Description

Skill Title

Domain and Range Constraints Skill Description This skill helps you figure out if your function has a constraint or not, and if it does helps you figure out what it is.

Examples

a.)F(x)= 3-x/5+x b.) x+2/√x2-75+x <= 0 x2-7>=0 X< -5 x2=7Therefore , domain excludes -5. X=±√7 Therefore, domain excludes +√7and - √7.

Application

When you are confused about a function and you have a number + or – a variable or a radicand on your denominator, then you probably have a constraint. With this skill you can find the constraint with no problem.

Description of the process The steps to follow to find the domains constraint is as follows: First, make sure you have a number + or – a variable or a radicand on your denominator. Second, if you have a number + or – a variable the first thing you will do is take the denominator and equal it to 0. Third , add or subtract the number over to the other side. And Walla you get your constraint! To find the constraint when you have a radicand you do as follows:o First take the function under the radicand and put it to < 0.o Second, subtract or add the number to the other side of the < sign.o Third find your constraint. o There, you’ve found your constraint.

Reference Sheet: Transformations State the term: Transformations

State the mathematical definition of the term.

A transformation is a general term for four specific ways to manipulate the shape of a point, a line, or shape. The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation. Explain the meaning of the term in your own words.

Basically what the term means, is a change in the original graph, which is y = [x]. By rotating, reflecting, translating, or dilating the change in the graph occurs.Show an example of the term and show something that looks like an example but is not. Explain what the difference between the two examples is.

Difference: In the first image you have just an image of a straight line, which is simply a function, nothing is being done to do to this line. On the other hand you have a reflection, a rotation, and a translation. These are basic examples of what happens in a tranformation. You have the point of origin and you change the location, without changing the figure.

Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea.

Translations (a translation is considered a 'direct isometry' because it not only maintains congruence, but it also, unlike reflections and rotations, preserves its orientation. • Turns - Rotates• Flips -Reflection• Dilations- Stretching

How to Write a Skill Description

Skill Title (A title gives you a short way to remember what the skill is)

Transformations

Skill Description: Use the point of origin, and rotate, reflect, dilate, and translate the image.

Examples (Use one or two challenging and relevant examples) Graph the following functions on the same graph. y=[x] y=-[x] y=-2[x

Application (Explain how, if you were looking at a word problem, you would know that this skill could help you solve the problem)

Reference Sheet: Transformations

State the term: Types Of Transformations

State the mathematical definition of the term. Stretching- extend the image(figure,equation)

When [a]> 1, stretch, along y-axis

Shrinking- the image(line,equation), becomes smaller

When 0 < [a]< 1, shrink along y-axis

Reflecting- the image(line,equation) mirrors over the x or y-axis.

When a < 0, reflect over x-axis

Explain the meaning of the term in your own words. 1. Multyplying a function by number > 1, stretches graph vertically. I t will be narrower. 2. Multyplying function by negative number reflects graph(makes the same image on the x-axis) over x-axis. 3. Multyplying by number smaller than 1, widens the graph. It shrinks the graph vertically

An example of the term and diff erence Example Parent Graph ---------------------------y=[x] Function Description F(x)=[x] Parent G(x)=-3[x] Reflecting over x-axis, shrinking H(x)=0.4[x] Stretching (wider) Non-Example Function y = x2 this represents a quadratic equation, which forms a parabola on the graph Diff erence: there are no straight lines on the grapg due to the fact that squaring a term, or the x, changes the whole graph.

Reference Sheet: Transformations

State the term: Types Of Transformations

State the mathematical definition of the term. Stretching- extend the image(figure,equation)

When [a]> 1, stretch, along y-axis

Shrinking- the image(line,equation), becomes smaller

When 0 < [a]< 1, shrink along y-axis

Reflecting- the image(line,equation) mirrors over the x or y-axis.

When a < 0, reflect over x-axis

Explain the meaning of the term in your own words. 1. Multyplying a function by number > 1, stretches graph vertically. I t will be narrower. 2. Multyplying function by negative number reflects graph(makes the same image on the x-axis) over x-axis. 3. Multyplying by number smaller than 1, widens the graph. It shrinks the graph vertically

An example of the term and diff erence Example Parent Graph ---------------------------y=[x] Function Description F(x)=[x] Parent G(x)=-3[x] Reflecting over x-axis, shrinking H(x)=0.4[x] Stretching (wider) Non-Example Function y = x2 this represents a quadratic equation, which forms a parabola on the graph Diff erence: there are no straight lines on the grapg due to the f act that squaring a term, or the x, changes the whole graph.

How to Write a Skill Description

Skill T itle Shrinking, stretching,reflecting

Skill Description

Describe how each graph is related to the parent graph.

Examples

y = [x]-1 1 translat ion (vertical) y = [x-1] 1 translation (horizontal) y = [x-1] +3 1 translation, 3 translation,(combined)

Application

You realize what kind of function you have. You decide what kind of transformation is occurring, by seeing what is inside of the absolute

Reference Sheet: Geometric Transformations State the term: Geometric Transformations

State the mathematical definition of the term.

It is a change made to a figure. The original figure is known as the parent graph.

Explain the meaning of the term in your own words.

This is when you have a distinct figure and and it changes its position like transition, rotation, ect.

Show an example of the term and show something that looks like an example but is not. Explain what the difference between the two examples is.

Difference: the difference between the two examples is the way they turn around on the grid. The first example it translates on the grid from one point to another. It moved 5 units up and 3 units down. For the second example it rotated the triangle 270 degrees which is different because instead of moving units it rotated in a specific degree.

Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea.

Another way that I will remember this concept is by the dividing squares on the normal grid and that it usually moves from one square to another or it will overlap with another.

(the point in the middle with the 4 divisions help me know that it might be talking about this )

How to Write a Skill DescriptionSkill Title The Geo that transforms Skill DescriptionThe skill is when it either moves around the grid with specific ways like rotation, transition, translation and dilation. This moves up or down the grid from the original plotting's of the points that were originally plotted. Examples EXAMPLE: Given triangle ABC where A (–2, 0), B (0, 4) and C (2, 1) Increase the size of the triangle by a factor of 1.5. Then, sketch the image.

EXAMPLE: Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2),. Reflect the triangle across the y-axis. Then, sketch the image.

Application: I would be able to use this in a word problem by knowing that the question would ask me what know what I would be wanting. Another way I would know that this is the skill I would be using this skill is by using if I needed to get new points on the plot and it tells me what the specific movement it did. Description of the process The first thing you do in this process is that you plot the points that are originally handed to you and that you have to plot them. Next you plot them then if it asks you to translate, rotate, or any other function you put that on the grid and you get the points that are the new points that I ploted. Than you get the points and than you put them as your answer if that’s what its asking you to do, if not you do otherwise.

Reference Sheet:Family of Functions State the term: Family of Functions

Mathematical definition of the term.Is a family of objects whse definitions depend on a set of perameters . Expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type. Meaning of the term in your own words.The meaning of this term is the specific way that the term is put out and the result in the graph and the possible way of the set of functions solved.An example of the term and the difference.

Example problem 1 Family of functions math Find the value of (f.g) (x) where f(x) = x 2 + 15 and f (g (x)) = f (2x+ 4). Solution: (f.g) (x) = f (g (x)) = f (2x+ 4) (f.g) (x) = (2x + 4) 2 + 15 = (4x 2 + 16 + 16x) + 15 (f.g) (x) = 4x 2 +16x +16. Example problem 2 Fami..{ fc(x) } = {x2 - c}. { fn(x) } = { n / (x - n) },.

Difference:

The difference of the two is that one is solving a problem and using the correct way of using it and the other is not for the fact that it is not right.

Show another way that you can think about this term or something that will help you remember the concept. You may want to show a picture, graph, diagram, or drawing that will help you remember the big idea.

Example problem 1 Family of functions math Find the value of (f .g) (x) where f (x) = x 2 + 15 and f (g (x)) = f (2x+ 4). Solution: (f .g) (x) = f (g (x)) = f (2x+ 4) (f .g) (x) = (2x + 4) 2 + 15 = (4x 2 + 16 + 16x) + 15 (f .g) (x) = 4x 2 +16x +16. Example problem 2 Fami.. { f c(x) } = {x2 - c}. { f n(x) } = { n / (x - n) },.

How to Write a Skill Description

Skill T itle The functions of diff erent units

Skill Description

This skill is good for the parabolas and be able to solve by graphing. This is also good for helping plot your graph.

Examples :

Application I would know by the way, how the problem is formed and what it is asking me to do. Or what the question or problem had there for me to work with. For example of the problem has the coordinates and its asking me to simply plot them then I will do just that but if its asking me to do otherwise than I will have to know what to use.

*Glossary*

Compound functions: A compound function is a function obtained from two given functions, where the range of one function is contained in the domain of the second function, by assigning to an element in the domain of the first function that element in the range of the second function whose inverse image is the image of the element. Domain Constraints: allow us to test the values inserted into the database and to test the queries to make sure comparisons made are appropriate. Domain: the set of all possible inputs of the function Evaluating Functions: When evaluating a function, simply replace / substitute the input for “x” in the rule Family of Functions: Is a family of objects whse definitions depend on a set of perameters . Expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type.

Function: a relation in which each element of the domain is paired with exactly one element in the range

Function operations: A ddition (f+g) (x)= f(x)+g(x) M ultiplication (f*g) (x) = f(x)*g(x) Subtraction (f -g) (x) = f(x)-g(x) D ivision (f/g) (x) = f(x)/g(x), g(x) not equal to 0 Geometric Transformations: It is a change made to a figure. The original figure is known as the parent graph. Range Constraints: a constraint is a rule that restricts the contents of a database relation. In relational algebra, every query result is in the form of a relation, so constraints may appear in the form of a statement involving one or more relational algebra expressions. Range: the set of all possible outputs of the function Relations: a set of pairs of input and output values Transformations: a general term for four specific ways to manipulate the shape of a point, a line, or shape. The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation. Types Of Transformations: Stretching- extend the image(figure,equation) When [a]> 1, stretch, along y-axis Shrinking- the image(line,equation), becomes smaller When 0 < [a]< 1, shrink along y-axis Reflecting- the image(line,equation) mirrors over the x or y-axis. When a < 0, reflect over x-axis Using the vertical line test: To determine whether it is a function or not