# Graph x=y^2-4y+1

x=y2-4y+1
Find the properties of the given parabola.
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
Select a few x values, and plug them into the equation to find the corresponding y values. The x values should be selected around the vertex.
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)
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
f(-2)=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)
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
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)
f(-1)=2+-1⋅-2
Multiply -1 by -2.
f(-1)=2+2
f(-1)=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)
f(-1)=2–1⋅-2
Multiply -1 by -2.
f(-1)=2-2
f(-1)=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