NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... ·...

30
Pre Calculus 1 of 30 Linear Functions Slope-Intercept Point-Slope b mx y + = 1 1 ) ( y x x m y + = 2 4 3 = x y 3 ) 1 ( 2 3 + + = x y

Transcript of NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... ·...

Page 1: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 1 of 30

Linear Functions Slope-Intercept Point-Slope

bmxy += 11 )( yxxmy +−=

2

43

−= xy 3)1(23

++−= xy

Page 2: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 2 of 30

Slope

12

12

xxyy

xy

runrisem

−=

ΔΔ

==

Parallel Line 21 mm = Perpendicular Lines 2

11m

m −=

21

21=

221

−=

Page 3: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 3 of 30

Functions

Definition: A rule that assigns a unique output (y) for every input (x). Range: Set of Output (y-values) of the Function Domain: Set of Input (x-values) of the Function

To Find Domain: The domain is usually all real numbers (ℝ) or −∞,∞ except for rational, even roots and logarithmic functions.

Rational Functions (Fractions)

421)(−

=x

xf

All real numbers except values of x that make the denominator equal to zero.

042 ≠−x D: 2≠x or ),2()2,( ∞∪−∞

Even Root Functions xxf 312)( −=

All Positive Numbers & Zero 0123 ≥+x

D: 4≤x or ]4,(−∞

Logarithmic Functions )4(log)( 3 += xxf

Only Positive Numbers 04 >+x

D: 4−>x or ),4[ ∞−

Page 4: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 4 of 30

Even Function Odd Function Symmetric about the y-axis Symmetric about the Origin

)()( xfxf −= )()( xfxf −=−

12)1()1(12)1()1(

2)(

2

2

2

−=−−=−−=−=

−=

ff

xxf

2)1(2)1(2)1(2)1(

2)(

3

3

3

−=−=−==

=

ff

xxf

Page 5: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 5 of 30

Transformation of Functions

Cxf ±)(

)( Cxf ∓

)(xf−

)( xf −

1),( <axaf

1),( >axaf

Page 6: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 6 of 30

Inverse Operations

triginversetrictrigonomenonentialexarithmiclo

rootspowerdivisiontionmultiplica

nsubtractioaddition

pg

↔↔↔↔↔

xxaxxxx

a1

33

sinsinlog

−↔↔↔

÷↔×−↔+

Inverse Functions

abfbaf

==

− )()(

1 25)(52)(

1 +=

−=− xxf

xxf

3251)1(

15)3(2)3(1 =

+=

=−=−ff

Steps to find Inverse Function 1. Switch all x’s and y’s 2. Solve for y

Horizontal Line Test

g(x) has an inverse

f(x) does not have an inverse

Page 7: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 7 of 30

Absolute Value Function

⎩⎨⎧

<−

≥==

00,

)(xxxx

xxf

Domain: ),( ∞−∞ Range: ),0[ ∞ Even Function Note: Corner at the Origin

Greatest Integer Function xxf =)(

Domain: ),( ∞−∞ Range: set of integers Note: Jump Discontinuity at every integer

value of x

Page 8: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 8 of 30

Quadratic (Square) Function

2)( xxf =

Domain: ),( ∞−∞ Range: ),0[ ∞ Even Function

Square (Even) Root Function

xxf =)(

Domain: ),0[ ∞ Range: ),0[ ∞ Note: Square (Even) Root cannot have

Negative Numbers in the Radical

Page 9: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 9 of 30

Cubic Function

3)( xxf =

Domain: ),( ∞−∞ Range: ),( ∞−∞ Odd Function

Cubic (Odd) Root Function 3)( xxf =

Domain: ),( ∞−∞ Range: ),( ∞−∞ Odd Function Note: Cubic (Odd) Root have an Undefined

(Vertical) Slope at 0=x

Page 10: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 10 of 30

Exponential Function

xaxf =)(

Domain: ),( ∞−∞ Range: ),0( ∞ Note: The number e (a common base) is an

irrational number, approximately 2.718…

Logarithmic Function

xxf alog)( =

Domain: ),0( ∞ Range: ),( ∞−∞ Note: ln (natural log) is log with base e.

Inside a log must always be positive.

Page 11: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 11 of 30

Sine Function

xxf sin)( =

Domain: ),( ∞−∞ Range: ]1,1[− Odd Function Period: 2π

Cosine Function

xxf cos)( =

Domain: ),( ∞−∞ Range: ]1,1[− Even Function Period: 2π

Page 12: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 12 of 30

Tangent Function

xxf tan)( =

Domain: All Real Numbers x ≠ 0.5π + nπ Vertical Asymptote: x ≠ 0.5π + nπ Range: ),( ∞−∞ Odd Function Period: π

Rational (Reciprocal) Function

xxf 1)( =

Domain: ),0()0,( ∞∪−∞ Range: ),0()0,( ∞∪−∞ Note: Denominator can never equal zero.

Page 13: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 13 of 30

Multiply Polynomials Distributing Method

Distribute all each term of the 1st polynomial to each term of the 2nd polynomial.

8414)122(7

+

+

xx

20262010126)42)(53(

2

2

−−

−+−

−+

xxxxx

xx

xxxxxxxxxx

xxxx

3132718183018353

)6)(353(

234

23423

22

−−−−

−−−−+

−−+

xxxxxxxxlog20log12106)log42)(53(

23

2

−−+

−+

Special Products

Sum & Difference

22

))((bababa

−+

Square of Binomial

22

2

2)(baba

ba+−

− 22

2

2)(baba

ba++

+

Page 14: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 14 of 30

Factoring Polynomials Common Factors

Find common factors and factor them out.

)6(148414+

+

xx

)52(42048

2

23

−−

−−

xxxxxx

)93(98127922

3223

yxyxxyxyyxyx

+−

+−

1=a Find factors of c that add up to b. Rewrite these factors as a product.

2082 −− xx 210 ⋅−

)2)(10( +− xx

962 +− xx 33 −⋅− )3)(3( −− xx

36162 ++ xx 123 ⋅

)12)(3( ++ xx

1≠a MUST find common factors first! Find factors of ac that add up to b. Then, divide those factors by a and reduce. Rewrite the reduced fractions as a product with the denominator multiplied by x and numerator added to x.

8166 2 +− xx )483(2 2 +− xx

322

32

36

1243

−⋅−

−⋅

−=⋅

)23)(2(2 −− xx

Page 15: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 15 of 30

Special Factoring Difference of 2 Squares

))((

22

bababa−+

Perfect Trinomial

2

22

)(2bababa

+−

2

22

)(2bababa

+

++

Standard to Vertex Form/Complete the Square

If 1=a , then:

423

25

12425

4255

125

2

2

2

squarehalf

+⎟⎠

⎞⎜⎝

⎛ +=

+−++=

++=

xy

xxy

xxy

If 1≠a , then:

`

213

232

1129

4932

1162

2

2

bounce

2

squarehalf

+⎟⎠

⎞⎜⎝

⎛ +=

+−⎟⎠

⎞⎜⎝

⎛ ++=

++=

xy

xxy

xxy

Page 16: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 16 of 30

Quadratic Formula If cbxax ++= 20 is not factorable use either complete the square or quadratic formula to solve:

aacbbx

242 −±−

=

Solve for x: 450 2 ++= xx

1,4235

295

)1(2)4)(1(4)5()5( 2

−−=

±−=

±−=

−±−=x

Complex (Imaginary) Numbers Numbers formed by multiplying a real number time i, where i, is the square root of negative 1.

1

11

4

3

2

=−=−=−=

iii

ii

4164424242424

==−⋅−

−=−−

=−

−=−

=

ii

Solve for x: 1250 2 ++= xx

i

i

x

522

5242204)1(2

)9)(1(4)4()4( 2

±−=

±−=

−±−=

−±−=

Page 17: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 17 of 30

max 1 31zeros 24

108)( 4

≥=≥=

+−=

n-n

xxxg

Polynomials: 012

21

1 ...)( axaxaxaxaxf nn

nn +++++= −

n, the largest exponent, is the degree (order) of the polynomial.

Leading Coefficient Test an is the coefficient of xn

Zeros (or x-intercept or roots) n is the maximum number of real zeros. There could be less than n. To find zeros, set the function equal to 0 and solve.

Maximums & Minimums

n–1 determines the possible number of both hills (maximums) and valleys (minimums) the function may have.

max 1min 2 31zeros 34

8)( 24

+≥=≥=−=

n-n

xxxf

Page 18: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 18 of 30

Rational Functions Horizontal Asymptote

1. Multiply (if necessary). 2. Take leading coefficient from numerator and

denominator. 3. Simplify.

a. If x remains on bottom, y = 0. b. If x’s cancel, y = coefficients. c. If x remains on top, no horizontal

asymptote.

Vertical Asymptote 1. Factor (if possible). 2. Simplify. 3. Set denominator equal to zero.

Graphing 1. Graph asymptotes as dashed lines. 2. Test points for direction. Note: Any section

in the center must be tested with at least two points.

16)4(3)( 2

2

+=xxxf

Horizontal Asymptote

3

331648243

2

2

2

2

=

==−

++=

yxx

xxxy

Vertical Asymptote

440

4)4(3

)4)(4()4(3 2

=−=

+=

−+

+=

xx

xx

xxxy

Page 19: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 19 of 30

Properties of Exponents & LogarithmsProduct Rule

Add the two exponents. baba xxx +=

Quotient Rule Subtract the smallest exponent from the largest.

abba

b

a

xx

xx

−−=

1or

Power Rule Multiply the exponents.

( ) abba xx = Negative Rule

If on bottom, goes to the top. If on top, goes to bottom.

aa

xx 1

=− aa x

x=

1

Radicals (Root) ↔ Rational Exponent Radical or root becomes the denominator in the exponent.

ba

b a

bb

xx

xx

=

=1

Logs Product Rule

abba logloglog =+ Quotient

baba logloglog =−

Power Rule abab loglog =

Page 20: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 20 of 30

Exponential Growth & Decay Growth

trAy )1( +=

4.124)20.01(50 5 =+=y

9.1916)20.01(50 20 =+=y

y = current amount at t A = initial amount r = rate t = time

Decay trAy )1( −=

4.16)20.01(50 5 =−=y

6.0)20.01(50 20 =−=y

Compound Interest

nt

nrPy ⎟⎠

⎞⎜⎝

⎛ += 1

40.080,22$408.01000,10

104

=⎟⎠

⎞⎜⎝

⎛ +=⋅

y

46.253,22$36508.01000,10

10365

=⎟⎠

⎞⎜⎝

⎛ +=⋅

y

y = current amount at t P = Principle (initial

deposit) r = rate t = time n = number of times

compounded

Compound Continuously rtPey =

41.255,22$000,10 1008.0 == ⋅ey

Page 21: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 21 of 30

Trigonometric Values

..tan

..cos

.

.sin

adjopp

hypadj

hypopp

=

=

=

θ

θ

θ

..cot

..sec

.

.csc

oppadj

adjhyp

opphyp

=

=

=

θ

θ

θ

θ

Page 22: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 22 of 30

Common Trigonometric Values

0 𝜋2

𝜋 𝜋2

2𝜋

𝜋3

2𝜋3

4𝜋3

5𝜋3

𝜋4

3𝜋4

5𝜋4

7𝜋4

𝜋6

5𝜋6

7𝜋6

11𝜋6

I II III IV

Examples

sin 𝜃 =32

𝜃 =𝜋3,2𝜋3

tan 𝜃 = −33

𝜃 =5𝜋6,11𝜋6

Degrees 0º 30º 45º 60º 90º

Radians 0 𝜋6

𝜋4

𝜋3

𝜋2

θsin 02= 0

12=12

22

32

42= 1

θcos 42= 1

32

22

12=12

02= 0

θtan 0 13=

33

22= 1 3

1= 3 undefined

Page 23: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 23 of 30

Trigonometric Identities

Reciprocal Identities

xx

xx

sin1csc

csc1sin

=

=

xx

xx

cos1sec

sec1cos

=

=

xx

xx

tan1cot

cot1tan

=

=

Quotient Identities

xxx

cossintan =

xxx

sincoscot =

Pythagorean Identities

1cossin 22 =+ xx

xx 22 sectan1 =+ xx 22 csccot1 =+

Page 24: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 24 of 30

Law of Sines

Cc

Bb

Aa

sinsinsin==

Area of an Oblique Triangle

BacCabAbcArea sin21sin

21sin

21

===

Law of Cosines

Abccba cos2222 −+= Baccab cos2222 −+= Cabbac cos2222 −+=

Heron’s Area Formula

))()(( csbsassArea −−−= ,

where 2cbas ++

=

Alternative Form

bcacbA

2cos

222 −+=

acbcaB

2cos

222 −+=

abcbaC

2cos

222 −+=

Page 25: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 25 of 30

Vectors

Components

𝒗 = 𝑞! − 𝑝!, 𝑞! − 𝑝! = 𝑣!, 𝑣!

𝒗 = 𝑣!𝒊 + 𝑣!𝒋

𝒗 = 𝒗 cos 𝜃 𝒊 + 𝒗 sin 𝜃 𝒋

Magnitude & Direction

𝒗 = 𝑣!! + 𝑣!!

𝜃 = tan!!𝑣!𝑣!

𝒖 = −1 − 2, 3 − (−5) = (−3, 8)

𝒖 = −3𝒊 + 8𝒋

𝒖 = (−3)! + 8! = 73 = 8.54

𝜃 = tan!!8−3

= 110°

Page 26: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 26 of 30

Conics General Equation 𝐴𝑥! + 𝐵𝑥𝑦 + 𝐶𝑦! + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 A, B, C, D, E, F are constants

Circle 𝑥! + 𝑦! with equal coefficients Ex: 3𝑥! + 3𝑦! + 2𝑥 − 3𝑦 = 0

Ellipse 𝑥! + 𝑦! with unequal coefficients Ex: 2𝑥! + 4𝑦! + 2𝑥 + 64 = 0

Parabola Either 𝑥! or 𝑦! Ex: 3𝑥! − 4𝑥 + 𝑦 − 5 = 0

Hyperbola Either 𝑥! − 𝑦! or 𝑦! − 𝑥! Ex: 3𝑥! − 𝑦! + 6𝑥 − 3𝑦 + 81 = 0

Page 27: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 27 of 30

Circle

Fact: Each point on a circle is equidistance from the center by r units.

(𝑥 − ℎ)! + (𝑦 − 𝑘)! = 𝑟!

Use complete the square to rewrite any conic section into its standard form.

𝑥! + 𝑦! − 4𝑥 + 6𝑦 − 23 = 0𝑥! − 4𝑥 + 4 + 𝑦! + 6𝑦 + 9 = 23 + 4 + 9

(𝑥 − 2)! + (𝑦 + 3)! = 36

9𝑥! − 𝑦! − 36𝑥 + 6𝑦 − 12 = 09(𝑥! − 4𝑥 + 4) − (𝑦! − 6𝑦 + 9) = 12 + 4 + 9

9(𝑥 − 2)! − (𝑦 − 3)! = 25

(𝑥 − 2)!

259

−(𝑦 − 3)!

25= 1

Page 28: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 28 of 30

Ellipse Fact: The sum of the distances from the foci to any point on the ellipse is 2𝑎.

(𝑥 − ℎ)!

𝑏!+(𝑦 − 𝑘)!

𝑎!= 1

(𝑥 − ℎ)!

𝑎!+(𝑦 − 𝑘)!

𝑏!= 1

𝑐 = 𝑎! − 𝑏! 𝑒 = !!< 1

Page 29: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 29 of 30

Parabola Fact: Each point on the graph (blue line) is equidistance from focus and directrix. Parabolas have a reflective property that focuses energy to one point, the focus. Thus, they are used as satellite dishes to focus low energy radio waves.

𝑦 − 𝑘 ! = 4𝑝 𝑥 − ℎ

(𝑥 − ℎ)! = 4𝑝(𝑦 − 𝑘)

Page 30: NC Pre Calculus - Mrs. Young's Math Classesmrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/... · of Pre Calculus 17 of 30 1 3 1 max 4 2 zeros ( ) 4 8 10 n-n g x x x Polynomials:

of

Pre Calculus 30 of 30

Hyperbola Fact: The difference of the distances from the foci to any point on the curve is 2𝑎.

(𝑥 − ℎ)!

𝑎! −(𝑦 − 𝑘)!

𝑏! = 1

(𝑦 − ℎ)!

𝑎! −(𝑥 − 𝑘)!

𝑏! = 1

𝑐 = 𝑎! + 𝑏! 𝑒 = !!> 1