# Graph 16x^2-9y^2+64x-54-161=0 16×2-9y2+64x-54-161=0
Find the standard form of the hyperbola.
Move all terms not containing a variable to the right side of the equation.
Add 54 to both sides of the equation.
16×2-9y2+64x-161=54
Add 161 to both sides of the equation.
16×2-9y2+64x=54+161
Add 54 and 161.
16×2-9y2+64x=215
16×2-9y2+64x=215
Complete the square for 16×2+64x.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=16,b=64,c=0
Consider the vertex form of a parabola.
a(x+d)2+e
Substitute the values of a and b into the formula d=b2a.
d=642(16)
Simplify the right side.
Cancel the common factor of 64 and 2.
Factor 2 out of 64.
d=2⋅322⋅16
Cancel the common factors.
Factor 2 out of 2⋅16.
d=2⋅322(16)
Cancel the common factor.
d=2⋅322⋅16
Rewrite the expression.
d=3216
d=3216
d=3216
Cancel the common factor of 32 and 16.
Factor 16 out of 32.
d=16⋅216
Cancel the common factors.
Factor 16 out of 16.
d=16⋅216(1)
Cancel the common factor.
d=16⋅216⋅1
Rewrite the expression.
d=21
Divide 2 by 1.
d=2
d=2
d=2
d=2
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 64 to the power of 2.
e=0-40964⋅16
Multiply 4 by 16.
e=0-409664
Divide 4096 by 64.
e=0-1⋅64
Multiply -1 by 64.
e=0-64
e=0-64
Subtract 64 from 0.
e=-64
e=-64
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
16(x+2)2-64
16(x+2)2-64
Substitute 16(x+2)2-64 for 16×2+64x in the equation 16×2-9y2+64x=215.
16(x+2)2-64-9y2=215
Move -64 to the right side of the equation by adding 64 to both sides.
16(x+2)2-9y2=215+64
Add 215 and 64.
16(x+2)2-9y2=279
Divide each term by 279 to make the right side equal to one.
16(x+2)2279-9y2279=279279
Simplify each term in the equation in order to set the right side equal to 1. The standard form of an ellipse or hyperbola requires the right side of the equation be 1.
(x+2)227916-y231=1
(x+2)227916-y231=1
This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.
(x-h)2a2-(y-k)2b2=1
Match the values in this hyperbola to those of the standard form. The variable h represents the x-offset from the origin, k represents the y-offset from origin, a.
a=3314
b=31
k=0
h=-2
The center of a hyperbola follows the form of (h,k). Substitute in the values of h and k.
(-2,0)
Find c, the distance from the center to a focus.
Find the distance from the center to a focus of the hyperbola by using the following formula.
a2+b2
Substitute the values of a and b in the formula.
(3314)2+(31)2
Simplify.
Use the power rule (ab)n=anbn to distribute the exponent.
Apply the product rule to 3314.
(331)242+(31)2
Apply the product rule to 331.
3231242+(31)2
3231242+(31)2
Simplify the numerator.
Raise 3 to the power of 2.
931242+(31)2
Rewrite 312 as 31.
Use axn=axn to rewrite 31 as 3112.
9(3112)242+(31)2
Apply the power rule and multiply exponents, (am)n=amn.
9⋅3112⋅242+(31)2
Combine 12 and 2.
9⋅312242+(31)2
Cancel the common factor of 2.
Cancel the common factor.
9⋅312242+(31)2
Divide 1 by 1.
9⋅31142+(31)2
9⋅31142+(31)2
Evaluate the exponent.
9⋅3142+(31)2
9⋅3142+(31)2
9⋅3142+(31)2
Simplify by cancelling exponent with radical.
Raise 4 to the power of 2.
9⋅3116+(31)2
Multiply 9 by 31.
27916+(31)2
Rewrite 312 as 31.
Use axn=axn to rewrite 31 as 3112.
27916+(3112)2
Apply the power rule and multiply exponents, (am)n=amn.
27916+3112⋅2
Combine 12 and 2.
27916+3122
Cancel the common factor of 2.
Cancel the common factor.
27916+3122
Divide 1 by 1.
27916+311
27916+311
Evaluate the exponent.
27916+31
27916+31
27916+31
To write 31 as a fraction with a common denominator, multiply by 1616.
27916+31⋅1616
Combine 31 and 1616.
27916+31⋅1616
Combine the numerators over the common denominator.
279+31⋅1616
Simplify the numerator.
Multiply 31 by 16.
279+49616
Add 279 and 496.
77516
77516
Rewrite 77516 as 77516.
77516
Simplify the numerator.
Rewrite 775 as 52⋅31.
Factor 25 out of 775.
25(31)16
Rewrite 25 as 52.
52⋅3116
52⋅3116
Pull terms out from under the radical.
53116
53116
Simplify the denominator.
Rewrite 16 as 42.
53142
Pull terms out from under the radical, assuming positive real numbers.
5314
5314
5314
5314
Find the vertices.
The first vertex of a hyperbola can be found by adding a to h.
(h+a,k)
Substitute the known values of h, a, and k into the formula and simplify.
(-2+3314,0)
The second vertex of a hyperbola can be found by subtracting a from h.
(h-a,k)
Substitute the known values of h, a, and k into the formula and simplify.
(-2-3314,0)
The vertices of a hyperbola follow the form of (h±a,k). Hyperbolas have two vertices.
(-2+3314,0),(-2-3314,0)
(-2+3314,0),(-2-3314,0)
Find the foci.
The first focus of a hyperbola can be found by adding c to h.
(h+c,k)
Substitute the known values of h, c, and k into the formula and simplify.
(-2+5314,0)
The second focus of a hyperbola can be found by subtracting c from h.
(h-c,k)
Substitute the known values of h, c, and k into the formula and simplify.
(-2-5314,0)
The foci of a hyperbola follow the form of (h±a2+b2,k). Hyperbolas have two foci.
(-2+5314,0),(-2-5314,0)
(-2+5314,0),(-2-5314,0)
Find the eccentricity.
Find the eccentricity by using the following formula.
a2+b2a
Substitute the values of a and b into the formula.
(3314)2+(31)23314
Simplify.
Multiply the numerator by the reciprocal of the denominator.
(3314)2+(31)24331
Use the power rule (ab)n=anbn to distribute the exponent.
Apply the product rule to 3314.
(331)242+(31)24331
Apply the product rule to 331.
3231242+(31)24331
3231242+(31)24331
Simplify the numerator.
Raise 3 to the power of 2.
931242+(31)24331
Rewrite 312 as 31.
Use axn=axn to rewrite 31 as 3112.
9(3112)242+(31)24331
Apply the power rule and multiply exponents, (am)n=amn.
9⋅3112⋅242+(31)24331
Combine 12 and 2.
9⋅312242+(31)24331
Cancel the common factor of 2.
Cancel the common factor.
9⋅312242+(31)24331
Divide 1 by 1.
9⋅31142+(31)24331
9⋅31142+(31)24331
Evaluate the exponent.
9⋅3142+(31)24331
9⋅3142+(31)24331
9⋅3142+(31)24331
Simplify by cancelling exponent with radical.
Raise 4 to the power of 2.
9⋅3116+(31)24331
Multiply 9 by 31.
27916+(31)24331
Rewrite 312 as 31.
Use axn=axn to rewrite 31 as 3112.
27916+(3112)24331
Apply the power rule and multiply exponents, (am)n=amn.
27916+3112⋅24331
Combine 12 and 2.
27916+31224331
Cancel the common factor of 2.
Cancel the common factor.
27916+31224331
Divide 1 by 1.
27916+3114331
27916+3114331
Evaluate the exponent.
27916+314331
27916+314331
27916+314331
To write 31 as a fraction with a common denominator, multiply by 1616.
27916+31⋅16164331
Combine 31 and 1616.
27916+31⋅16164331
Combine the numerators over the common denominator.
279+31⋅16164331
Simplify the numerator.
Multiply 31 by 16.
279+496164331
Add 279 and 496.
775164331
775164331
Rewrite 77516 as 77516.
77516⋅4331
Simplify the numerator.
Rewrite 775 as 52⋅31.
Factor 25 out of 775.
25(31)16⋅4331
Rewrite 25 as 52.
52⋅3116⋅4331
52⋅3116⋅4331
Pull terms out from under the radical.
53116⋅4331
53116⋅4331
Simplify the denominator.
Rewrite 16 as 42.
53142⋅4331
Pull terms out from under the radical, assuming positive real numbers.
5314⋅4331
5314⋅4331
Simplify terms.
Cancel the common factor of 31.
Factor 31 out of 531.
31⋅54⋅4331
Factor 31 out of 331.
31⋅54⋅431⋅3
Cancel the common factor.
31⋅54⋅431⋅3
Rewrite the expression.
54⋅43
54⋅43
Cancel the common factor of 4.
Cancel the common factor.
54⋅43
Rewrite the expression.
5(13)
5(13)
Combine 5 and 13.
53
53
53
53
Find the focal parameter.
Find the value of the focal parameter of the hyperbola by using the following formula.
b2a2+b2
Substitute the values of b and a2+b2 in the formula.
3125314
Simplify.
Cancel the common factor of 312 and 31.
Factor 31 out of 312.
31315314
Cancel the common factors.
Factor 31 out of 531.
313131⋅54
Cancel the common factor.
313131⋅54
Rewrite the expression.
3154
3154
3154
Multiply the numerator by the reciprocal of the denominator.
315⋅14
Multiply 315⋅14.
Multiply 315 and 14.
315⋅4
Multiply 5 by 4.
3120
3120
3120
3120
The asymptotes follow the form y=±b(x-h)a+k because this hyperbola opens left and right.
y=±43⋅(x-(-2))+0
Simplify to find the first asymptote.
Remove parentheses.
y=43⋅(x-(-2))+0
Simplify 43⋅(x-(-2))+0.
Simplify the expression.
Add 43⋅(x-(-2)) and 0.
y=43⋅(x-(-2))
Multiply -1 by -2.
y=43⋅(x+2)
y=43⋅(x+2)
Apply the distributive property.
y=43x+43⋅2
Combine 43 and x.
y=4×3+43⋅2
Multiply 43⋅2.
Combine 43 and 2.
y=4×3+4⋅23
Multiply 4 by 2.
y=4×3+83
y=4×3+83
y=4×3+83
y=4×3+83
Simplify to find the second asymptote.
Remove parentheses.
y=-43⋅(x-(-2))+0
Simplify -43⋅(x-(-2))+0.
Simplify the expression.
Add -43⋅(x-(-2)) and 0.
y=-43⋅(x-(-2))
Multiply -1 by -2.
y=-43⋅(x+2)
y=-43⋅(x+2)
Apply the distributive property.
y=-43x-43⋅2
Combine x and 43.
y=-x⋅43-43⋅2
Multiply -43⋅2.
Multiply 2 by -1.
y=-x⋅43-2(43)
Combine -2 and 43.
y=-x⋅43+-2⋅43
Multiply -2 by 4.
y=-x⋅43+-83
y=-x⋅43+-83
Simplify each term.
Move 4 to the left of x.
y=-4⋅x3+-83
Move the negative in front of the fraction.
y=-4×3-83
y=-4×3-83
y=-4×3-83
y=-4×3-83
This hyperbola has two asymptotes.
y=4×3+83,y=-4×3-83
These values represent the important values for graphing and analyzing a hyperbola.
Center: (-2,0)
Vertices: (-2+3314,0),(-2-3314,0)
Foci: (-2+5314,0),(-2-5314,0)
Eccentricity: 53
Focal Parameter: 3120
Asymptotes: y=4×3+83, y=-4×3-83
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