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
