© 21st Century Math Projects - cppasd.com - L25-30... · To save the city, the F(x) Men have had...

© 21st Century Math Projects


Project Title: Superhero Transformations

Standard Focus: Geometry & Spatial Sense, Patterns,

Algebra & Functions Time Range: 3-4 Days

Supplies: Scissors

Topics of Focus:

- Graphing Transformations (Translations and Reflections)

- Families of Functions

Benchmarks:

Seeing Structure in

Expressions A-SSE 1b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

Interpreting

Functions F-IF

7. Graph functions expressed symbolically and show key features of the graph, by hand in

simple cases and using technology for more complicated cases.★

Building Functions F-BF 1b. Combine standard function types using arithmetic operations.

Building Functions F-BF

3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for

specific values of k; find the value of k given the graphs. Experiment with cases and illustrate

an explanation of the effects on the graph using technology. Include recognizing even and

odd functions from their graphs and algebraic expressions for them.

Congruence G-CO

2. Represent transformations in the plane describe transformations as functions that take

points in the plane as inputs and give other points as outputs. Compare transformations that

preserve distance and angle to those that do not

Procedures:

A) Task 1: Hand out the Superhero “Cards” and have students cut out the functions. This will let them move the functions

freely on the graphs and turn into a hands-on activity. All of the graphs are of the same scale so the cut-outs will work for every

task.

B) Task 2: Hand out “F(x) Men”. Students will understand the relationship between vertical and horizontal translations on a

coordinate plane and their algebra. This is a task that intends to scaffold the concepts that will be in the more challenging

problems.

C) Task 3: Hand out “Join the Squad”. Students will use vertical and horizontal translations with multiple functions to zap the

enemies into submission. Each Mission has slightly different requirements so encourage students to read carefully.

D) Task 4: Hand out “Reflections”. This introduces students to flipping functions horizontally or vertically and how that relates

to changes in the functions. Doing this activity with a graphing calculate to check their work, could be a useful addition.

E) Task 5: Hand out “The Boss Challenge”. Students must put all of their skills together in order to translation and reflect

functions in such a way to eliminate all enemies.


The F(x) Men (Cut out the functions for each Superhero)

Family: Linear

Function: f(x) = x

Super Power: Nuclear

Fusion Blasts

from her eyes

Lady Straightedge is the direct,

no-nonsense leader of the F(x)

Men. She understands the

fastest way to get from A to B.

Family: Quadratic

Function: f(x) = x2

Super Power: Parabolic

Kinetic Rays

Pawabawa is well rounded and

perfectly symmetrical. He is

able to absorb the negative

energies from his enemies and

turn them into positive rays of

their power squared.

Family: Exponential

Function: f(x) = 2x

Super

Power:

Ginormous

Robo Suit

Robo-Grow escalates conflict

and smashes henchmen into

dust in mere seconds. Good

thing she’s on our team.

Depending on the situation,

Robo-Grow can shrink or

expand as necessary.


The F(x) Men (Cut out the functions for each Superhero)

Family: Absolute

Value

Function: f(x) = |x|

Super

Power:

Finger

Beams

Captain Abs is an upbeat,

glass half-full, goodie two

shoes. He’ll turn any positive

into a negative. If you need a

shoulder to cry on or you

need to defeat a Boss,

Captain Abs is your guy.

Family: Square Root

Function: f(x) = √𝑥

Super

Power:

Ninja skills

with sword

You can always count on

Radical Girl to keep it real.

Down-to-Earth, Radical Girl

wields her sword with the

same tenacity that she

discards Lames, Haters and

Ignorant People.

Family: Cubing

Function: f(x) = x3

Super

Power:

Curvy Laser

Gun

Bipolar Tommy can go from

negative to positive in a

second – and back again.

That’s why we love him.

Although he’s a little Jekyll

and Hyde, he’s an essential

member of the team.


Under the careful watch of the superhero team the F(x)

Men, Cartesianville is safe from all evildoers. But this

could change at a moment’s notice.

Each hero has special abilities related to a mathematical

function – but along with it comes certain limitations.

Under no circumstances can they touch a Baddie or they

will lose all their powers! Another challenge they all share is that their powers only work when

they are facing due north (right up the y-axis) and this makes it difficult to zap their targets. To hit

a target, they need to move to a new location from the origin – a mathematical translation. They

program their new coordinates and POOF! Here’s how it works:

Of course translations can be combined to make more elaborate transformations-- WAIT! Oh No!

What’s that?! Alien Baddies in the west side Home Depot? Lady Straightedge do you copy?

Vertical translations occur when a function moves up or

down the y-axis. In the general case y = f(x), a vertical

translation will result in y = f(x) + d, where d is the number of

units shifted up or down. Notice the d is outside the function.

Horizontal translations occur when a function moves

left or right along the x-axis. In the general case y = f(x), a

horizontal translation will result in y = f(x - c), where c is the

number of units shifted left or right. Notice the c is inside

the function.

(Hint: When you write this correctly, it may feel like you

are going the opposite direction.)


Although she was selecting a new tub of grout to refinish the lines in her

bathroom floor, Lady Straightedge never misses a chance to zap Alien

baddies. Her nuclear fusion glasses blast in f(x) = x lines.

Lady Straightedge will only use vertical

translations (moving up and down the y-axis).

She has determined the order she will eliminate

her targets and numbered them 1 - 4.

For each Alien Baddie, complete the chart

below. She needs to know what equation to

program into her system to launch her powers!

(Use the cut out of her linear power to help)

Alien # Describe the shift in words

Where does she

need to stand?

(Ordered Pair)

Equation to

Program

1

2

3

4

Write a general equation to describe any vertical translation of the linear function f(x) = x:

On the other side of town, there seems to be a disturbance with some Evil Ninjas at the park. Please

don’t zap any kids!


While he was shooting some hoops, Pawabawa hustled to see four Evil

Ninjas in the park. From where he stands (he calls it the vertex) he zaps kinetic

rays that follow a path of f(x) = x2.

Since all of the Ninja are on the northern side of

the park, Pawabawa will only use horizontal

translations (moving left and right on the x-axis).

Note the number next to each Ninja and

complete the chart below. In some cases you may

think there is more than one equation, but make

sure you DON’T ZAP ANY KIDS! That’d be bad.

(Use the cut out of his quadratic power to help)

Ninja # Describe the shift in words

Where does he

need to stand?

(Ordered Pair)

Equation to

Program

1

2

3

4

Write a general equation to describe any horizontal translation of the quadratic function f(x) = x2:

But wait! What’s that?!? A gang of Angry Bandits have invaded the University’s bioscience lab!


Robo-Grow was in the middle of a guinea pig breeding experiment, but has

no problem jumping in her suit to obliterate five Angry Bandits. Her Robo-

Grow suit blasts charges that model the function f(x) = 2x.

Since her suit is so powerful, Robo-Grow can

only blast TWO CHARGES. She must figure out

a way to eliminate all the Angry Bandits with

only two shots. She will have to translate both

horizontally and vertically.

Circle or highlight the two groups of Angry

Bandits and complete the chart below. (Use the

cut out of her exponential power to help)

Now luckily, all seems calm. At least the children will be able to sleep tonight in Cartesianville, but

beneath the surface, a greater evil seems to be stirring… To be continued…

Bandit

Group

#

Describe the shift in words (Which shift is

inside the function? Which is outside?)

Where does she

need to stand?

(Ordered Pair)

Equation to

Program

1

2

Write a general equation to describe any translation of the exponential function f(x) = 2x :


A mysterious new enemy has Baddies crawling all over

the city. These enemies are equipped with their own evil

rays. If the heroes touch the rays or the Baddies

themselves, they are toast! To save the city, the F(x) Men

have had to bring in a few new recruits to help!

Four Possessed Geishas have infiltrated

Cartesianville’s nuclear power plant! Our two

heroes, Lady Straightedge and Bipolar Tommy are

standing at the origin and see the chaos. They

cannot pass through the light beams the villains

have set or else they’ll be dust.

Using both of their powers, they must hit all

four Possessed Geishas. They can only

program one shot each! Determine how

they need to move to accomplish the task.

1. Where do they need to stand on the

coordinate plane to complete the mission?

Hero Ordered Pair

Lady Straightedge

Bipolar Tommy

2. To use their powers, the heroes must

program their suits with their new

coordinates. Be sure to watch the signs and

simplify.

Hero Equation to Program

Lady Straightedge

Bipolar Tommy

Lady Straightedge Bipolar Tommy

Linear Cubic

f(x) = x f(x) = x3


Five Mischievous Cyborgs are scattered around ready

to blow up the dam and flood the city! It’s only a matter

of time until the entire city is underwater.

Often undoing each other’s work, Pawabawa

and Radical Girl have to set aside their personal

differences to zap the Mischievous Cyborgs and

avoid the Death Ray. Time is of the essence and

they only get one shot each! Determine which

equation they need to program to save the city!



Hero Ordered Pair

Pawabawa

Radical Girl

2. To use their powers, the heroes must program their suits with their new coordinates. Be sure to

watch the signs and simplify when appropriate.

Hero Equation of

Powers

Pawabawa

Radical Girl

3. To settle their dispute, which hero eliminated more Mischievous Cyborgs?

Pawabawa Radical Girl

Quadratic Square Root

f(x) = x2 f(x) = √𝑥


Nine Scallywag Pirates have set up a row of explosives

that will destroy the public transit system! Thousands

of citizens are in danger!

Since there are nine Pirates, it’s impossible that

the heroes will be able to take them all out at

the same time. They estimate they only have 4

chances to eliminate all of the Pirates.

Determine the equations they will need to

program to eliminate all of the pirates.



Attempt Hero Ordered

Pair

2. Robo-Grow and Captain Abs must program their suits with their new coordinates. Be sure to


Attempt Hero Equation of

Powers

3. Is it possible or is the train station going to be blown to smithereens?

Robo-Grow Captain Abs

Exponential Absolute Value

f(x) = 2x f(x) = |x|


Although the F(x) Men have been able to fend off this

new evil force, they are about to encounter something

that will flip every formula they’ve ever learned on its

head! A mystical spell has been cast and it has made

their superpowers go haywire. Some powers have flipped vertically and others have flipped

horizontally! They must adapt to these new challenges and face their Inner and Outer Demons

to save the city. Here is some mathematical background to help:

Other than the difference with horizontal translations with horizontal reflections, all other

translations work the same way when combined with reflections. It’s up to our heroes to

overcome these challenges and keep the citizens of Cartesianville safe.

Vertical reflections occur when a function is reflected

across the x-axis. In the general case y = f(x), a vertical

reflection will result in y = - f(x). Notice the (-) is outside the

function.

Horizontal reflections occur when a function is reflected

across the y-axis. In the general case y = f(x), a horizontal

reflection will result in y = f(- x). Notice the (-) is inside the

function.

(Note: when translating horizontally reflected functions,

horizontal translations work opposite. For example, (-x+3)2

will be flipped across the y-axis and shifted three units to the

right. This is because (-x+3)2 = (-(x - 3))2


Just as the F(x) Men zap a skulk

of Rabid Foxes, they realize

everything zaps backwards!

South is North! After putting

their heads together, they notice

if they place a negative sign outside of their function everything seems to work. But now they only

have time to fire one blast apiece! Can they zap all the Foxes in time? And they better not step on the

Cosecant Lava Beams!

(Use the cut outs of the powers to help)



Hero Ordered Pair

Lady Straightedge

Pawabawa

Captain Abs


program their suits with their coordinates.

Watch the signs!

3. The F(x) Men must describe this new phenomenon with the rest of the team. Explain in your own

words how translations and reflections “outside the function” impact a graph. What are the

similarities between them?

Lady Straightedge Pawabawa Captain Abs

Linear Quadratic Absolute Value

f(x) = -x f(x) = -x2 f(x) = -|x|

Hero Equation of

Powers

Lady Straightedge

Pawabawa

Captain Abs


Across the city, the other F(x) Men

have something stranger going on

when they try to zap a mob of

Ornery Devils. Their powers come

out reverse! A mirror image over

the y-axis.

After tinkering, they realize if they put a

negative sign inside their function, it flips their

graph the right way. But they had a new problem,

when they thought they were shifting left, they

were now shifting right. Everyone has time for

one blast and they need to knock out all these

Ornery Little Devils and avoid their gamma rays.


1. Where do they need to stand on the coordinate

plane to complete the mission?

Hero Ordered Pair

Robo-Grow

Bipolar Tommy

Radical Girl

2. To use their powers, the heroes must program their

suits with their coordinates. Watch the signs!

3. In the case of a horizontal reflection, not every member of the F(x) Men would have their functions

look different. Whose functions would still appear the same? Why is this the case?

Robo-Grow Bipolar Tommy Radical Girl

Exponential Cubic Square Root

f(x) = 2-x f(x) = -x3 f(x) = √−𝑥

Hero Equation of

Powers

Robo-Grow

Bipolar Tommy

Radical Girl


The F(x) Men have thwarted the threats to

Cartesianville and in the process have learned how to

flip their powers horizontally and vertically at will.

But from above there arose such a clatter and the Evil

Boss pulling the strings is finally revealed… it is the

Dysfunctional Santa. With the help of his Rabies

Infected Reindeer, Santa has a weapon so powerful

he’ll melt the North Pole and flood the Earth!

Santa doesn’t play by the rules and uses

Pixie Dust Death Beams that defy the laws

of mathematical functions! He’s on the

naughty list this year!

To defeat Santa, the F(x) Men have one

blast each to zap him, all of his Reindeer,

avoid the Pixie Dust Death Beams and

NOT ZAP ANY LITTLE GIRLS. Simple

eh? You need to at least get as many as you

can! If the F(x) Men are not successful it will

not be a very Merry Christmas.

(Use the cut outs of the powers. Functions may

need to be flipped vertically or horizontally)

Superhero Where do they need to stand?

(Ordered Pair)

Equation of Powers

Lady Straightedge

Pawabawa

Robo-Grow

Captain Abs

Radical Girl

Bipolar Tommy

The fate of Cartesianville is in your hands. Good luck, heroes.


Thank you for being my Math Friend!

If you liked this

21st Century Math Project

You might like others. (Click the logo)

Math it Up.

Boomdiggy.

http://www.teacherspayteachers.com/Store/21st-Century-Math-Projects/Category/CSI-Whodunnits


Although she was selecting a new tub of grout to refinish the lines in her

bathroom floor, Lady Straightedge never misses a chance to zap Alien

baddies. Her nuclear fusion glasses blast in f(x) = x lines.

Lady Straightedge will only use vertical

translations (moving up and down the y-axis).

She has determined the order she will eliminate

her targets and numbered them 1 - 4.

For each Alien Baddie, complete the chart

below. She needs to know what equation to

program into her system to launch her powers!

(Use the cut out of her linear power to help)

Alien # Describe the shift in words

Where does she

need to stand?

(Ordered Pair)

Equation to

Program

1 Shift down three units (0, -3) Y = X - 3

2 Shift up two units (0, 2) Y = X + 2

3 Shift up ten units. (0, 10) Y = x +10

4 Shift down nine units (0,-9) Y = x – 9

Write a general equation to describe any vertical translation of the linear function f(x) = x+d

On the other side of town, there seems to be a disturbance with some Evil Ninjas at the park. Please

don’t zap any kids!


While he was shooting some hoops, Pawabawa hustled to see four Evil

Ninjas in the park. From where he stands (he calls it the vertex) he zaps kinetic

rays that follow a path of f(x) = x2.

Since all of the Ninja are on the northern side of

the park, Pawabawa will only use horizontal

translations (moving left and right on the x-axis).

Note the number next to each Ninja and

complete the chart below. In some cases you may

think there is more than one equation, but make

sure you DON’T ZAP ANY KIDS! That’d be bad.

(Use the cut out of his quadratic power to help)

Ninja # Describe the shift in words

Where does he

need to stand?

(Ordered Pair)

Equation to

Program

1 Shift left 10 units (-10,0) Y = (x+10)2

2 Shift left 2 units. (-2, 0) Y = (x+2)2

3 Shift right 3 units (3,0) Y = (x-3)2

4 Shift right 7 units. (7,0) Y = (x-7)2

Write a general equation to describe any horizontal translation of the quadratic function f(x) = (x-

c)2:

But wait! What’s that?!? A gang of Angry Bandits have invaded the University’s bioscience lab!


Robo-Grow was in the middle of a guinea pig breeding experiment, but has

no problem jumping in her suit to obliterate five Angry Bandits. Her Robo-

Grow suit blasts charges that model the function f(x) = 2x.

Since her suit is so powerful, Robo-Grow can

only blast TWO CHARGES. She must figure out

a way to eliminate all the Angry Bandits with

only two shots. She will have to translate both

horizontally and vertically.

Circle or highlight the two groups of Angry

Bandits and complete the chart below. (Use the

cut out of her exponential power to help)

Now luckily, all seems calm. At least the children will be able to sleep tonight in Cartesianville, but

beneath the surface, a greater evil seems to be stirring… To be continued…

Bandit

Group

#

Describe the shift in words (Which shift is

inside the function? Which is outside?)

Where does she

need to stand?

(Ordered Pair)

Equation to

Program

1 Shift up 1 (outside) and three to the left (inside) (-3, 1) Y = 2x+3 + 1

2 Shift three to the right (inside) and down 7

(outside).

(3, 7) Y = 2x-3 - 7

Write a general equation to describe any translation of the exponential function f(x) = 2x-c +d


A mysterious new enemy has Baddies crawling all over

the city. These enemies are equipped with their own evil

rays. If the heroes touch the rays or the Baddies

themselves, they are toast! To save the city, the F(x) Men

have had to bring in a few new recruits to help!

Four Possessed Geishas have infiltrated Cartesianville’s

nuclear power plant! Our two heroes, Lady

Straightedge and Bipolar Tommy are standing at the

origin and see the chaos. They cannot pass through the

light beams the villains have set or else they’ll be dust.

Using both of their powers, they must hit all

four Possessed Geishas. They can only program

one shot each! Determine how they need to

move to accomplish the task.



Hero Ordered Pair

Lady Straightedge (-3, 1)

Bipolar Tommy (5, -2)


program their suits with their new coordinates.

Be sure to watch the signs and simplify.

Hero Equation to

Program

Lady Straightedge (x+3) + 1 = x + 4

Bipolar Tommy (x-5)3 – 2

Lady Straightedge Bipolar Tommy

Linear Cubic

f(x) = x f(x) = x3


Five Mischievous Cyborgs are scattered around ready

to blow up the dam and flood the city! It’s only a matter

of time until the entire city is underwater.

Often undoing each other’s work, Pawabawa

and Radical Girl have to set aside their personal

differences to zap the Mischievous Cyborgs and

avoid the Death Ray. Time is of the essence and

they only get one shot each! Determine which

equation they need to program to save the city!



Hero Ordered Pair

Pawabawa (-4, -8)

Radical Girl (-5, 0)

2. To use their powers, the heroes must program their suits with their new coordinates. Be sure to


Hero Equation of

Powers

Pawabawa (x+4)2 – 8

Radical Girl √𝑥 + 5

3. To settle their dispute, which hero eliminated more Mischievous Cyborgs? Pawabawa got 3 and

Radical Girl got 2.

Pawabawa Radical Girl

Quadratic Square Root

f(x) = x2 f(x) = √𝑥


Nine Scallywag Pirates have set up a row of explosives

that will destroy the public transit system! Thousands

of citizens are in danger!

Since there are nine Pirates, it’s impossible that

the heroes will be able to take them all out at

the same time. They estimate they only have 4

chances to eliminate all of the Pirates.

Determine the equations they will need to

program to eliminate all of the pirates.



Attempt Hero Ordered

Pair

1 Robo-Grow (3, -5)

2 Captain Abs (-4, -2)

3 Captain Abs (-2, -9)

4 Robo-Grow (-4, 0)

2. Robo-Grow and Captain Abs must program their suits with their new coordinates. Be sure to


Attempt Hero Equation of

Powers

1 Robo-Grow 2x-3 - 5

2 Captain Abs |x + 4| - 2

3 Captain Abs |x + 2| - 9

4 Robo-Grow 2x+4

3. Is it possible or is the train station going to be blown to smithereens? Yes, it is possible.

Robo-Grow Captain Abs

Exponential Absolute Value

f(x) = 2x f(x) = |x|


Although the F(x) Men have been able to fend off this

new evil force, they are about to encounter something

that will flip every formula they’ve ever learned on its

head! A mystical spell has been cast and it has made

their superpowers go haywire. Some powers have flipped vertically and others have flipped

horizontally! They must adapt to these new challenges and face their Inner and Outer Demons to

save the city. Here is some mathematical background to help:

Other than the difference with horizontal translations with horizontal reflections, all other

translations work the same way when combined with reflections. It’s up to our heroes to overcome

these challenges and keep the citizens of Cartesianville safe.

Vertical reflections occur when a function is reflected

across the x-axis. In the general case y = f(x), a vertical

reflection will result in y = - f(x). Notice the (-) is

outside the function.

Horizontal reflections occur when a function is reflected

across the y-axis. In the general case y = f(x), a horizontal

reflection will result in y = f(- x). Notice the (-) is inside the

function.

(Note: when translating horizontally reflected functions,

horizontal translations work opposite. For example, (-x+3)2

will be flipped across the y-axis and shifted three units to

the right. This is because (-x+3)2 = (-(x - 3))2


Just as the F(x) Men zap a skulk

of Rabid Foxes, they realize

everything zaps backwards!

South is North! After putting

their heads together, they notice

if they place a negative sign outside of their function everything seems to work. But now they only

have time to fire one blast apiece! Can they zap all the Foxes in time? And they better not step on the

Cosecant Lava Beams!




Hero Ordered Pair

Lady Straightedge (4, -2)

Pawabawa (8,7)

Captain Abs (0, -1)


program their suits with their coordinates.

Watch the signs!

3. The F(x) Men must describe this new phenomenon with the rest of the team. Explain in your own

words how translations and reflections “outside the function” impact a graph. What are the

similarities between them?

Lady Straightedge Pawabawa Captain Abs

Linear Quadratic Absolute Value

f(x) = -x f(x) = -x2 f(x) = -|x|

Hero Equation of

Powers

Lady Straightedge -x – 4 – 2 = -x -6

Pawabawa -(x-8)2 + 7

Captain Abs -|x| - 1


Across the city, the other F(x) Men

have something stranger going on

when they try to zap a mob of

Ornery Devils. Their powers come

out reverse! A mirror image over

the y-axis.

After tinkering, they realize if they put a

negative sign inside their function, it flips their

graph the right way. But they had a new

problem, when they thought they were shifting

left, they were now shifting right. Everyone has

time for one blast and they need to knock out all

these Ornery Little Devils and avoid their gamma

rays. (Use the cut outs of the powers to help)

1. Where do they need to stand on the coordinate

plane to complete the mission?

Hero Ordered Pair

Robo-Grow (-5, -6)

Bipolar Tommy (7, 3)

Radical Girl (6, 2)

2. To use their powers, the heroes must program their

suits with their coordinates. Watch the signs!

3. In the case of a horizontal reflection, not every member of the F(x) Men would have their functions

look different. Whose functions would still appear the same? Why is this the case?

Absolute Value and the Quadratic function would appear the same since they are symmetrical.

Robo-Grow Bipolar Tommy Radical Girl

Exponential Cubic Square Root

f(x) = 2-x f(x) = -x3 f(x) = √−𝑥

Hero Equation of

Powers

Robo-Grow 2-x – 5 - 6

Bipolar Tommy (-x+7)3 + 3

Radical Girl √−𝑥 + 6 + 2


The F(x) Men have thwarted the threats to Cartesianville

and in the process have learned how to flip their powers

horizontally and vertically at will. But from above there

arose such a clatter and the Evil Boss pulling the strings is

finally revealed… it is the Dysfunctional Santa. With the

help of his Rabies Infected Reindeer, Santa has a weapon

so powerful he’ll melt the North Pole and flood the Earth!

Santa doesn’t play by the rules and uses Pixie

Dust Death Beams that defy the laws of

mathematical functions! He’s on the naughty

list this year!

To defeat Santa, the F(x) Men have one blast

each to zap him, all of his Reindeer, avoid

the Pixie Dust Death Beams and NOT ZAP

ANY LITTLE GIRLS. Simple eh? You need

to at least get as many as you can! If the F(x)

Men are not successful it will not be a very

Merry Christmas.

(Use the cut outs of the powers. Functions

may need to be flipped vertically or

horizontally)

Superhero Where do they need to stand?

(Ordered Pair)

Equation of Powers

Lady Straightedge (-7, 3) ( or a few others) X + 10

Pawabawa (-2, -7) (x +2) 2 - 7

Robo-Grow (4, -7) 2x-4 - 7

Captain Abs (3, 6) -|x-3| +6

Radical Girl (6, -6) √−𝑥 + 6 − 6

Bipolar Tommy (-4, 4) (x +4) 2 +4

The fate of Cartesianville is in your hands. Good luck, heroes.


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