Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter7.4r

Text File | 1995-04-09 | 8KB | 322 lines

 320
à 7.4è The Hyperbola
äè Please find the intercepts of the following hyperbolas.
#âSêïxìèyìêêêèyìèxì
#êêè── - ──ï=ï1êêè ── - ──ï=ï1
êêè9è 4êêêè 4è 1
#êêê xìï=ï9êêêïyìï=ï4
êêêïxï=ï± 3êêê yï=ï± 2
êêThe intercepts areêêThe intercepts are
êê(3, 0) and (-3, 0)êê(0, 2) and (0, -2)
#éS The equation, xì/9 - yì/4 = 1, has only x-intercepts.ïIf you
let y = 0, then you can solve for x.

#êê xìè0ì
#êê ── - ──ï=ï1
êê 9è 4êêêë The x-intercepts are
#êêë xìï=ï9êêê(3, 0) and (-3, 0)
#êêêê ┌─
#êêêxï=ï± á9
êêêxï=ï± 3

The equation does not have any y-intercepts.ïOn the other hand, the
#equation, yì/4 - xì/1 = 1, has only y-intercepts.ïLetting x = 0 allows
you to solve for y.

#êê yìè0ì
#êê ── - ──ï=ï1
êê 4è 1êêêë The y-intercepts are
#êêë yìï=ï4êêê(0, 2) and (0, -2)
#êêêê ┌─
#êêêyï=ï± á4
êêêyï=ï± 2

This equation does not have any x-intercepts.ïIf you let y = 0 and
solve for x, then you end up with the square root of a negative number.
This is an indication that the curve does not cross the x-axis.ïThus,
there are no x-intercepts.
# 1êêêêêë xìèyì
#êë Find the intercepts of the hyperbola ── - ──ï=ï1
êêêêêêê 9è 1

êA)ï(0, 3), (0, -3)êêêC)ï(3, 0), (-3, 0)

êB)ï(0, 1), (0, -1)êêêD)ïå
#üêêêèxìè0ì
#êêêêï── - ──ï=ï1
êêêêï9è 1
#êêêêêxìï=ï9
#êêêêêêï┌─
#êêêêê xï=ï± á9ï=ï± 3
êêèThe x-intercepts are (3, 0) and (-3, 0)
Ç C
# 2êêêêêë yìèxì
#êë Find the intercepts of the hyperbola ── - ──ï=ï1
êêêêêêê 1è 4

ê A)ï(0, 2), (0, -2)êêëC)ï(2, 0), (-2, 0)

ê B)ï(0, 1), (0, -1)êêëD)ïå
#üêêêèyìè0ì
#êêêêï── - ──ï=ï1
êêêêï1è 9
#êêêêêyìï=ï1
#êêêêêêï┌─
#êêêêê yï=ï± á1ï=ï± 1
êêèThe x-intercepts are (0, 1) and (0, -1)
Ç B
 3
#êë Find the intercepts of the hyperbola xì - yìï=ï1


êA)ï(1, 0), (-1, 0)êêêC)ï(0, 1), (0, -1)

êB)ï(2, 0), (-2, 0)êêêD)ïå
#üêêêèxì - 0ìï=ï1
#êêêêêxìï=ï1
#êêêêêêï┌─
#êêêêê xï=ï± á1ï=ï± 1
êêèThe x-intercepts are (1, 0) and (-1, 0)
Ç A
äè Find the equations of the asymptotes that the following
hyperbolas approach.
#âSê xìèyìêêêè yìèxì
#êêï── - ──ï=ï1êêë── - ──ï=ï1
êêï9è 4êêêë4è 1
êêaï=ï3, bï=ï2êêïaï=ï1, bï=ï2
êêêë2xêêêêï2x
#ë Asymptotesïyï=ï± ──êèAsymptotesïyï=ï± ──
êêêë3êêêêè1
éS An asymptote is a line that a curve gets closer and closer to.
Each hyperbola has two asymptotes. These will be helpful in drawing the
graphs of the hyperbolas.ïTo find the equations of the asymptotes, you
should solve for y.

#êêêè yìèxì
#êêêè ── - ──ï=ï1
#êêêè bìèaì

#êêêè yìëxì
#êêêè ──ï=ï── + 1
#êêêè bìëaì

#êêêêè bìxì
#êêêè yìï=ï──── + bì
#êêêêëaì
#êêêêë ┌──────────
#êêêêë │bìxì
#êêêêë │──── + bì
#êêêëyï=ï± á aì

#As x gets larger, the bì term becomes insignificant, and y approaches
#êï┌─────────
#êï│bìxìêëb
#êï│──── + bìï=ï± ─ x.
# yï=ï± á aìêë aêêêb
#Thus, the equations of the asymptotes are y = ± ─ x.
êêêêêêë a
#In the example, xì/9 - yì/4 = 1, a is 3, and b is 2.ïThe asymptotes are
seen to be y = ± 2/3 x.
#In the example, yì/4 - xì/1 = 1, a is 1, and b is 2.ïThe asymptotes are
seen to be y = ± 2/1 x.
# 4êêêêêêêxìèyì
#èFind the equations of the asymptotes of the hyperbola, ── - ──ï=ï1
êêêêêêêêï9è 1

êïA)ïyï=ï± 9êêêè C)ïyï=ï± 1/3 x

êïB)ïyï=ï± 3xêêêèD)ïå
#üêêêèxìèyì
#êêêêï── - ──ï=ï1
êêêêï9è 1
êêêëaï=ï3 and bï=ï1
êêêêêè1
#êêêêïyï=ï± ─ x
êêêêêè3
Ç C
# 5êêêêêêêyìèxì
#èFind the equations of the asymptotes of the hyperbola, ── - ──ï=ï1
êêêêêêêêï1è 4

êïA)ïyï=ï± 2xêêêêC)ïyï=ï± x

êïB)ïyï=ï± 1/2 xêêêè D)ïå
#üêêêèyìèxì
#êêêêï── - ──ï=ï1
êêêêï1è 4
êêêëaï=ï2 and bï=ï1
êêêêêè1
#êêêêïyï=ï± ─ x
êêêêêè2
Ç B
 6
#èFind the equations of the asymptotes of the hyperbola, xì - yìï=ï1


êïA)ïyï=ï± 2xêêêêC)ïyï=ï± x

êïB)ïyï=ï± 1/2 xêêêè D)ïå
ü
#êêêêïxì - yìï=ï1
êêêëaï=ï2 and bï=ï1
êêêêêè1
#êêêêïyï=ï± ─ x
êêêêêè1
Ç C
äè Find the coordinates of the corners of the fundamental
rectangle for the given hyperbolas.
#âêêê xìèyì
#êêêê ── - ──ï=ï1
êêêê 9è 4
êêêëaï=ï3 and bï=ï2

êêThe corners of the fundamental rectangle are
êêè(3, 2), (3, -2), (-3, 2), and (-3, -2)
éS The fundamental rectangle is very helpful in drawing the graph
of a hyperbola.ïThis is because the asymptotes coincide with the
diagonals of the rectangle.ïTo graph a hyperbola, we will draw the
fundamental rectangle, draw the diagonals of the rectangle, plot the
intercepts, then draw smooth curves through the intercepts. The curves
should approach the asymptotes.ïThe corners of the fundamental
rectangle are given by (a, b), (a, -b), (-a, b), and (-a, -b) where a
and b are numbers from the following equations.
#êêëxìèyìêëyìèxì
#êêë── - ──ï=ï1ïorï── - ──ï=ï1.
#êêëaìèbìêëbìèaì
 7è Find the corners of the fundamental rectangle for the
#êè xìèyì
#hyperbola, ── - ──ï=ï1.
êè 9è 1

èA)ï(3, 1), (3, -1), (-3, 1), (-3, -1)êè C)ï(1, 3), (-1, 3)
èB)ï(9, 1), (-9, 1)êêêêïD)ïå
ü


êêè(3, 1), (3, -1), (-3, 1), and (-3, -1)
Ç A
 8è Find the corners of the fundamental rectangle for the
#êè yìèxì
#hyperbola, ── - ──ï=ï1.
êè 1è 4

 A) (2, 1),(1, 2),(-1, 2),(1, -2)ëC) (2, 1),(2, -1),(-2, 1),(-2, -1)
 B) (1, 2),(1, -2),(-1, 2),(-1, -2)èD) å
ü


êêè(2, 1), (2, -1), (-2, 1), and (-2, -1)
Ç C
 9è Find the corners of the fundamental rectangle for the
#hyperbola, xì - yìï=ï1.

 A) (1, 1),(1, -1),(-1, 1),(-1, -1)ëC) (1, 2),(1, -1),(1, -2),(1, 1)

 B) (2, 1),(-2, 1),(1, 2),(-1, -2)ë D) å
ü


êêè(1, 1), (1, -1), (-1, 1), and (-1, -1)
Ç A
äè Please graph the following hyperbolas.
â


êêêëPlease see Details.
#éSëyìèxì
#ëGraph ── - ── = 1.
êë4è 1

Intercepts (0, 2)(0, -1)êêêè Corners of (1, 2)(1, -2)
Asymptotesïy = ± 2xêêêê rectangleï(-1,2)(-1,-2)
@fig7401.bmp,200,45
# 10êGraph the hyperbola yì/1 - xì/4 = 1
@fig7402.bmp,28,225
@fig7403.bmp,232,225
@fig7404.bmp,450,225
#üëyìèxì
#è Graph ── - ──ï=ï1
êè 1è 4

Intercepts (0,1)(0,-1)êêêë Corners of (2,1)(2,-1)
Asymptotesïy = ± 1/2 xêêêërectangleï(-2,1)(-2,-1)
@fig7405.bmp,3450,572
Ç B
#ï11ê Graph the hyperbola xì - yìï=ï1
@fig7406.bmp,28,225
@fig7407.bmp,232,225
@fig7408.bmp,450,225
#üGraph xì - yìï=ï1


Intercepts (1, 0)(-1, 0)êêêëCorners of
Asymptotesïy = ± xêêêêèrectangle (1,1)(1,-1)
êêêêêêêêï(-1,1)(-1,-1)
@fig7409.bmp,3450,572
Ç A
äè Please identify each of the following equations as a
parabola, circle, ellipse, or hyperbola.
âS

#êè 1)ïxì + yìï=ï4êThis equation is a circle.

#êè 2)ï3xì - 4yìï=ï12è This equation is a hyperbola.
éS The parabola, circle, ellipse, and hyperbola are four curves
that are sometimes called the conic sections.ïThey are called this,
because if you slice a cone and look at the sectional view, it will
always be one of ç curves.ïThere are some exceptions.ïFor example,
if you slice through the vertex of the cone, you will only get a point.
Generally, however, when you slice a cone, you get one of ç curves.
It is always true that all four of ç curves can be described by one
general quadratic equation.

#êêè axì + byì + cxy + dx + ey + fï=ï0

Thus, every conic section has an equation of this form.ïNot every equa-
tion of this form represents a conic section, however, since some equa-
tions of this form have no graph.ïIf c = 0 in this general equation,
and the equation represents a conic section, then you can predict the
kind of conic section by looking at the coefficients.

è1)ïIf a = b, then your conic is a circle.
#è2)ïIf a ƒ b and they have the same sign, you will get an ellipse.
è3)ïIf a or b is zero, then you will get a parabola.
è4)ïIf a and b have different signs, then you will get a hyperbola.

#In the equation, xì + yì = 4, a = b = 1, and the equation represents a
#circle.ïIn the equation, 3xì - 4yì = 12, a = 3 and b = -4.ïSince ç
two numbers have different signs, the equation is a hyperbola.
 12

#êêë Identify 4xì + 9yì + 3xï=ï5.

èA)ïparabolaêB)ïcircleêC)ïellipseêD)ïhyperbola
ü


 Since a = 4 and b = 9 are not equal but have the same sign, this equa-
 tion is an ellipse.
Ç C
 13

#êêëIdentify xì + 3x + y + 5ï=ï0.

èA)ïparabolaêB)ïcircleêC)ïellipseêD)ïhyperbola
ü


êê Since b = 0, this equation is a parabola.
Ç A
 14

#êêëIdentify 2xì + 2yì + 2y - 3yï=ï10.

èA)ïparabolaêB)ïcircleêC)ïellipseêD)ïhyperbola
ü


êêSince a = b = 2, this equation is a circle.
Ç B