5.2: Vertex Form

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# 5.2: vertex form - PowerPoint PPT Presentation

5.2: Vertex Form. How do you graph and apply quadratic functions? How do you locate a quadratic function that is shifted?. VERTEX FORM (RECALL). From a previous chapter, recall that the standard form of a parabola is y = x 2 . Any translation of that graph can be interpreted by:

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5.2: Vertex Form

How do you graph and apply quadratic functions?

How do you locate a quadratic function that is shifted?

VERTEX FORM (RECALL)
• From a previous chapter, recall that the standard form of a parabola is y = x2.
• Any translation of that graph can be interpreted by:
• Y = a(x – h)2 + k
• things inside parenthesis with x move graph left/right (backwards) h units
• Constant separate from the parenthesis will move graph up/down k units.
• #’s in front of the squared variable :
• Fractions make graph wider
• Whole numbers make graph narrower.
Refresh

Graphic organizer (vertex form)

Warm-up

Note Taking Guide

p. 63

Complete Ex 1 #1-3 and

Checkpoint 1

Did you know?
• Dartboards are made out of horsehairs.
• \'Crack\' gets it name because it crackles when you smoke it.
• Marijuana is Spanish for \'Mary Jane.\'
• One of the many Tarzans, Karmuela Searlel, was mauled to death on the set by a raging elephant.
• Slinkys were invented by an airplane mechanic; he was playing with engine parts and realized the possible secondary use of one of the springs.

U.S. Interstates which go north-south are numbered sequentially starting from the west with odd numbers, and Interstates which go east-west are numbered sequentially starting from the south with even numbers.

• Ballroom dancing is a major at Brigham Young University.
• Professional ballerinas use about twelve pairs of toe shoes per week.
• There are 336 dimples on a regulation golf ball.
• A walla-walla scene is one where extras pretend to be talking in the background -- when they say "walla-walla" it looks like they are actually talking.
• Cranberry Jell-o is the only jell-o flavor that comes from real fruit, not artificial flavoring.
• Don MacLean\'s song "American Pie" was written about Buddy Holly, The Big Bopper, and Ritchie Valens. All three were on the same plane that crashed.
• The Declaration of Independence was written on hemp paper
Homework
• Check homework
• Go over
Matching Activity
• Get into partners
• Using the cards provided, match the equations to their axis of symmetry
• Pass out Graphic Organizer #7
• Complete Vertex form on front
• Complete two problems on back also

Note Taking Guide

p. 64 – “Notes” – Example 2

You do:

Checkpoint 2, 3

Summary
• In your notes, summarize –
• What is the most important thing when graphing quadratic equation in vertex form?
Guided Practice

In books:

p. 65 Selected Problems (2 – 24) Even

Assignment/Homework
• Note Taking Guide
• P. 67-68
• #1-15, skip 4-6
Warm-up

Error Analysis: Look at the following problem. Determine if it is done correctly or incorrectly. Then graph the parabola. Find the vertex and axis of symmetry.

y = (x – 2)2 + 4

p = 2, q = -4

Vertex = (2, -4)

Interesting Facts
• In ancient Rome, it was considered a sign of leadership to be born with a crooked nose
• The word "checkmate" in chess comes from the Persian phrase "Shah-Mat," which means the king is dead
• If you yelled for 8 years, 7 months and 6 days, you would have produced enough sound energy to heat one cup of coffee
• A common drink for Tibetans is Butter Tea which is made out of butter, salt, and brick tea

Bourbon was first made by a Baptist minister from Bourbon County in Kentucky in 1789. That is where it got its name

• The Pentagon, in Arlington, Virginia, has twice as many bathrooms as is necessary. When it was built in the 1940s, the state of Virginia still had segregation laws requiring separate toilet facilities for blacks and whites
• McDonald restaurants serve food and drink to an amazing 43 million customers on a daily basis
• The game Monopoly was once very popular in Cuba; however, Fidel Castro ordered that all games be destroyed
• Nearly half of all Americans suffer from symptoms of burnout

Test over standard and vertex form on Friday (8/21)

(Section 3.1-3.2 in books)

HW
• HW check - Verbal:
• Vertex
• Axis of Symmetry
• 2nd part: x-intercepts
Today’s LEQ:

How do you analyze and graph quadratic functions of the forms f(x) = ax2 + bx + c ?

Graphic Organizers:

Finish 2nd problem on front of GO #7

New GO – Extra Examples

Note Taking Guide (N.T.G. ) p. 57-58

Minimum or Maximum

*Write answers to these problems in “notes” section in N.T.G. on p. 58 (reminders)

• If a graph opens up, it has a _________
• If a graph opens down, it has a _________
• The min/max in a parabola always occurs at the _______
Quick Test

Do these graphs have a minimum or a maximum?

Y = x2 + 1

Y = -x2 -4

Y = x2 _ 2x

Y = x2 + 2x + 1

Y = 3x2 – 6x + 4

Y = -2x2 + 5x - 3

Completing a Table of Values

Ex like in homework

“Copy and complete the table of values for the function”

Y = -3x2

Summary

What is the first thing you need to remember to graph a quadratic in standard form?

(what is the main thing that is different from vertex form?)

Warm-up

In N.T.G. (Note Taking Guide), complete p. 58

Checkpoints # 1-3

Did you know?
• There are more beetles than any other kind of creature in the world.
• Velcro was invented by a Swiss guy who was inspired by the way burrs attached to clothing.
• Polar bears\' fur is not white, it\'s clear. Polar bear skin is actually black. Their hair is hollow and acts like fiber optics, directing sunlight to warm their skin.
• Polar bears camouflage themselves more completely during a hunt by covering their black noses with their paws.
• The amount of tropical rainforest cut down each year is an area the size of Tennessee.

The face of a penny can hold about thirty drops of water.

• Medieval knights put sharkskin on their sword handles to give them a more secure grip; they would dig the sharp scales into their palms.
• The only planet without a ring is earth.
• Cleopatra used pomegranate seeds for lipstick.
• Cleopatra\'s last name was Ptolemy, and she was Greek rather than Egyptian.
• The raised reflective dots in the middle of highways are called Botts dots.
• The Amazon rainforest produces half the world\'s oxygen supply.
• Reindeer like to eat bananas.
• Chia Pets are only sold in December.
Today: Fun with Calculators
• Make sure you have a calculator.
• You might want to write in your notes how/where on the calculator you find the buttons to solve the problems (so you can look back at it later)
• I have 6 Examples that I am going to do with you (write them in your notes)…
Graphing on the calculatorComplete Examples in Notes

Graph the function.

Label: domain, range, intervals of increase, intervals of decrease, zeros, vertex, axis of symmetry.

Y = 2x2 + 3

Graphing on the calculatorComplete Examples in Notes

Graph the function.

Label: domain, range, intervals of increase, intervals of decrease, zeros, vertex, axis of symmetry.

Y + 1 – 2x= -x2

Graphing on the calculatorComplete Examples in Notes

Graph the function.

Label: domain, range, intervals of increase, intervals of decrease, zeros, vertex, axis of symmetry.

Y = ½ x2 - 2

Graphing on the calculatorComplete Examples in Notes

Graph the function.

Label: domain, range, intervals of increase, intervals of decrease, zeros, vertex, axis of symmetry.

Y = -5x2 + 4x + 2

Graphing on the calculatorComplete Examples in Notes

Graph the function.

Label: domain, range, intervals of increase, intervals of decrease, zeros, vertex, axis of symmetry.

Y =-2x2 – 4x + 3

Graphing on the calculatorComplete Examples in Notes

Graph the function.

Label: domain, range, intervals of increase, intervals of decrease, zeros, vertex, axis of symmetry.

Complete Ex3 & Checkpoint 4

(p.65 in N.T.G)

Summary – Turn in!

Using GO #7 and/or graphing calculator, graph the functions.

Label: domain, range, intervals of increase, intervals of decrease, zeros, vertex, axis of symmetry.

1) Y + 3 = 2x2 + 4x

2) Y = 2(x – 2)(x – 3)

*Show graphs – used the calculator? Draw a rough “sketch” on your paper.

Warm-up
• Textbooks
• P.59
• #36-39 (all one word problem)
• You should be able to do 36 – 37 without even reading the problem
• #38-39 you have to determine what the variables mean by reading the problem
Interesting Facts

The Saguaro Cactus, found in South-western United States does not grow branches until it is 75 years old.

In Belgium, 172,000 tons of chocolate are produced in a year

The word Nike comes from Greek Mythology. Nike is the goddess of victory and was often depicted as a small winged figure whom the goddess Athene carried

The biggest bug in the world is the Goliath Beetle which can weigh up to 3.5 ounces and be 4.5 inches long

Leaving the water running while brushing your teeth can waste four gallons of water in a minute

The word Spain means "the land of rabbits."

In 1936, the first practical helicopter was invented. It was the German Focke-Wulf Fw 61.

The word tulip comes from the Turkish word for turban

Psychokinesis refers to the ability of moving objects through psychic power

In Britain, one out of every four potatoes is eaten in the form of french fries

In the movie Psycho by Alfred Hitchcock, chocolate syrup was used for blood in the shower scene

• No candy today – trash on the floor yesterday
• You will have a quiz tomorrow over this domain, range, important points, etc.
• A lot of people in this class are losing points… remember, you must have 1 point remaining in order to drop a low test grade….
• Some of you are going to have to earn points back…or hope you don’t need to drop any bad tests!
Practice (partial study guide)
• Count off #1 – 8
• Split up into 8 groups
• Your group will be assigned 1 problem
• Each group should allect one specialist to show the rest of the class how to do the problem.
• Everyone should turn in all either problems, completed.
• Try to do these by hand! (Use the calc to check only)
Warm-up

Gather things to turn in:

• Two Practice Problems (half sheet of paper) (May have already turned in)
• TOTD from Wednesday (two problems copied from slide show-HW)
• 8 Problems from yesterday
• Warm-ups since Monday 8/10
• Staple them in this order with your name on top page.
Did you know?

During WWII, because a lot of players were called to duty, the Pittsburgh Steelers and Philadelphia Eagles combined to become The Steagles

The titan arum flower is the largest flower in the world and gives off a horrible odor that smells like rotting flesh when it blooms

Every day, the average person swallows about a quart of snot

When the only queen ant dies, so does the entire colony, because no new workers are born

The average American kid will eat approximately 1.500 peanut butter sandwiches by high school graduation

Goats do not have upper front teeth

It has been suggested that shepherds are responsible for inventing the game golf. It is said that they used to use their staffs to hit the stones

There are about 6,800 languages in the world

Studies have shown that by putting on slow background music it can make a person eat food at a slower rate

Today’s Lesson…
• You probably already know how to do these problems because we’ve been doing them “along”
• Today, the topic is about changing quadratic equations form one form to the other
• Key Questions:
• How do you find the vertex form of a function written in standard form.
• How do you transfer functions from vertex to standard form?
Vertex vs. Standard Form

KQ1: How do you find the vertex form of a function written in standard form?

KQ2: How do you transfer functions from vertex to standard form?

Which is better, standard or vertex form?

Standard Form Vertex Form

Y = -3x2 – 12x – 8 y = -3(x + 2)2 + 4

Graphic Organizers…

Steps: Standard Vertex
• Find the vertex
• It’s in standard form, so use –b/2a to find x, then plug that in to find y.
• So the vertex is (h,k)
• Now write the general vertex form.
• Y = a(x – h)2 + k
• The “a” in front of the vertex form is the same as it was in the standard form equation.
Ex1.

Write the function y = 2x2 + 10x + 7 in vertex form.

x-coordinate: -b/2a -10/2(2) = -10/4

=-5/2 or - 2.5

y-coordinate: 2(-2.5)2 + 10(-2.5) + 7

-5.5

Substitute the vertex point (-2.5,-5.5) into the vertex form y = a(x – h)2 + k a from above

y = 2(x + 2.5)2 – 5.5

Ex2.

Write y = -3x2 +12x + 5 in vertex form.

x-intercept: -b/2a = -12/2(-3) = -12/-6 = 2

y-intercept: y = -3(2)2 + 12(2) + 5 = 17

Re-write in vertex form:

y = -3(x – 2)2 + 17

From vertex to standard form:

To change an equation from vertex to standard form, you have to multiply out the function.

y = 3(x -1)2 + 12

y = 3(x – 1)(x – 1) + 12

y = 3(x2 – 2x + 1) + 12

y = 3x2 – 6x + 3 + 12

y = 3x2 – 6x + 15

So, what do you think?
• Which is better, standard or vertex form?
• How do you change equations from one form to another?
• Vertex to standard
• Standard to vertex
Quiz
• Again, this quiz is very similar to your test format. You may use a calculator or work through these problems by hand.
• You may use your notes!
• If you are talking during the quiz, you will receive an automatic zero…no warnings.
• If you have a question, raise your hand!
Assignment/Homework

Due Monday…

N.T.G.

p. 60-62 p. 67-69

#1-24 #1-29

Some have already been completed; complete the rest and be prepared to turn in on Monday! (I will have you take out those pages)

Warm-up
• For the function

f(x) = (x + 5)2,

as x  ∞ f(x)  ?

• For the function

f(x) = x2 -2x + 1,

what is the range?

Did you know?
• A group of people that are hired to clap at a performance are called a claque
• In 1926, a waiter in Budapest committed suicide. He left his suicide note in the form of a crossword and the police had to get help from the public to solve it
• In an year, an average American kid eats 46 slices of pizza
• Bubble gum contains rubber
• When Kleenex was first introduced to the market in 1924, it was marketed as a make up or cold cream remover

In the year 1900, for a women to be a telephone operator she had to be between the ages of 17 and 26 and not be married

• Babies that are exposed to cats and dogs in their first year of life have a lower chance of developing allergies when they grow older
• The accent that Mike Myers used for the character Shrek came from the accent that his mother would use when she was telling him bedtime stories when he was a child
• In the Netherlands, there are special traffic lanes for bicycles. There are approximately 17,000 kms of cycle lanes with special bicycle traffic lights.
• It was believed by Ancient Hindus that the world was a sphere and rested on the back of four elephants, which stood on a turtle
NEW RULES
• SEATING CHARTS!
• NEW PASS rule
• 5th Block today!
• If you talked during quiz Friday, you have a zero, come during 5th block to re-take the quiz
Assignment/Homework

Due Monday…after Test tomorrow!

(so you can use these notes to study!)

N.T.G.

p. 60-62 p. 67-69

#1-24 #1-29

Some have already been completed; complete the rest and be prepared to turn in!

(I will have you take out those pages)

Return Papers

• Go over quizzes
• Folders/Re-collect Quizzes
• Study Guide

Quiz 1.1 (compare all to parent graph y = x2)

• 1-sheet-of-paper rule!
• Work on “cheat sheets”
Warm-up
• For the function y = -x2 – 6x – 7, find the vertex.
• If the graph y = ax2 + bx + c opens up, is a>0 or a<0?
• Which function does NOT have a maximum?
• Y = -x2 – 5x – 6
• Y = -x2 – x – 6
• Y = 3x2 – 15x + 2
• Y = 49 – x2
• What is the vertex of y = -3(x – 2)2 – 4?
Did you know?
• If you have 3 quarters, 4 dimes, and 4 pennies, you have \$1.19. You also have the largest amount of money in coins without being able to make change for a dollar. The numbers \'172\' can be found on the back of the U.S. \$5 dollar bill in the bushes at the base of the Lincoln Memorial.
• President Kennedy was the fastest random speaker in the world with upwards of 350 words per minute.
• In the average lifetime, a person will walk the equivalent of 5 times around the equator.
• Odontophobia is the fear of teeth.
• The 57 on Heinz ketchup bottles represents the number of varieties of pickles the company once had.
• In the early days of the telephone, operators would pick up a call and use the phrase, "Well, are you there?". It wasn\'t until 1895 that someone suggested answering the phone with the phrase "number please?"
• The surface area of an average-sized brick is 79 cm squared.
According to suicide statistics, Monday is the favored day for self-destruction.
• Cats sleep 16 to 18 hours per day.
• The most common name in the world is Mohammed.
• It is believed that Shakespeare was 46 around the time that the King James Version of the Bible was written. In Psalms 46, the 46th word from the first word is shake and the 46th word from the last word is spear.
• Karoke means "empty orchestra" in Japanese.
• The Eisenhower interstate system requires that one mile in every five must be straight. These straight sections are usable as airstrips in times of war or other emergencies.
• The first known contraceptive was crocodile dung, used by Egyptians in 2000 B.C.

### Questions before test?

20 questions

Be sure to make your graphs very carefully; they will be graded on accuracy!

When in doubt, show all work!

1.1 Test: Standard/Vertex Form
• You will need: calculator, cheat sheet, something to write with
• If you need scratch paper, I will pass it out; it must be turned in with your test!
• You have the remainder of class to take the test
• NO TALKING until EVERYONE is finished
• Talking during test will result in a grade of zero (I may not necessarily even warn you) and you will have to re-take an alternate test during 5th block
• Once you have finished your test, make sure workbook pages 60-69 are complete.
• You do not have to tear them out, just turn in your whole notebook
Warm-up
• Convert f(x) = x2 – 10x + 6 to vertex form.
• Convert f(x) = 3(x – 1)2 + 8 to standard form. Show formulas used for the conversion.
Did you know?
• Facts:The East Japan Railway Company (JR-East), as part of research aimed at developing more environmentally friendly train stations, is testing an experimental system that produces electricity as people pass through ticket gates.
• World Facts:It is common mistake to say that The Great wall of China is visible from outer space. It is too thin to be noticed from such a great distance. Only two man made structures visible from space are: The Pyramids of Giza and the Hoover Dam.
• Internet Facts:Yahoo the complex internet organism has complicated name. Word "Yahoo" is shortcut for "Yet Another Hierarchical Officious Oracle". It was coined by PhD candidates at Stanford University: David Filo and JerryYang.

Science Facts: Closest spiral galaxy to our Milky Way galaxy is Andromeda and even traveling at the speed of light it would take 2 million years to reach it. There are 100 billion galaxies that we can observe. Prediction is that there is much more, but Hubble Space Telescope couldn\'t detect them. The Andromeda Galaxy and the Milky Way is predicted to merge within the next 5 billion years. In the event like this smaller galaxies are consumed by larger ones. Andromeda galaxy is twice as large as Milky Way.

• Politics Facts:Thirteen years our buddy Bill Gates from Winblows was the richest man in the world. However, as of 2008 the top five richest people are: 1. Warren Buffett (Stocks and investments), 2.Carlos Slim Helu (Mexican telecom mogul), 3. William Gates III (Windows and more), 4. Lakshmi Mittal (World\'s largest steelmaker - India), 5. Mukesh Ambani (Petrochemicals giant Reliance Industries – India).
• Sport Facts:In ancient Japan, public contests were held to see who in a town could fart the loudest and longest. Winners were awarded many prizes and received great recognition.

STILL IN PROGRESS

:-Sp

20 minutes
• Finish Tests
• Make sure workbooks are turned in – p 60-69.
• If you do not finish in 20 minutes, plan to come to 5th block to finish the test or tomorrow morning at 7:00 am (you will need a pass)

Part 1:5 minutes

• Fingers closed – make a “mitten”
• YOU WANT IT TO LOOK LIKE A PARABOLA
• Select 10 coordinates from the graph

(try to select points at “bulls eye” or grid intersection); write them in box1

Part 2:
• Graphing Calculator
• STAT  5, Enter (so we’re on the same page)
• STAT  1, top of list, clear, down arrow, then…
• STAT  1, Enter (to enter in your coordinates)
• Put x-coordinate of your point into L1
• Put y-coordinate of your point into L2
• STAT  Calc  5, Enter
• Write down the quadratic equation for your hand – round decimals to hundredth’s place
Part 3
• 2nd Y=
• 1
• “ON”
• Type: scattered points
• X-list: L1 (2nd 1)
• Y-list: L2 (2nd  2)
• Mark: circle with open center
• 2nd  MODE
• Now, go to y= and put in your quadratic equation from part 2; graph; draw the screen
Part 4:

*Not on paper

SUMMARY

Write and answer the following question:

How well does your graph fit your points? Give reasons you think it does/does not fit.

TURN IN

GP/Assignment/Pairs
• Textbook
• Example problems p. 61 #34-39
• On a piece of paper, to turn in, DO:
• P. 59 #35,40,41
• 1) Make an x,y-chart put into L1, L2 in your calculator, do stat  calc  5