unit 4.3 conics: hyperbolas p. 333 · 2020. 8. 13. · foci at (± 3 , 0) and vertices at (± 2 ,...

Unit 4.3 – Conics: Hyperbolas P. 333

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The Conjugate axis is perpendicular to the Transverse axis, and runs through the center. Think of the hyperbola’s

transverse axis as similar to the ellipse’s major axis, and the hyperbola’s conjugate axis as similar to the ellipse’s

minor axis.

In an ellipse:a is the distance between the center and the vertices.c is the distance between the center and the foci.

Same here, except the foci are on the OPPOSITE sides of the vertices.b is used to graph the asymptotes. Can you guess where it would be located?

Differences betweenEllipse and Hyperbola:

Ellipse: Terms are addedHyperbola: Terms are subtracted

Ellipse: a > bHyperbola: No such rule

Ellipse: b is a clear visibly defined distanceHyperbola: b is not

Ellipse, finding the foci: c2 = a2 – b2. Hyperbola, finding the foci: c2 = a2 + b2.

The Conjugate axis is horizontal.

Conjugate axis

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The Conjugate axis is vertical.

1) Using the terms’ signs, do either first:a) Identify 𝒂𝟐 and 𝒃𝟐, and calculate ab) Determine whether the transverse axis is horizontal or vertical

2) Locate the vertices (a units from the center)3) Calculate c (using the formula)4) Locate the foci (c units from the center)

Difference betweenEllipse and Hyperbola:

Ellipse: 𝒂𝟐 is the larger of the 2 denominatorsHyperbola: 𝒂𝟐 is the denominator of the term that’s positive

Consequently, we know that the other denominator is 𝒃𝟐.

Find the vertices, locate the foci, and graph the following hyperbola:

It’s

c will be smaller on the left and larger on the right.b does not need to be graphed.

1) Using the terms’ signs, do either first:a) Identify 𝒂𝟐 and 𝒃𝟐, and calculate ab) Determine whether the transverse axis

is horizontal or vertical2) Locate the vertices (a units from the center)3) Calculate c (using the formula)4) Locate the foci (c units from the center)

Find the vertices, locate the foci, and graph the following hyperbola:

It’s

c will be smaller on the bottom and larger on the top.b does not need to be graphed.

1) Using the terms’ signs, do either first:a) Identify 𝒂𝟐 and 𝒃𝟐, and calculate ab) Determine whether the transverse axis


2 2

19 7

x y− =

CW # 1

1. 2.

3. If the distance between the two foci is 10…

a. What is the distance between one of the foci and the center?b. What is the distance between one of the foci and the closest vertex?

For the following two equations, is the transverse axis horizontal or vertical? How do you know?

The vertices (distance a) and foci (distance c) are on the transverse axis, whether horizontal or vertical.

CW # 2Find the vertices, locate the foci, and graph the following hyperbola:

a. b.1) Using the terms’ signs, do either first:

a) Identify 𝒂𝟐 and 𝒃𝟐, and calculate ab) Determine whether the transverse axis


Find the standard form of the equation of the hyperbola with foci at (± 3 , 0) and vertices at (± 2 , 0).

Solution

From the graph, you can determine that c = 3, because the foci are three units from the center.Moreover, a = 2 because the vertices are two units from the center. So, it follows that b2 = c2 – a2 = 32 – 22 = 9 – 4 = 5

Because the vertices and foci are on a horizontal line,the transverse axis is horizontal, so the standard form of the equation is

Finally, substitute 4 for 𝒂𝟐 and 5 for 𝒃𝟐 to obtain

1) Identify a and c2) Calculate 𝒂𝟐 and 𝒃𝟐

3) Determine which standard formof the equation to use, based ontransverse axis

4) Substitute into that

CW # 3Find the standard form of the equation of the hyperbola with foci at (0 , ± 6) and vertices at (0 , ± 3). Graph it.

1) Identify a and c2) Calculate 𝒂𝟐 and 𝒃𝟐

3) Determine which standard formof the equation to use, based ontransverse axis

4) Substitute into that

Furthermore, the asymptotes pass through the corners of a rectangle of dimensions 2a by 2b.

For the given hyperbola…1) Is the transverse axis horizontal or vertical?2) Where are the vertices?3) Where are the b points?4) What are the asymptotes?5) Where are the foci?6) What are the dimensions (and area) of the rectangle drawn between its branches?

Solution

1) Since the x2 fraction is positive, the transverse axis is horizontal.Its denominator is a2, meaning 4, so a = 2. Therefore the vertices are located at (–2 , 0) and (2 , 0).

2) Consequently the y2 denominator is b2, meaning 12, so b = 12 = 2 3

The b points are located at (0 , 2 3) and (0 , −2 3).

The asymptotes are y = ±𝑏

𝑎𝑥, so they are y = ±

2 3

2𝑥 = ± 3𝑥

3) The foci with a horizontal transverse axis are at (–c , 0) and (c , 0).

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 or 𝒄 = 𝒂𝟐 + 𝒃𝟐 = 𝟒 + 𝟏𝟐 = 𝟏𝟔 = 𝟒So the foci are at (–4 , 0) and (4 , 0).

4) The dimensions of the rectangle are 2a by 2b, meaning 𝟒 𝐛𝐲 𝟒 𝟑. The area is 𝟏𝟔 𝟑

CW # 4For the given hyperbola…1) Is the transverse axis horizontal or vertical?2) Where are the vertices?3) Where are the b points?4) What are the asymptotes?5) Where are the foci?6) What are the dimensions (and area) of the rectangle drawn between its branches?

Graphing Hyperbolas Centered At The OriginThey’re graphed using vertices and asymptotes.

1) Determine whether the transverse axis is horizontal or vertical.2) Identify 𝒂𝟐 and 𝒃𝟐, then calculate a and b.3) Locate and plot the vertices.4) Locate and plot the b points.5) Locate and plot the four points that make up the rectangle drawn

between its branches.If the transverse axis is horizontal, the points are:

Upper left: (–a , b), Lower left: (–a , –b), Upper right: (a , b), Lower right: (a , –b)

If the transverse axis is vertical, the points are:Upper left: (–b , a), Lower left: (–b , –a),Upper right: (b , a), Lower right: (b , –a)

6) Draw the rectangle, then draw an X through its corners, continuing outward in all four directions. These are the positive and negative asymptotes. We don’t have to calculate them!

7) Draw the 2 branches of the hyperbola by starting at each vertex and approach the asymptotes.

Graph this hyperbola,which we analyzed before.

Previously we determined that:

1) a = 2 and b = 2 32) The transverse axis is horizontal.3) The vertices are at (–2 , 0) and (2 , 0).

4) The b points are at (0 , 2 3 ) and (0 ,−2 3 ).

Continuing:5) The four points that make up the rectangle drawn

between its branches are:

Upper left: (–a , b) = (−2 , 2 3 )

Lower left: (–a , –b) = (−2 ,−2 3 )

Upper right: (a , b) = ( 2 , 2 3 )

Lower right: (a , –b) = ( 2 ,−2 3 )6) Draw the rectangle and the X through all four corners.7) Draw the hyperbola itself.

HyperbolaPositive asymptoteNegative asymptoteUpper left point

Lower left point

Upper right point

Lower right point

Upper horizontal lineLower horizontal lineLeft vertical lineRight vertical line

Theseare the functionsI gaveDesmosto graphthe previous hyperbola.

CW # 5Graph this hyperbola, which we analyzed before.

1) Determine whether the transverse axis is horizontal or vertical.2) Identify 𝒂𝟐 and 𝒃𝟐, then calculate a and b.3) Locate and plot the vertices.4) Locate and plot the b points.5) Locate and plot the four points that make up the rectangle drawn






Previously (CW #4) we determined that:1) a = 2 and b = 32) The transverse axis is vertical.3) The vertices are at (0 , –2) and (0 , 2).4) The b points are at (3 , 0) and (–3 , 0). Continue with Step #5...

For hyperbola, we won’t be discussing directrix, eccentricity, or latus rectum. Aren’t you thrilled?

CW # 61) Determine whether the transverse axis is horizontal or vertical.2) Identify 𝒂𝟐 and 𝒃𝟐, then calculate a and b.3) Locate and plot the vertices.4) Locate and plot the b points.5) Locate and plot the four points that make up the rectangle drawn






1) Determine whether the transverseaxis is horizontal or vertical

2) Identify asymptote forms3) Given a or b, calculate the other 4) Using a and b, write the equation

CW # 71) Determine whether the transverse axis is horizontal or vertical2) Identify asymptote forms3) Given a or b, calculate the other 4) Using a and b, write the equation

Then graph it.

An interesting application of conic sections involves the

orbits of comets in our solar system. Comets can have

elliptical, parabolic, or hyperbolic orbits. The center of

the sun is a focus of each of these orbits, and each orbit

has a vertex at the point where the comet is closest to

the sun. Undoubtedly, many comets with parabolic or

hyperbolic orbits have not been identified. You get to

see such comets only once. Comets with elliptical orbits,

such as Halley’s comet, are the only ones that remain in

our solar system. Of the 610 comets identified prior to

1970, 245 have elliptical orbits, 295 have parabolic

orbits, and 70 have hyperbolic orbits.

CW # 8

Unit 4.4 – Conics: Hyperbolas And Their Translations

0) Determine the location of the center, which is at (h , k). 1) Determine whether the transverse axis is horizontal or vertical.2) Identify 𝒂𝟐 and 𝒃𝟐, then calculate a and b.3) Locate and plot the vertices.4) Locate and plot the b points.5) Locate and plot the four points that make up the rectangle drawn

between its branches.6) Draw the rectangle, then draw an X through its corners, continuing

outward in all four directions. These are the positive and negative asymptotes. We don’t have to calculate them!


0) h = 2 k = 3 Center = (2 , 3)1) Transverse is horizontal2) 𝒂𝟐 = 𝟏𝟔, a = 4 𝒃𝟐 = 𝟗, b = 33) Vertices at (–2 , 3) and (6 , 3) 4) b points at (2 , 6) and (2 , 0)5) The four points are: UL: (–2 , 6), LL: (–2 , 0), UR: (6 , 6), LR: (6 , 0)6) Draw…and draw!

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 or

𝒄 = 𝒂𝟐 + 𝒃𝟐 = 𝟏𝟔 + 𝟗 = 𝟐𝟓 = 𝟓So the foci are at (–3 , 3) and (7 , 3).

The asymptotes for an non-translated

hyperbola are y = ±𝑏

𝑎𝑥,

so here they are y − 3 = ±3

4(𝑥 − 2)

CW # 1

0) h = k = Center = ( , )1) Transverse is 2) 𝒂𝟐 = , a = 𝒃𝟐 = , b = 3) Vertices at ( , ) and ( , ) 4) b points at ( , ) and ( , )5) The four points are:

UL: ( , ), LL: ( , ), UR: ( , ), LR: ( , )

6) Draw…and draw!

c = Foci at ( , ) and ( , )

Asymptotes at

Continuing…

0) h = 3 k = 5 Center = ( 3 , 5 )1) Transverse is Vertical2) 𝒂𝟐 = 4 , a = 2 𝒃𝟐 = 25 , b = 53) Vertices at (3 , 3) and (3 , 7) 4) b points at (–2 , 5) and (8 , 5)5) The four points are:

UL: (–2 , 7), LL: (–2 , 3), UR: (8 , 7), LR: (8 , 3)

6) Draw…and draw!

𝒄 = 𝒂𝟐 + 𝒃𝟐 = 𝟒 + 𝟐𝟓 = 𝟐𝟗 ≈ 𝟓. 𝟒

Foci at (3 , 5 + 29) and (3 , 5 − 29)

The asymptotes for a non-translated hyperbola are y = ±𝑎

𝑏𝑥,

so here they are y − 5 = ±2

5(𝑥 − 3) (Yes, you could convert it to slope-intercept form)

CW # 2

0) h = k = Center = ( , )1) Transverse is 2) 𝒂𝟐 = , a = 𝒃𝟐 = , b = 3) Vertices at ( , ) and ( , ) 4) b points at ( , ) and ( , )5) The four points are:

UL: ( , ), LL: ( , ), UR: ( , ), LR: ( , )

6) Draw…and draw!

c = Foci at ( , ) and ( , )

Asymptotes at

h = 2 k = 1 Center = (2 , 1)

Transverse is vertical

a = 3, 𝒂𝟐 = 𝟗 c = 4, 𝒄𝟐 = 𝟏𝟔

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐

𝒃𝟐 = 𝒄𝟐 − 𝒂𝟐

𝒃𝟐 = 𝟏𝟔 − 𝟗 = 𝟕

CW # 3

h = k = Center = ( , )

Transverse is _____________

a = , 𝒂𝟐 = c = , 𝒄𝟐 =

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐

𝒃𝟐 = 𝒄𝟐 − 𝒂𝟐

𝒃𝟐 =

Find the standard form equation of the graphed hyperbola.What are the equations of the asymptotes?What is the value of y when x = 8 ?

h = 2 k = –2 Center = (2 , –2)Transverse is horizontala = 1, 𝒂𝟐 = 𝟏Looking at the graph, if you draw a vertical line intersecting the right vertex,you can see that it also intersects the positive asymptote at (4 , 0), meaningthe upper horizontal line is at y = 0, meaning b = 2.

So the standard form equation is (𝑥 − 2)2

1−

𝑦 + 2 2

4= 1

The asymptotes for this non-translated hyperbola are

y − k = ±𝑏

𝑎(𝑥 − ℎ) so here they are

y − (−2) = ±2

1(𝑥 − 2) or y + 2 = ± 2(𝑥 − 2)

(8 − 2)2

1−

𝑦 + 2 2

4= 1 36 −

𝑦 + 2 2

4= 1 144 − 𝑦 + 2 2 = 4

− 𝑦 + 2 2 = −140 𝑦 + 2 2 = 140 𝑦 + 2 = 140

𝐲 = 𝟗. 𝟖𝟑

CW # 4Find the standard form equation of the graphed hyperbola.What are the equations of the asymptotes?What is the value of x when y = 15 ?

h = k = Center = ( , )Transverse is a = , 𝒂𝟐 =Looking at the graph, if you draw a ______________line intersecting the __________ vertex, you can see that it also intersects the positive asymptote at ( , ), meaning theupper _______________ line is at _________, meaning b = ____So the standard form equation is

The asymptotes for this non-translated hyperbola are

y − k = ± (𝑥 − ℎ) so here they are

Equation:

unit 4.3 conics: hyperbolas p. 333 · 2020. 8. 13. · foci at (± 3 , 0) and vertices at (± 2 ,...

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