y+1=-112⋅(x-3)2

Simplify -112⋅(x-3)2.

Rewrite (x-3)2 as (x-3)(x-3).

y+1=-112⋅((x-3)(x-3))

Expand (x-3)(x-3) using the FOIL Method.

Apply the distributive property.

y+1=-112⋅(x(x-3)-3(x-3))

Apply the distributive property.

y+1=-112⋅(x⋅x+x⋅-3-3(x-3))

Apply the distributive property.

y+1=-112⋅(x⋅x+x⋅-3-3x-3⋅-3)

y+1=-112⋅(x⋅x+x⋅-3-3x-3⋅-3)

Simplify and combine like terms.

Simplify each term.

Multiply x by x.

y+1=-112⋅(x2+x⋅-3-3x-3⋅-3)

Move -3 to the left of x.

y+1=-112⋅(x2-3⋅x-3x-3⋅-3)

Multiply -3 by -3.

y+1=-112⋅(x2-3x-3x+9)

y+1=-112⋅(x2-3x-3x+9)

Subtract 3x from -3x.

y+1=-112⋅(x2-6x+9)

y+1=-112⋅(x2-6x+9)

Apply the distributive property.

y+1=-112×2-112(-6x)-112⋅9

Simplify.

Combine x2 and 112.

y+1=-x212-112(-6x)-112⋅9

Cancel the common factor of 6.

Move the leading negative in -112 into the numerator.

y+1=-x212+-112(-6x)-112⋅9

Factor 6 out of 12.

y+1=-x212+-16(2)(-6x)-112⋅9

Factor 6 out of -6x.

y+1=-x212+-16(2)(6(-x))-112⋅9

Cancel the common factor.

y+1=-x212+-16⋅2(6(-x))-112⋅9

Rewrite the expression.

y+1=-x212+-12(-x)-112⋅9

y+1=-x212+-12(-x)-112⋅9

Combine -12 and x.

y+1=-x212–x2-112⋅9

Cancel the common factor of 3.

Move the leading negative in -112 into the numerator.

y+1=-x212–x2+-112⋅9

Factor 3 out of 12.

y+1=-x212–x2+-13(4)⋅9

Factor 3 out of 9.

y+1=-x212–x2+-13⋅4⋅(3⋅3)

Cancel the common factor.

y+1=-x212–x2+-13⋅4⋅(3⋅3)

Rewrite the expression.

y+1=-x212–x2+-14⋅3

y+1=-x212–x2+-14⋅3

Combine -14 and 3.

y+1=-x212–x2+-1⋅34

Multiply -1 by 3.

y+1=-x212–x2+-34

y+1=-x212–x2+-34

Simplify each term.

Move the negative in front of the fraction.

y+1=-x212–x2+-34

Multiply –x2.

Multiply -1 by -1.

y+1=-x212+1×2+-34

Multiply x2 by 1.

y+1=-x212+x2+-34

y+1=-x212+x2+-34

Move the negative in front of the fraction.

y+1=-x212+x2-34

y+1=-x212+x2-34

y+1=-x212+x2-34

Move all terms not containing y to the right side of the equation.

Subtract 1 from both sides of the equation.

y=-x212+x2-34-1

To write -1 as a fraction with a common denominator, multiply by 44.

y=-x212+x2-34-1⋅44

Combine -1 and 44.

y=-x212+x2-34+-1⋅44

Combine the numerators over the common denominator.

y=-x212+x2+-3-1⋅44

Simplify the numerator.

Multiply -1 by 4.

y=-x212+x2+-3-44

Subtract 4 from -3.

y=-x212+x2+-74

y=-x212+x2+-74

Move the negative in front of the fraction.

y=-x212+x2-74

y=-x212+x2-74

y=-x212+x2-74

Rewrite the equation in vertex form.

Complete the square for -x212+x2-74.

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

a=-112,b=12,c=-74

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=122(-112)

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

d=12⋅12⋅(-1(112))

Cancel the common factor of 1 and -1.

Rewrite 1 as -1(-1).

d=12⋅-1⋅-12⋅(-1(112))

Move the negative in front of the fraction.

d=12⋅(-12(112))

d=12⋅(-12(112))

Combine 2 and 112.

d=12⋅(-1212)

Cancel the common factor of 2 and 12.

Factor 2 out of 2.

d=12⋅(-12(1)12)

Cancel the common factors.

Factor 2 out of 12.

d=12⋅(-12⋅12⋅6)

Cancel the common factor.

d=12⋅(-12⋅12⋅6)

Rewrite the expression.

d=12⋅(-116)

d=12⋅(-116)

d=12⋅(-116)

Multiply.

Multiply the numerator by the reciprocal of the denominator.

d=12⋅(-(1⋅6))

Multiply 6 by 1.

d=12⋅(-1⋅6)

Multiply -1 by 6.

d=12⋅-6

d=12⋅-6

Cancel the common factor of 2.

Factor 2 out of -6.

d=12⋅(2(-3))

Cancel the common factor.

d=12⋅(2⋅-3)

Rewrite the expression.

d=-3

d=-3

d=-3

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

Simplify each term.

Simplify the numerator.

Apply the product rule to 12.

e=-74-12224⋅(-1(112))

One to any power is one.

e=-74-1224⋅(-1(112))

Raise 2 to the power of 2.

e=-74-144⋅(-1(112))

e=-74-144⋅(-1(112))

Simplify the denominator.

Multiply 4 by -1.

e=-74-14-4(112)

Combine -4 and 112.

e=-74-14-412

e=-74-14-412

Reduce the expression by cancelling the common factors.

Cancel the common factor of -4 and 12.

Factor 4 out of -4.

e=-74-144(-1)12

Cancel the common factors.

Factor 4 out of 12.

e=-74-144⋅-14⋅3

Cancel the common factor.

e=-74-144⋅-14⋅3

Rewrite the expression.

e=-74-14-13

e=-74-14-13

e=-74-14-13

Move the negative in front of the fraction.

e=-74-14-13

e=-74-14-13

Multiply the numerator by the reciprocal of the denominator.

e=-74-(14⋅(-1⋅3))

Multiply -1 by 3.

e=-74-(14⋅-3)

Combine 14 and -3.

e=-74–34

Move the negative in front of the fraction.

e=-74+34

Multiply –34.

Multiply -1 by -1.

e=-74+1(34)

Multiply 34 by 1.

e=-74+34

e=-74+34

e=-74+34

Combine the numerators over the common denominator.

e=-7+34

Add -7 and 3.

e=-44

Divide -4 by 4.

e=-1

e=-1

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

-112⋅(x-3)2-1

-112⋅(x-3)2-1

Set y equal to the new right side.

y=-112⋅(x-3)2-1

y=-112⋅(x-3)2-1

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

a=-112

h=3

k=-1

Since the value of a is negative, the parabola opens down.

Opens Down

Find the vertex (h,k).

(3,-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⋅(-112)

Simplify.

Cancel the common factor of 1 and -1.

Rewrite 1 as -1(-1).

-1(-1)4⋅(-112)

Move the negative in front of the fraction.

-14(112)

-14(112)

Combine 4 and 112.

-1412

Cancel the common factor of 4 and 12.

Factor 4 out of 4.

-14(1)12

Cancel the common factors.

Factor 4 out of 12.

-14⋅14⋅3

Cancel the common factor.

-14⋅14⋅3

Rewrite the expression.

-113

-113

-113

Multiply the numerator by the reciprocal of the denominator.

-(1⋅3)

Multiply 3 by 1.

-1⋅3

Multiply -1 by 3.

-3

-3

-3

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,-4)

(3,-4)

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=2

y=2

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

Direction: Opens Down

Vertex: (3,-1)

Focus: (3,-4)

Axis of Symmetry: x=3

Directrix: y=2

Direction: Opens Down

Vertex: (3,-1)

Focus: (3,-4)

Axis of Symmetry: x=3

Directrix: y=2

Replace the variable x with 2 in the expression.

f(2)=-(2)212+22-74

Simplify the result.

Simplify each term.

Raise 2 to the power of 2.

f(2)=-412+22-74

Cancel the common factor of 4 and 12.

Factor 4 out of 4.

f(2)=-4(1)12+22-74

Cancel the common factors.

Factor 4 out of 12.

f(2)=-4⋅14⋅3+22-74

Cancel the common factor.

f(2)=-4⋅14⋅3+22-74

Rewrite the expression.

f(2)=-13+22-74

f(2)=-13+22-74

f(2)=-13+22-74

Divide 2 by 2.

f(2)=-13+1-74

f(2)=-13+1-74

Simplify the expression.

Write 1 as a fraction with a common denominator.

f(2)=-13+33-74

Combine the numerators over the common denominator.

f(2)=-1+33-74

f(2)=-1+33-74

Add -1 and 3.

f(2)=23-74

To write 23 as a fraction with a common denominator, multiply by 44.

f(2)=23⋅44-74

To write -74 as a fraction with a common denominator, multiply by 33.

f(2)=23⋅44-74⋅33

Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.

Multiply 23 and 44.

f(2)=2⋅43⋅4-74⋅33

Multiply 3 by 4.

f(2)=2⋅412-74⋅33

Multiply 74 and 33.

f(2)=2⋅412-7⋅34⋅3

Multiply 4 by 3.

f(2)=2⋅412-7⋅312

f(2)=2⋅412-7⋅312

Combine the numerators over the common denominator.

f(2)=2⋅4-7⋅312

Simplify the numerator.

Multiply 2 by 4.

f(2)=8-7⋅312

Multiply -7 by 3.

f(2)=8-2112

Subtract 21 from 8.

f(2)=-1312

f(2)=-1312

Move the negative in front of the fraction.

f(2)=-1312

The final answer is -1312.

-1312

-1312

The y value at x=2 is -1312.

y=-1312

Replace the variable x with 1 in the expression.

f(1)=-(1)212+12-74

Simplify the result.

One to any power is one.

f(1)=-112+12-74

To write 12 as a fraction with a common denominator, multiply by 66.

f(1)=-112+12⋅66-74

Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.

Multiply 12 and 66.

f(1)=-112+62⋅6-74

Multiply 2 by 6.

f(1)=-112+612-74

f(1)=-112+612-74

Combine the numerators over the common denominator.

f(1)=-1+612-74

Add -1 and 6.

f(1)=512-74

To write -74 as a fraction with a common denominator, multiply by 33.

f(1)=512-74⋅33

Write each expression with a common denominator of 12, by multiplying each by an appropriate factor of 1.

Multiply 74 and 33.

f(1)=512-7⋅34⋅3

Multiply 4 by 3.

f(1)=512-7⋅312

f(1)=512-7⋅312

Combine the numerators over the common denominator.

f(1)=5-7⋅312

Simplify the numerator.

Multiply -7 by 3.

f(1)=5-2112

Subtract 21 from 5.

f(1)=-1612

f(1)=-1612

Cancel the common factor of -16 and 12.

Factor 4 out of -16.

f(1)=4(-4)12

Cancel the common factors.

Factor 4 out of 12.

f(1)=4⋅-44⋅3

Cancel the common factor.

f(1)=4⋅-44⋅3

Rewrite the expression.

f(1)=-43

f(1)=-43

f(1)=-43

Move the negative in front of the fraction.

f(1)=-43

The final answer is -43.

-43

-43

The y value at x=1 is -43.

y=-43

Replace the variable x with 4 in the expression.

f(4)=-(4)212+42-74

Simplify the result.

Simplify each term.

Raise 4 to the power of 2.

f(4)=-1612+42-74

Cancel the common factor of 16 and 12.

Factor 4 out of 16.

f(4)=-4(4)12+42-74

Cancel the common factors.

Factor 4 out of 12.

f(4)=-4⋅44⋅3+42-74

Cancel the common factor.

f(4)=-4⋅44⋅3+42-74

Rewrite the expression.

f(4)=-43+42-74

f(4)=-43+42-74

f(4)=-43+42-74

Divide 4 by 2.

f(4)=-43+2-74

f(4)=-43+2-74

To write 2 as a fraction with a common denominator, multiply by 33.

f(4)=-43+2⋅33-74

Combine 2 and 33.

f(4)=-43+2⋅33-74

Combine the numerators over the common denominator.

f(4)=-4+2⋅33-74

Simplify the numerator.

Multiply 2 by 3.

f(4)=-4+63-74

Add -4 and 6.

f(4)=23-74

f(4)=23-74

To write 23 as a fraction with a common denominator, multiply by 44.

f(4)=23⋅44-74

To write -74 as a fraction with a common denominator, multiply by 33.

f(4)=23⋅44-74⋅33

Multiply 23 and 44.

f(4)=2⋅43⋅4-74⋅33

Multiply 3 by 4.

f(4)=2⋅412-74⋅33

Multiply 74 and 33.

f(4)=2⋅412-7⋅34⋅3

Multiply 4 by 3.

f(4)=2⋅412-7⋅312

f(4)=2⋅412-7⋅312

Combine the numerators over the common denominator.

f(4)=2⋅4-7⋅312

Simplify the numerator.

Multiply 2 by 4.

f(4)=8-7⋅312

Multiply -7 by 3.

f(4)=8-2112

Subtract 21 from 8.

f(4)=-1312

f(4)=-1312

Move the negative in front of the fraction.

f(4)=-1312

The final answer is -1312.

-1312

-1312

The y value at x=4 is -1312.

y=-1312

Replace the variable x with 5 in the expression.

f(5)=-(5)212+52-74

Simplify the result.

Raise 5 to the power of 2.

f(5)=-2512+52-74

To write 52 as a fraction with a common denominator, multiply by 66.

f(5)=-2512+52⋅66-74

Multiply 52 and 66.

f(5)=-2512+5⋅62⋅6-74

Multiply 2 by 6.

f(5)=-2512+5⋅612-74

f(5)=-2512+5⋅612-74

Combine the numerators over the common denominator.

f(5)=-25+5⋅612-74

Simplify the numerator.

Multiply 5 by 6.

f(5)=-25+3012-74

Add -25 and 30.

f(5)=512-74

f(5)=512-74

To write -74 as a fraction with a common denominator, multiply by 33.

f(5)=512-74⋅33

Multiply 74 and 33.

f(5)=512-7⋅34⋅3

Multiply 4 by 3.

f(5)=512-7⋅312

f(5)=512-7⋅312

Combine the numerators over the common denominator.

f(5)=5-7⋅312

Simplify the numerator.

Multiply -7 by 3.

f(5)=5-2112

Subtract 21 from 5.

f(5)=-1612

f(5)=-1612

Cancel the common factor of -16 and 12.

Factor 4 out of -16.

f(5)=4(-4)12

Cancel the common factors.

Factor 4 out of 12.

f(5)=4⋅-44⋅3

Cancel the common factor.

f(5)=4⋅-44⋅3

Rewrite the expression.

f(5)=-43

f(5)=-43

f(5)=-43

Move the negative in front of the fraction.

f(5)=-43

The final answer is -43.

-43

-43

The y value at x=5 is -43.

y=-43

Graph the parabola using its properties and the selected points.

xy1-432-13123-14-13125-43

xy1-432-13123-14-13125-43

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (3,-1)

Focus: (3,-4)

Axis of Symmetry: x=3

Directrix: y=2

xy1-432-13123-14-13125-43

Graph y+1=-1/12*(x-3)^2