# Graph (x-1)(x-5) (x-1)(x-5)
Find the properties of the given parabola.
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
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.
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
-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
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
-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
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)     