# Graph f(x)=x^2+k f(x)=x2+k
Move all terms containing variables to the left side of the equation.
Subtract x2 from both sides of the equation.
y-x2=k
Subtract k from both sides of the equation.
y-x2-k=0
Move y.
-x2-k+y=0
-x2-k+y=0
This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.
(y-k)2a2-(x-h)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=1
b=1
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 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.
(1)2+(1)2
Simplify.
One to any power is one.
1+(1)2
One to any power is one.
1+1
Add 1 and 1.
2
2
2
Find the vertices.
The first vertex of a hyperbola can be found by adding a to k.
(h,k+a)
Substitute the known values of h, a, and k into the formula and simplify.
(0,1)
The second vertex of a hyperbola can be found by subtracting a from k.
(h,k-a)
Substitute the known values of h, a, and k into the formula and simplify.
(0,-1)
The vertices of a hyperbola follow the form of (h,k±a). Hyperbolas have two vertices.
(0,1),(0,-1)
(0,1),(0,-1)
Find the foci.
The first focus of a hyperbola can be found by adding c to k.
(h,k+c)
Substitute the known values of h, c, and k into the formula and simplify.
(0,2)
The second focus of a hyperbola can be found by subtracting c from k.
(h,k-c)
Substitute the known values of h, c, and k into the formula and simplify.
(0,-2)
The foci of a hyperbola follow the form of (h,k±a2+b2). Hyperbolas have two foci.
(0,2),(0,-2)
(0,2),(0,-2)
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.
122
Simplify.
One to any power is one.
12
Multiply 12 by 22.
12⋅22
Combine and simplify the denominator.
Multiply 12 and 22.
222
Raise 2 to the power of 1.
2212
Raise 2 to the power of 1.
22121
Use the power rule aman=am+n to combine exponents.
221+1
Add 1 and 1.
222
Rewrite 22 as 2.
Use axn=axn to rewrite 2 as 212.
2(212)2
Apply the power rule and multiply exponents, (am)n=amn.
2212⋅2
Combine 12 and 2.
2222
Cancel the common factor of 2.
Cancel the common factor.
2222
Divide 1 by 1.
221
221
Evaluate the exponent.
22
22
22
22
22
The asymptotes follow the form y=±a(x-h)b+k because this hyperbola opens up and down.
y=±1⋅x+0
Simplify 1⋅x+0.
Add 1⋅x and 0.
y=1⋅x
Multiply x by 1.
y=x
y=x
Simplify -1⋅x+0.
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: (0,1),(0,-1)
Foci: (0,2),(0,-2)
Eccentricity: (0,2),(0,-2)
Focal Parameter: 22
Asymptotes: y=x, y=-x
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