x=y2-4y+1

Rewrite the equation in vertex form.

Complete the square for y2-4y+1.

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

a=1,b=-4,c=1

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=-42(1)

Simplify the right side.

Cancel the common factor of 4 and 2.

Factor 2 out of 4.

d=-2⋅22⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=-2⋅22(1)

Cancel the common factor.

d=-2⋅22⋅1

Rewrite the expression.

d=-21

Divide 2 by 1.

d=-1⋅2

d=-1⋅2

d=-1⋅2

Multiply -1 by 2.

d=-2

d=-2

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

Simplify each term.

Cancel the common factor of (-4)2 and 4.

Rewrite -4 as -1(4).

e=1-(-1⋅4)24(1)

Apply the product rule to -1(4).

e=1-(-1)2⋅424(1)

Raise -1 to the power of 2.

e=1-1⋅424(1)

Multiply 42 by 1.

e=1-424(1)

Factor 4 out of 42.

e=1-4⋅44⋅1

Cancel the common factors.

Factor 4 out of 4⋅1.

e=1-4⋅44(1)

Cancel the common factor.

e=1-4⋅44⋅1

Rewrite the expression.

e=1-41

Divide 4 by 1.

e=1-1⋅4

e=1-1⋅4

e=1-1⋅4

Multiply -1 by 4.

e=1-4

e=1-4

Subtract 4 from 1.

e=-3

e=-3

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

(y-2)2-3

(y-2)2-3

Set x equal to the new right side.

x=(y-2)2-3

x=(y-2)2-3

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

a=1

h=-3

k=2

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

Opens Right

Find the vertex (h,k).

(-3,2)

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.

(-114,2)

(-114,2)

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

y=2

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=-134

x=-134

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

Direction: Opens Right

Vertex: (-3,2)

Focus: (-114,2)

Axis of Symmetry: y=2

Directrix: x=-134

Direction: Opens Right

Vertex: (-3,2)

Focus: (-114,2)

Axis of Symmetry: y=2

Directrix: x=-134

Substitute the x value -2 into f(x)=2+-1(-3-x). In this case, the point is (-2,3).

Replace the variable x with -2 in the expression.

f(-2)=2+-1(-3-(-2))

Simplify the result.

Simplify each term.

Multiply -1 by -2.

f(-2)=2+-1(-3+2)

Add -3 and 2.

f(-2)=2+-1⋅-1

Multiply -1 by -1.

f(-2)=2+1

Any root of 1 is 1.

f(-2)=2+1

f(-2)=2+1

Add 2 and 1.

f(-2)=3

The final answer is 3.

y=3

y=3

Convert 3 to decimal.

=3

=3

Substitute the x value -2 into f(x)=2–1(-3-x). In this case, the point is (-2,1).

Replace the variable x with -2 in the expression.

f(-2)=2–1(-3-(-2))

Simplify the result.

Simplify each term.

Multiply -1 by -2.

f(-2)=2–1(-3+2)

Add -3 and 2.

f(-2)=2–1⋅-1

Multiply -1 by -1.

f(-2)=2-1

Any root of 1 is 1.

f(-2)=2-1⋅1

Multiply -1 by 1.

f(-2)=2-1

f(-2)=2-1

Subtract 1 from 2.

f(-2)=1

The final answer is 1.

y=1

y=1

Convert 1 to decimal.

=1

=1

Substitute the x value -1 into f(x)=2+-1(-3-x). In this case, the point is (-1,3.41421356).

Replace the variable x with -1 in the expression.

f(-1)=2+-1(-3-(-1))

Simplify the result.

Simplify each term.

Multiply -1 by -1.

f(-1)=2+-1(-3+1)

Add -3 and 1.

f(-1)=2+-1⋅-2

Multiply -1 by -2.

f(-1)=2+2

f(-1)=2+2

The final answer is 2+2.

y=2+2

y=2+2

Convert 2+2 to decimal.

=3.41421356

=3.41421356

Substitute the x value -1 into f(x)=2–1(-3-x). In this case, the point is (-1,0.58578643).

Replace the variable x with -1 in the expression.

f(-1)=2–1(-3-(-1))

Simplify the result.

Simplify each term.

Multiply -1 by -1.

f(-1)=2–1(-3+1)

Add -3 and 1.

f(-1)=2–1⋅-2

Multiply -1 by -2.

f(-1)=2-2

f(-1)=2-2

The final answer is 2-2.

y=2-2

y=2-2

Convert 2-2 to decimal.

=0.58578643

=0.58578643

Graph the parabola using its properties and the selected points.

xy-32-23-21-13.41-10.59

xy-32-23-21-13.41-10.59

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: (-3,2)

Focus: (-114,2)

Axis of Symmetry: y=2

Directrix: x=-134

xy-32-23-21-13.41-10.59

Graph x=y^2-4y+1