x-5=(y-1)2

Rewrite (y-1)2 as (y-1)(y-1).

x-5=(y-1)(y-1)

Expand (y-1)(y-1) using the FOIL Method.

Apply the distributive property.

x-5=y(y-1)-1(y-1)

Apply the distributive property.

x-5=y⋅y+y⋅-1-1(y-1)

Apply the distributive property.

x-5=y⋅y+y⋅-1-1y-1⋅-1

x-5=y⋅y+y⋅-1-1y-1⋅-1

Simplify and combine like terms.

Simplify each term.

Multiply y by y.

x-5=y2+y⋅-1-1y-1⋅-1

Move -1 to the left of y.

x-5=y2-1⋅y-1y-1⋅-1

Rewrite -1y as -y.

x-5=y2-y-1y-1⋅-1

Rewrite -1y as -y.

x-5=y2-y-y-1⋅-1

Multiply -1 by -1.

x-5=y2-y-y+1

x-5=y2-y-y+1

Subtract y from -y.

x-5=y2-2y+1

x-5=y2-2y+1

x-5=y2-2y+1

Add 5 to both sides of the equation.

x=y2-2y+1+5

Add 1 and 5.

x=y2-2y+6

x=y2-2y+6

Rewrite the equation in vertex form.

Complete the square for y2-2y+6.

Use the form ax2+bx+c, to find the values of a, b, and c.

a=1,b=-2,c=6

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=-22(1)

Simplify the right side.

Cancel the common factor of 2.

Cancel the common factor.

d=-22⋅1

Divide 1 by 1.

d=-1⋅1

d=-1⋅1

Multiply -1 by 1.

d=-1

d=-1

Find the value of e using the formula e=c-b24a.

Simplify each term.

Raise -2 to the power of 2.

e=6-44⋅1

Multiply 4 by 1.

e=6-44

Divide 4 by 4.

e=6-1⋅1

Multiply -1 by 1.

e=6-1

e=6-1

Subtract 1 from 6.

e=5

e=5

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

(y-1)2+5

(y-1)2+5

Set x equal to the new right side.

x=(y-1)2+5

x=(y-1)2+5

Use the vertex form, x=a(y-k)2+h, to determine the values of a, h, and k.

a=1

h=5

k=1

Since the value of a is positive, the parabola opens right.

Opens Right

Find the vertex (h,k).

(5,1)

Find p, the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

14a

Substitute the value of a into the formula.

14⋅1

Cancel the common factor of 1.

Cancel the common factor.

14⋅1

Rewrite the expression.

14

14

14

Find the focus.

The focus of a parabola can be found by adding p to the x-coordinate h if the parabola opens left or right.

(h+p,k)

Substitute the known values of h, p, and k into the formula and simplify.

(214,1)

(214,1)

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

y=1

Find the directrix.

The directrix of a parabola is the vertical line found by subtracting p from the x-coordinate h of the vertex if the parabola opens left or right.

x=h-p

Substitute the known values of p and h into the formula and simplify.

x=194

x=194

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Right

Vertex: (5,1)

Focus: (214,1)

Axis of Symmetry: y=1

Directrix: x=194

Direction: Opens Right

Vertex: (5,1)

Focus: (214,1)

Axis of Symmetry: y=1

Directrix: x=194

Substitute the x value 6 into f(x)=x-5+1. In this case, the point is (6,2).

Replace the variable x with 6 in the expression.

f(6)=(6)-5+1

Simplify the result.

Simplify each term.

Subtract 5 from 6.

f(6)=1+1

Any root of 1 is 1.

f(6)=1+1

f(6)=1+1

Add 1 and 1.

f(6)=2

The final answer is 2.

y=2

y=2

Convert 2 to decimal.

=2

=2

Substitute the x value 6 into f(x)=-x-5+1. In this case, the point is (6,0).

Replace the variable x with 6 in the expression.

f(6)=-(6)-5+1

Simplify the result.

Simplify each term.

Subtract 5 from 6.

f(6)=-1+1

Any root of 1 is 1.

f(6)=-1⋅1+1

Multiply -1 by 1.

f(6)=-1+1

f(6)=-1+1

Add -1 and 1.

f(6)=0

The final answer is 0.

y=0

y=0

Convert 0 to decimal.

=0

=0

Substitute the x value 7 into f(x)=x-5+1. In this case, the point is (7,2.41421356).

Replace the variable x with 7 in the expression.

f(7)=(7)-5+1

Simplify the result.

Subtract 5 from 7.

f(7)=2+1

The final answer is 2+1.

y=2+1

y=2+1

Convert 2+1 to decimal.

=2.41421356

=2.41421356

Substitute the x value 7 into f(x)=-x-5+1. In this case, the point is (7,-0.41421356).

Replace the variable x with 7 in the expression.

f(7)=-(7)-5+1

Simplify the result.

Subtract 5 from 7.

f(7)=-2+1

The final answer is -2+1.

y=-2+1

y=-2+1

Convert -2+1 to decimal.

=-0.41421356

=-0.41421356

Graph the parabola using its properties and the selected points.

xy51626072.417-0.41

xy51626072.417-0.41

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: (5,1)

Focus: (214,1)

Axis of Symmetry: y=1

Directrix: x=194

xy51626072.417-0.41

Graph x-5=(y-1)^2