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Graphing a Quadratic Function

Video vocabulary

job

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US / dʒɒb /

UK / dʒɑ:b /

A1
  • noun
  • Something that must be done
  • Regular work at which you earn money

just

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US / dʒʌst /

UK / dʒʌst /

A1
  • adjecitve
  • Proper or appropriate; as is deserved
  • Doing or being what is right or fair
  • Legally correct; having enough evidence
  • adverb
  • Almost; nearly
  • Exactly at the time of
  • Exactly (the same as, like)

line

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US / laɪn /

UK / laɪn /

A1
  • noun
  • Border that divides two places or regions
  • Long crease in the skin, often from age
  • Mark that is long, straight and very thin
  • Row of words, letters or numbers that are written
  • Telephone connection
  • People waiting one behind the other; queue
  • Long thin wire, e.g. for electrical signal
  • verb
  • To put material along the inside of something

much

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US / mʌtʃ /

UK / mʌtʃ /

A1
  • adjecitve
  • Very; to a high level or degree
  • adverb
  • A lot; large amount; a high degree of
  • Nearly; basically
  • On many occasions; often
  • noun
  • A large amount; a great quantity

positive

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US / ˈpɒzətɪv /

UK / ˈpɑzɪtɪv /

A2
  • adjecitve
  • Showing agreement or support for something
  • Being sure about something; knowing the truth
  • Having the charge produced by electrons
  • Being good or useful
  • Showing that a chemical is present in a test

set

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US / set /

UK / sɛt /

A1
  • adjecitve
  • Prepared for something; ready
  • noun
  • Complete group of something
  • Place where a television show or movie is filmed
  • Group of games in a tennis match
  • Device that receives radio or television signals
  • verb
  • To make a clock state or ring at a particular time
  • To decide upon or choose something
  • To make something ready for use (e.g. table)
  • To decide on a price for something
  • To become hard (of glue, concrete etc.)
  • To be located in a specific place or time
  • To put (e.g. a vase) down carefully in a place
  • To put someone in a certain condition or state
  • (Of the sun) to go lower than the horizon

well

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US / wel /

UK / wɛl /

A1
  • adjecitve
  • Being in a fortunate, good condition or state
  • Healthy and in good condition
  • adverb
  • In an acceptable, proper, or successful manner
  • In a complete, total, or full manner
  • A great and considerable degree
  • In a successful way, in a good or satisfactory way
  • exclamation
  • Used to begin or continue a particular statement
  • Used to show you want to start saying something
  • Used to express that you are waiting for a reply
  • You say this to change the topic of a conversation
  • Used to indicate you are making a final remark
  • noun
  • Long deep hole in the ground to get water
  • verb
  • To cause your eyes to fill with tears

start

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US / stɑ:t /

UK / stɑrt /

A1
  • noun
  • First time or place that a thing exists; beginning
  • First opportunity to achieve something, e.g. a job
  • Sudden action or movement because you are scared
  • other
  • Beginning of something in place or time
  • verb
  • To do, be or happen for the first time; begin
  • To turn something on

yourself

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US / jɔ:ˈself /

UK / jʊrˈsɛlf, jɔr-, jor-, jə- /

A1
  • pronoun
  • Reflexive form of 'you', used for an emphasis

for

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US / fə(r) /

UK / fɔr,fə /

A1
  • preposition
  • Used to show the purpose, or need of something

time

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US / taɪm /

UK / taɪm /

A1
  • noun
  • Speed at which music is played; tempo
  • Point as shown on a clock, e.g. 3 p.m
  • Number of hours, minutes needed to do something
  • Occasion when something happens
  • Period or occasion that something occurred
  • Period in history or the past
  • Something measured in minutes, hours, days, etc.
  • How long an event takes; duration
  • verb
  • To check speed at which music is performed
  • To choose a specific moment to do something
  • To measure how long an event takes, e.g. a race
  • To schedule something to occur at a specific moment

why

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US / waɪ /

UK / hwaɪ, waɪ /

A1
  • adverb
  • Question word to ask for a reason
  • exclamation
  • (Used to express surprise or approval)

those

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US / ðəʊz /

UK / ðoz /

A1
  • determiner
  • The people or things we are talking about
  • pronoun
  • The objects far from the speaker

arbitrarily

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US / 'ɑ:bɪtrəlɪ /

UK / ˌɑrbəˈtrɛrəlɪ /

B1
  • adverb
  • Without using reason, often unfairly; on a whim

graph

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US / græf /

UK / ɡræf /

A2
  • noun
  • Image expressing numerical data in lines or shapes
  • verb
  • To show data using images

draw

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US / drɔ: /

UK / drɔ /

A1
  • noun
  • Something that attracts people to visit a place
  • A lottery or prize
  • Result of a game, contest where the score is equal
  • verb
  • To attract attention to someone or something
  • To get closer to or approach something or someone
  • To influence a person's involvement in something
  • To move an object by pulling
  • To create an image using pen or pencil and paper
  • To take one thing out of a container, etc.
  • (Of 2 teams) to finish a game with the same score

notice

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US / 'nəʊtɪs /

UK / ˈnotɪs /

A1
  • noun
  • Time by when you must leave, e.g. if you are fired
  • Paper with an announcement, e.g. pinned to a board
  • verb
  • To become aware by sight, touch, or hearing

could

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US / kəd /

UK / kʊd /

A1
  • other
  • Past form of 'can' to mean have ability
  • Used to ask for something politely, or to offer

know

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US / nəʊ /

UK / noʊ /

A1
  • verb
  • To be familiar with a person or place, thing
  • To feel sure or be convinced about something
  • To discover or be aware of something
  • To be able to distinguish
  • To have knowledge of things

with

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US / wɪð /

UK / wɪð, wɪθ /

A1
  • preposition
  • As an employee of a certain company
  • Against something or someone
  • By using something as a means
  • Due to; because of
  • Including
  • Possessing as a feature
  • Related to
  • (Of the manner/attitude by which you do something)
  • Used to express that people or things are together

have

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US / həv /

UK / hæv /

A1
  • verb
  • To drink, smoke, eat or use something
  • To experience the effects of something
  • To organize an event, or join in an activity
  • To own, possess, or hold something
  • To cause to happen or produce a particular effect

last

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US / lɑ:st /

UK / læst /

A1
  • adjecitve
  • Most recent; the latest
  • The one before
  • adverb
  • Final; with nothing following; after all others
  • Most recent; the latest
  • determiner
  • The one at the end
  • verb
  • To remain or continue for a certain period of time

fall

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US / fɔ:l /

UK / fɔl /

A1
  • noun
  • Season after summer and before winter; Autumn
  • A drop in amount; decrease
  • Dropping from a standing position to the ground
  • verb
  • To drop in amount; to decrease
  • To come down from a higher position suddenly
  • To go from standing to the ground, by accident

sample

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US / ˈsɑ:mpl /

UK / ˈsæmpl /

A2
  • noun
  • A small quantity of something used as a taster
  • verb
  • To try (e.g. food, drink), to see if you like it

negative

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US / 'neɡətɪv /

UK / ˈnɛɡətɪv /

A2
  • adjecitve
  • Being harmful, unwanted or unhelpful
  • In mathematics, being less than zero
  • Focusing on the bad aspects; pessimistic
  • noun
  • The opposite to a positive electrical charge
  • In grammar, containing words such as 'no' or 'not'
  • Reply to a question or statement that means 'no'
  • Image on camera film used to make a photo

relatively

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US / ˈrelətɪvli /

UK / ˈrɛlətɪvlɪ /

A2
  • adverb
  • In a manner comparing something to something else

see

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US / si: /

UK / si /

A1
  • verb
  • To go on a date with someone
  • To predict whether something is, or will be true
  • To understand what someone says
  • To use your eyes to look at something
  • To visit a place often for sightseeing or pleasure
  • To visit or meet with someone
  • To find out by waiting or looking
  • To watch (a game, movie or TV show)

scale

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US / skeɪl /

UK / skel /

A2
  • noun
  • Size, level, or amount when compared
  • Small hard plates that cover the body of fish
  • Device that is used to weigh a person or thing
  • Range of numbers from the lowest to the highest
  • Dimensions or size of something
  • verb
  • To change the size of but keep the proportions
  • To climb something large (e.g. a mountain)
  • To remove the scales of a fish

whether

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US / ˈweðə(r) /

UK / ˈhwɛðɚ, ˈwɛð- /

A1
  • conjunction
  • If something will happen or not

decent

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US / ˈdi:snt /

UK / ˈdisənt /

B1
  • adjecitve
  • Being fairly good; acceptable

can

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US / kæn /

UK / kən,kæn /

A1
  • noun
  • A metal container for storing food
  • other
  • Able to; have the skill to; have the time to
  • verb
  • To have permission to
  • To be possible to do
  • To preserve food by sealing it in a metal container

parabola

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US / pəˈræbələ /

UK / pəˈræbələ /

A1
  • noun
  • Plane curve formed by intersecting other planes

blue

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US / blu: /

UK / blu /

A1
  • adjecitve
  • Color of the clear sky
  • Feeling sad or down

pick

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US / pɪk /

UK / pɪk /

A1
  • noun
  • Sharp tool used for breaking the ground
  • Item selected from a group of choices
  • verb
  • To take fruit or vegetables from a plant or tree
  • To decide on a thing from various choices; select
  • To remove (flower, etc.) with your fingers

really

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US / ˈri:əli /

UK / ˈriəˌli, ˈrili /

A1
  • adverb
  • Used to stress something that is said; definitely
  • In a true or accurate manner
  • Very

would

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US / wʊd /

UK / wʊd /

A1
  • other
  • Used to indicate something imagined
  • Used to ask or request something politely
  • Verb used to indicate possibility, request etc.
  • Used to indicate a preference
  • Used to indicate a reaction
  • Indicates something that occurred repeatedly
  • Used to indicate willingness or ability to do

definition

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US / ˌdefɪˈnɪʃn /

UK / ˌdɛfəˈnɪʃən /

A2
  • noun
  • Explanation of word's meaning, as in dictionaries
  • The sharpness of an image on a screen
  • Visual clearness in a photo or on a screen

experience

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US / ɪk'spɪərɪəns /

UK / ɪkˈspɪriəns /

A1
  • noun
  • Thing a person has done or that happened to them
  • An event at which you learned something
  • Knowledge gained by living life, doing new things
  • verb
  • To gain knowledge by doing things

equal

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US / 'i:kwəl /

UK / ˈikwəl /

A1
  • adjecitve
  • Same in shape, size, or number
  • verb
  • To add up or be the same as

table

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US / ˈteɪbl /

UK / ˈtebəl /

A1
  • noun
  • Diagram that shows data in rows and columns
  • Raised flat surface you sit at to work or eat
  • verb
  • To leave an idea to be discussed at a later date

opening

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US / ˈəʊpnɪŋ /

UK / ˈoʊpnɪŋ /

A1
  • noun
  • Free job; job you can apply for
  • Action of starting something, e.g. a new business
  • verb
  • To make accessible or able to be used, e.g. road
  • To start the activities or services of a business
  • To make a space or hole appear in something
  • To move or change so that it is no longer closed
  • To start something, e.g. a new business

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

output

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US / 'aʊtpʊt /

UK / ˈaʊtˌpʊt /

A2
  • noun
  • Amount of something that is produced
  • verb
  • To produce data or information, as a computer

reasonable

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US / ˈri:znəbl /

UK / ˈrizənəbəl /

A2
  • adjecitve
  • Being somewhat good, but not great
  • Fair and sensible; appropriate

become

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US / bɪ'kʌm /

UK / bɪˈkʌm /

A1
  • verb
  • To begin to be; grow to be; develop into
  • To be suitable for a particular job or role

same

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US / seɪm /

UK / sem /

A1
  • adjecitve
  • Being equal in quality and amount
  • Identical; not different
  • Not changing; constant

but

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US / bət /

UK / bʌt,bət /

A1
  • conjunction
  • Used before you say something different, opposite

crazy

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US / 'kreɪzɪ /

UK / ˈkrezi /

A2
  • adjecitve
  • Mentally ill; mad
  • Not being sensible or practical
  • Really like someone or something very much
  • noun
  • Person acting in a manner that is strange and odd

dot

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US / dɒt /

UK / dɑt /

B1
  • noun
  • Small round mark
  • verb
  • To place small amounts/things in various places
  • To mark something by placing a small, round spot

want

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US / wɔnt /

UK / wɑnt, wɔnt /

A1
  • noun
  • Something you desire or wish for
  • State or condition of poverty
  • verb
  • To desire or wish for something; hope for a thing

any

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US / 'enɪ /

UK / ˈɛni /

A1
  • adjecitve
  • One, or some, no matter which
  • pronoun
  • One (thing) of many; some

think

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US / θɪŋk /

UK / θɪŋk /

A1
  • verb
  • To have an idea about something without certainty
  • To have an idea, opinion or belief about something

couple

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US / 'kʌpl /

UK / ˈkʌpəl /

A2
  • noun
  • Two people in a romantic relationship
  • Two of something; two people; a pair
  • verb
  • To join something to something else
  • (Two animals) to engage in sexual relations
  • To put two people or things together

finally

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US / ˈfaɪnəli /

UK / ˈfaɪnəli /

A1
  • adverb
  • used especially at the beginning of a sentence to introduce the last point or idea
  • After a long time or some difficulty

something

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US / ˈsʌmθɪŋ /

UK / ˈsʌmθɪŋ /

A1
  • adverb
  • In some degree or measure
  • noun
  • a person or thing of some value or consequence
  • pronoun
  • Thing that is not yet known or named
  • Approximately

and

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US / ənd /

UK / ənd, ən,ænd /

A1
  • conjunction
  • Used to refer to two or more things
  • Plus; in addition; on top of that
  • Used to introduce an action that follows another

another

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US / əˈnʌðə(r) /

UK / əˈnʌðɚ /

A1
  • adjecitve
  • One more, but not this
  • determiner
  • One more added
  • pronoun
  • One more (thing)

from

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US / frəm /

UK / frʌm, frɑm,frəm /

A1
  • preposition
  • Starting at a particular place, time or level

out

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US / aʊt /

UK / aʊt /

A1
  • adverb
  • No longer stylish (e.g. fashion)
  • Away from home, business, or the usual place
  • In a direction/movement away from inside or center
  • noun
  • Act when a batter cannot continue playing
  • verb
  • To tell people a secret

over

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US / 'əʊvə(r) /

UK / ˈovɚ /

A1
  • adverb
  • Finished; no longer happening
  • Towards the ground/away from a standing position
  • noun
  • A set of six consecutive balls bowled in cricket
  • other
  • Finished
  • preposition
  • Showing the relation when a thing is above another
  • On or covering something
  • Above or across something
  • More than you need
  • From one side to the other
  • In the direction of; toward

clean

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US / kli:n /

UK / klin /

A1
  • adjecitve
  • Not marked in any way; blank
  • Done according to the rules or standards; fair
  • Free from disease or sickness
  • Being free from dirt or marks; washed
  • Behaving well; not breaking any rules; legal
  • adverb
  • Completely; entirely
  • verb
  • To make free from dirt or marks; washed
  • To make neat and orderly e.g. a room
  • To remove the inner parts of an animal

call

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US / kɔ:l /

UK / kɔl /

A1
  • noun
  • A order or request for action
  • The sound an animal makes, often when in danger
  • A correct prediction, e.g. a decision in a game
  • A short visit to a place or person
  • A decision by a referee or umpire in a game
  • verb
  • To make a request or order for action
  • (Animal) to make sounds to warn of danger
  • To announce the date of an election
  • To make a judgment about something
  • To give someone or something a name
  • To guess in advance the result or outcome
  • To visit a place or person for a short time
  • To phone someone
  • To speak, announce or shout in a loud voice

integer

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US / ˈɪntɪdʒə(r) /

UK / ˈɪntɪdʒɚ /

B2
  • noun
  • Math a whole number

leave

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US / li:v /

UK / liv /

A1
  • noun
  • Permission to do something
  • Vacation time; time off work
  • verb
  • To go away from; depart
  • To gift property to someone after you die
  • To allow something to continue happening
  • To let something remain unused or available
  • To let something or someone remain somewhere

right

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US / raɪt /

UK / raɪt /

A1
  • adjecitve
  • Correct or true
  • The south side of you looking at the rising sun
  • adverb
  • Exactly in a place or time
  • Being appropriate for a particular event
  • In a straight or direct manner
  • noun
  • Something you are legally or morally allowed to do
  • Ideal of what is just and good
  • Political view based on conservatism
  • verb
  • To fix something previously wrong or unjust
  • To make upright, as a boat that has turned over

value

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US / 'vælju: /

UK / ˈvælju /

A1
  • noun
  • Importance you attach to something
  • Some information expressed as a number
  • Stated price of something; worth
  • verb
  • To place a high importance on something
  • To say how much money something is worth

first

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US / fɜ:st /

UK / fɚst /

A1
  • adjecitve
  • 1st; Coming before all others in time or place
  • noun
  • Something that has not happened before
  • Position of the winner of a competition

get

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US / ɡet /

UK / ɡɛt /

A1
  • verb
  • To become affected by illness or disease
  • To begin to feel or understand an emotion or idea
  • To send or bring someone or something home, etc.
  • To go somewhere to obtain something
  • To (cause to) do a particular thing
  • To obtain, receive or be given something
  • To currently have
  • To prepare for use
  • To understand something being said or read

case

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US / keɪs /

UK / kes /

A1
  • noun
  • Container used to carry things, e.g. clothes
  • Set of facts about a crime
  • Set of facts about an event
  • Example or instance of something
  • verb
  • To find out about a place, as in planning to steal

frankly

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US / ˈfræŋkli /

UK / ˈfræŋkli /

B2
  • adverb
  • Speaking honestly and directly

then

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US / ðen /

UK / ðɛn /

A1
  • adverb
  • A time that is not now
  • Following another thing in time or arrangement
  • Following on reasonably from what was just stated
  • At that time, not now

again

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US / ə'ɡen /

UK / əˈɡɛn /

A1
  • adverb
  • One more time, once more

even

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US / 'i:vn /

UK / ˈivən /

A1
  • adjecitve
  • Being equal in influence, power, possibility etc.
  • Being divided equally or fairly
  • (Of teams/contest) with similar chances of success
  • Smooth, level, or regular
  • Can be divided by 2; ending in 2, 4, 6, 8, or 0
  • At a regular, equally steady manner
  • Remaining steady and unchanging
  • adverb
  • (When this is surprising) as well; too
  • (Used to compare two things/stress the difference)
  • verb
  • To make something level; to flatten
  • To make something equal

like

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US / laɪk /

UK / laɪk /

A1
  • adjecitve
  • Being equal in status or kind to something else
  • Similar; the same
  • adverb
  • For example
  • preposition
  • Used as a filler; used to ad emphasis
  • In a way similar to the way something is done
  • Such as; for example
  • verb
  • To want to
  • To find something pleasing; to prefer something

what

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US / wɒt /

UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /

A1
  • pronoun
  • Thing or things that; the same kind as
  • Used to emphasize something you are about to say
  • adjecitve
  • Used to refer to a particular amount, or number
  • Used to place emphasis on a particular quality
  • adverb
  • To a certain degree
  • determiner
  • Question words used for asking for information
  • exclamation
  • Used to express excitement, shock, or surprise
  • Used to ask people to repeat something they said

sit

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US / sɪt /

UK / sɪt /

A1
  • verb
  • To take an exam
  • To be in a resting position on a chair
  • To provide seating for people
  • To remain in a still position while being painted

that

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US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
  • The person or thing being talked about
  • Used to identify something both the speakers know
  • adverb
  • Used to reinforce adjectives and adverbs
  • To such a degree
  • conjunction
  • Used to connect a noun clause
  • Used to introduce an adverbial clause
  • Used to introduce a noun clause
  • determiner
  • Used as a noun to refer to something
  • pronoun
  • Referring an object far away
  • Used to refer to the relative pronoun ‘which’

happen

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US / 'hæpən /

UK / ˈhæpən /

A1
  • verb
  • To take place or occur

each

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US / i:tʃ /

UK / itʃ /

A1
  • determiner
  • Every one of two or more things
  • pronoun
  • Per one person or thing

number

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US / ˈnʌmbə(r) /

UK / ˈnʌmbɚ /

A1
  • noun
  • Symbols such as 1, 2, 56, 793
  • Particular song or dance performed during a show
  • Total quantity or amount of things
  • verb
  • To put numbers on things
  • To assign a sequence within a group, series, set
  • To claim to be part of a larger group, series, set

square

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US / skweə(r) /

UK / skwɛr /

A2
  • adjecitve
  • Being fair or honest
  • Multiply a number by itself e.g. 2x2 or 6x6
  • noun
  • Shape with 4 equal length sides and 4 equal angles
  • The result of multiplying Length by the width
  • Open public area in the middle of the town
  • verb
  • To make something balance or be equal
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both

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US / bəuθ /

UK / boθ /

A1
  • determiner
  • Used to refer to two things at the same time
  • pronoun
  • Used to refer to two things at the same time

whereas

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US / ˌweərˈæz /

UK / hwɛrˈæz, wɛr- /

A2
  • conjunction
  • Taking into consideration the fact that

bit

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  • noun
  • Device put in a horse's mouth to control it
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  • verb
  • (E.g. of fish) to take bait and be caught

encourage

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US / ɪn'kʌrɪdʒ /

UK / ɛnˈkɚrɪdʒ, -ˈkʌr- /

A2
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  • To increase the chance of something happening
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nice

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  • Being attractive or of high quality or value
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look

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  • noun
  • Person's beauty or physical attractiveness
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  • To appear to be; seem

general

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A1
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  • Widespread, normal or usual
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  • Top ranked officer in the army

arbitrary

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UK / ˈɑrbɪˌtrɛri /

B1
  • adjecitve
  • (Of decisions) unsupported; without any evidence

comma

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B2
  • noun
  • Mark showing a pause or list in a sentence

work

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US / wɜ:k /

UK / wɚk /

A1
  • adjecitve
  • Activity completed in order to make money
  • noun
  • The product of some artistic or literary endeavor
  • Everything created by an author, artist, musician
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curve

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they

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  • pronoun
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take

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UK / tek /

A1
  • noun
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  • verb
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main

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  • adjecitve
  • Most important; most often used

paper

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  • adjecitve
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into

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one

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  • pronoun
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  • The thing or person previously referred to
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other

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than

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  • conjunction
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big

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  • adjecitve
  • Popular
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let

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little

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  • adjecitve
  • Being short in length of time
  • Not great in size, amount, or degree; small
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might

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the

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  • article
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how

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  • adjecitve
  • To what degree? in what amount?
  • adverb
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kind

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you

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this

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  • adjecitve
  • Used to indicate something already been discussed
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make

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  • noun
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  • verb
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verify

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B2
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half

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  • adjecitve
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straight

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  • adjecitve
  • Not having curves, bends, or angles
  • Not gay; heterosexual
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  • in a line; immediately; honestly and directly

input

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previous

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plus

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  • A benefit or advantage
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side

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  • adjecitve
  • In addition to the central or most important thing
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  • noun
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actually

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  • adverb
  • Used to add new (often different) information

use

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UK / ju:s /

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  • noun
  • Ability to employ something
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  • Act of spending money
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  • verb
  • To have; to eat, drink, consume etc.
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write

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  • verb
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point

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  • noun
  • An item to be discussed
  • Small spot or dot
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  • The size of the text on a page or screen
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  • Certain position in time or space
  • Idea or fact you try to convince people about
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here

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try

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  • noun
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exactly

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  • adverb
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once

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probably

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wait

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will

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  • noun
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  • Document saying who gets your money when you die
  • Strong desire to do something; determination
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  • other
  • Used with verbs to express the future
  • verb
  • To leave property to someone after your death
  • To use mental effort to make something happen
  • Am (is, are) likely to do
  • To influence someone to do something
  • To make something happen by strongly wanting it
  • Subtitles section
  • Whereas to graph the function, f of x is equal to negative 3x squared plus 8. So let's just try a couple of x values and see what values f of x takes on for each of those x values.
  • And then we can plot them and graph our function.
  • So let me draw a little table here.
  • So this will be x and let's call this f of x.
  • And we could set our y values to be equal to f of x.
  • So let's say that x is equal to negative 2. And I'm just picking these numbers arbitrarily.
  • You could have picked negative 3. You could have picked negative 1.5. I'm just picking numbers that are relatively close to 0 on both the negative and the positive side.
  • And numbers, I like using the integers because those will be relatively easy to work with.
  • Much better than if I was using something like negative pi or negative 1.3 or something crazy like that.
  • So these are just arbitrary points that I'm sampling from the domain.
  • So I can get the generalized shape of the, in this case it's probably going to be a curve.
  • The generalized shape of the curve and then I can connect the dots.
  • So when x is negative 2, what is f of x?
  • F of x is negative 3 times negative 2 squared plus 8. Which is equal to negative 2 squared is positive 4. Positive 4 times negative 3. Let me write this.
  • This is negative 3 times 4 plus 8. That's negative 12 plus 8 which is equal to negative 4. So the point negative 2, negative 4 is going to be on our graph.
  • Actually let me just start graphing it.
  • So let me draw my, I'll call this the y-axis.
  • And I'm just going to set our y values whatever the output of our function is.
  • So y is equal to f of x.
  • And then let me draw the x-axis.
  • So let me draw the x-axis like this.
  • So let me draw, actually I want to do it like this.
  • I'll draw the x-axis like that.
  • So there's my x-axis.
  • And I'm going to try the points x is equal to negative 2, x is equal to negative 1, x is equal to 0, x is equal to 1, and x is equal to 2. So this first point over here, 2, negative 4. And actually let me graph that.
  • So let's say that this right over here is negative 4. This would be positive 4. And this would be positive 8. So the first point is negative 2, negative 4. When x is negative 2, f of x.
  • When the input into our function is negative 2, the output is negative 4. Negative 2, negative 4 right over there.
  • That's the point negative 2, negative 4. Let's do another one.
  • What happens when x is equal to negative 1?
  • Then f of x is negative 3 times negative 1 squared.
  • Let me write it.
  • Negative 1 squared plus 8. Negative 1 squared is just 1. So it becomes negative 3 plus 8. This is going to be equal to 5. So we have the point negative 1, 5 on our graph.
  • Negative 1, 5 is going to sit.
  • So this would be 6 right over here.
  • So 5 is going to be right over there.
  • So we have negative 1, negative 1, 5 is going to be right over there.
  • Negative 1 comma 5. And then let's see what happens when x is equal to 0. When x is equal to 0, f of x is negative 3 times 0 squared plus 8. This part's just a 0, so it just equals 8. So we have the point 0,
  • 8. 0, 8 is right over there.
  • So that's 0 comma 8. And then let's see what happens when we go when x becomes positive 1. Then you have negative 3 times positive 1 squared plus 8. Positive 1 squared is the same thing as negative 1 squared.
  • So once again, you get negative 3 plus 8. It is equal to 5. So this is the point right over here.
  • It is 1 comma 5. And then finally, let's do one more point.
  • Let's see what happens when x is positive 2. When x is positive 2, f of x is negative 3 times positive 2 squared plus 8. This right over here is negative 3 times 4, which is negative 12 plus 8 is equal to negative 4. So when x is positive 2,
  • Y equals f of x is equal to negative 4. So this over here, and you might want to plot even more points to get the general shape of it, is going to be a downward opening curve, a downward opening parabola,
  • We could call it.
  • And it will look like this.
  • And I'll draw it as a dotted line just so that I can do a decent job of it.
  • It's going to look something like, let me draw it a little bit straighter.
  • So this curve, I think the main thing is I have my scale off a little bit.
  • So it's going to look something like that.
  • And so this point is right over here.
  • I'm going to have to make these points a little bit bigger so they fall out of my line.
  • But I want the curve to look somewhat reasonable.
  • That point is right over there.
  • And this last blue point is the point 2 comma negative 4. And if you're saying, wait, how do you know to curve it just like that using those points?
  • Frankly, that's just from previous experience knowing that this is a parabola.
  • If you really wanted to verify it for yourself, I really encourage you to take out some nice, clean graph paper and plot the points exactly.
  • And then you can even try points like negative 1 half and positive 1 half.
  • And you'll see that this curve shapes out like this.
  • And one thing you might notice is that the curve is kind of that the right side of this curve looks like the left side of this curve, that whatever the positive x value, we take on the same f of x value as the negative x value.
  • And that's because in our function definition, whatever our input is, we square it.
  • So whether we start with positive 2 or negative 2, when you square it, you get the same value.
  • Whether you start with positive 1 or negative 1, you square it, you get the same value.
  • And that's why f of positive 2 is equal to f of negative 2. And for really any other number, f of positive 1 is equal to f of negative 1.