(x-1)(x-5)

Rewrite the equation in vertex form.

Complete the square for (x-1)(x-5).

Expand (x-1)(x-5) using the FOIL Method.

Apply the distributive property.

x(x-5)-1(x-5)

Apply the distributive property.

x⋅x+x⋅-5-1(x-5)

Apply the distributive property.

x⋅x+x⋅-5-1x-1⋅-5

x⋅x+x⋅-5-1x-1⋅-5

Simplify and combine like terms.

Simplify each term.

Multiply x by x.

x2+x⋅-5-1x-1⋅-5

Move -5 to the left of x.

x2-5⋅x-1x-1⋅-5

Rewrite -1x as -x.

x2-5x-x-1⋅-5

Multiply -1 by -5.

x2-5x-x+5

x2-5x-x+5

Subtract x from -5x.

x2-6x+5

x2-6x+5

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

a=1,b=-6,c=5

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=-62(1)

Simplify the right side.

Cancel the common factor of 6 and 2.

Factor 2 out of 6.

d=-2⋅32⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=-2⋅32(1)

Cancel the common factor.

d=-2⋅32⋅1

Rewrite the expression.

d=-31

Divide 3 by 1.

d=-1⋅3

d=-1⋅3

d=-1⋅3

Multiply -1 by 3.

d=-3

d=-3

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

Simplify each term.

Raise -6 to the power of 2.

e=5-364⋅1

Multiply 4 by 1.

e=5-364

Divide 36 by 4.

e=5-1⋅9

Multiply -1 by 9.

e=5-9

e=5-9

Subtract 9 from 5.

e=-4

e=-4

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

(x-3)2-4

(x-3)2-4

Set y equal to the new right side.

y=(x-3)2-4

y=(x-3)2-4

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

a=1

h=3

k=-4

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

Opens Up

Find the vertex (h,k).

(3,-4)

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 y-coordinate k if the parabola opens up or down.

(h,k+p)

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

(3,-154)

(3,-154)

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

x=3

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting p from the y-coordinate k of the vertex if the parabola opens up or down.

y=k-p

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

y=-174

y=-174

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

Direction: Opens Up

Vertex: (3,-4)

Focus: (3,-154)

Axis of Symmetry: x=3

Directrix: y=-174

Direction: Opens Up

Vertex: (3,-4)

Focus: (3,-154)

Axis of Symmetry: x=3

Directrix: y=-174

Replace the variable x with 2 in the expression.

f(2)=((2)-1)((2)-5)

Simplify the result.

Subtract 1 from 2.

f(2)=1((2)-5)

Multiply (2)-5 by 1.

f(2)=(2)-5

Subtract 5 from 2.

f(2)=-3

The final answer is -3.

-3

-3

The y value at x=2 is -3.

y=-3

Replace the variable x with 1 in the expression.

f(1)=((1)-1)((1)-5)

Simplify the result.

Subtract 1 from 1.

f(1)=0((1)-5)

Subtract 5 from 1.

f(1)=0⋅-4

Multiply 0 by -4.

f(1)=0

The final answer is 0.

0

0

The y value at x=1 is 0.

y=0

Replace the variable x with 4 in the expression.

f(4)=((4)-1)((4)-5)

Simplify the result.

Subtract 1 from 4.

f(4)=3((4)-5)

Subtract 5 from 4.

f(4)=3⋅-1

Multiply 3 by -1.

f(4)=-3

The final answer is -3.

-3

-3

The y value at x=4 is -3.

y=-3

Replace the variable x with 5 in the expression.

f(5)=((5)-1)((5)-5)

Simplify the result.

Subtract 1 from 5.

f(5)=4((5)-5)

Subtract 5 from 5.

f(5)=4⋅0

Multiply 4 by 0.

f(5)=0

The final answer is 0.

0

0

The y value at x=5 is 0.

y=0

Graph the parabola using its properties and the selected points.

xy102-33-44-350

xy102-33-44-350

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (3,-4)

Focus: (3,-154)

Axis of Symmetry: x=3

Directrix: y=-174

xy102-33-44-350

Graph (x-1)(x-5)