y=a(x-0)2-5
Move all terms containing variables to the left side of the equation.
Subtract a(x-0)2 from both sides of the equation.
y-a(x-0)2=-5
Subtract 0 from x.
y-ax2=-5
y-ax2=-5
Flip the sign on each term of the equation so the term on the right side is positive.
-y+ax2=5
Divide each term by 5 to make the right side equal to one.
-y5+ax25=55
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.
ax25-y5=1
ax25-y5=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=5
b=5
k=0
h=0
The center of a hyperbola follows the form of (h,k). Substitute in the values of h and k.
(0,0)
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.
(5)2+(5)2
Simplify.
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
(512)2+(5)2
Apply the power rule and multiply exponents, (am)n=amn.
512⋅2+(5)2
Combine 12 and 2.
522+(5)2
Cancel the common factor of 2.
Cancel the common factor.
522+(5)2
Divide 1 by 1.
51+(5)2
51+(5)2
Evaluate the exponent.
5+(5)2
5+(5)2
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
5+(512)2
Apply the power rule and multiply exponents, (am)n=amn.
5+512⋅2
Combine 12 and 2.
5+522
Cancel the common factor of 2.
Cancel the common factor.
5+522
Divide 1 by 1.
5+51
5+51
Evaluate the exponent.
5+5
5+5
Add 5 and 5.
10
10
10
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.
(5,0)
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.
(-5,0)
The vertices of a hyperbola follow the form of (h±a,k). Hyperbolas have two vertices.
(5,0),(-5,0)
(5,0),(-5,0)
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.
(10,0)
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.
(-10,0)
The foci of a hyperbola follow the form of (h±a2+b2,k). Hyperbolas have two foci.
(10,0),(-10,0)
(10,0),(-10,0)
Find the eccentricity by using the following formula.
a2+b2a
Substitute the values of a and b into the formula.
(5)2+(5)25
Simplify.
Simplify the numerator.
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
(512)2+(5)25
Apply the power rule and multiply exponents, (am)n=amn.
512⋅2+(5)25
Combine 12 and 2.
522+(5)25
Cancel the common factor of 2.
Cancel the common factor.
522+(5)25
Divide 1 by 1.
51+(5)25
51+(5)25
Evaluate the exponent.
5+(5)25
5+(5)25
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
5+(512)25
Apply the power rule and multiply exponents, (am)n=amn.
5+512⋅25
Combine 12 and 2.
5+5225
Cancel the common factor of 2.
Cancel the common factor.
5+5225
Divide 1 by 1.
5+515
5+515
Evaluate the exponent.
5+55
5+55
Add 5 and 5.
105
105
Combine 10 and 5 into a single radical.
105
Divide 10 by 5.
2
2
2
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.
5210
Simplify.
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
(512)210
Apply the power rule and multiply exponents, (am)n=amn.
512⋅210
Combine 12 and 2.
52210
Cancel the common factor of 2.
Cancel the common factor.
52210
Divide 1 by 1.
5110
5110
Evaluate the exponent.
510
510
Multiply 510 by 1010.
510⋅1010
Combine and simplify the denominator.
Multiply 510 and 1010.
5101010
Raise 10 to the power of 1.
51010110
Raise 10 to the power of 1.
510101101
Use the power rule aman=am+n to combine exponents.
510101+1
Add 1 and 1.
510102
Rewrite 102 as 10.
Use axn=axn to rewrite 10 as 1012.
510(1012)2
Apply the power rule and multiply exponents, (am)n=amn.
5101012⋅2
Combine 12 and 2.
5101022
Cancel the common factor of 2.
Cancel the common factor.
5101022
Divide 1 by 1.
510101
510101
Evaluate the exponent.
51010
51010
51010
Cancel the common factor of 5 and 10.
Factor 5 out of 510.
5(10)10
Cancel the common factors.
Factor 5 out of 10.
5105⋅2
Cancel the common factor.
5105⋅2
Rewrite the expression.
102
102
102
102
102
The asymptotes follow the form y=±b(x-h)a+k because this hyperbola opens left and right.
y=±1⋅x+0
Add 1⋅x and 0.
y=1⋅x
Multiply x by 1.
y=x
y=x
Add -1⋅x and 0.
y=-1⋅x
Rewrite -1x as -x.
y=-x
y=-x
This hyperbola has two asymptotes.
y=x,y=-x
These values represent the important values for graphing and analyzing a hyperbola.
Center: (0,0)
Vertices: (5,0),(-5,0)
Foci: (10,0),(-10,0)
Eccentricity: 2
Focal Parameter: 102
Asymptotes: y=x, y=-x
Graph y=a(x-0)^2-5
