f(x)=x2+k

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 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

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)

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 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

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: (0,1),(0,-1)

Foci: (0,2),(0,-2)

Eccentricity: (0,2),(0,-2)

Focal Parameter: 22

Asymptotes: y=x, y=-x

Graph f(x)=x^2+k