f(x)=-92×2+13+6

Rewrite the equation in vertex form.

Simplify -92×2+13+6.

Simplify each term.

Combine x2 and 92.

y=-x2⋅92+13+6

Move 9 to the left of x2.

y=-9×22+13+6

y=-9×22+13+6

To write 6 as a fraction with a common denominator, multiply by 33.

y=-9×22+13+6⋅33

Combine 6 and 33.

y=-9×22+13+6⋅33

Combine the numerators over the common denominator.

y=-9×22+1+6⋅33

Simplify the numerator.

Multiply 6 by 3.

y=-9×22+1+183

Add 1 and 18.

y=-9×22+193

y=-9×22+193

y=-9×22+193

Complete the square for -9×22+193.

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

a=-92,b=0,c=193

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=02(-92)

Simplify the right side.

Cancel the common factor of 0 and 2.

Factor 2 out of 0.

d=2(0)2(-92)

Cancel the common factors.

Cancel the common factor.

d=2⋅02(-92)

Rewrite the expression.

d=0-92

d=0-92

d=0-92

Multiply the numerator by the reciprocal of the denominator.

d=0(-29)

Multiply 0(-29).

Multiply -1 by 0.

d=0(29)

Multiply 0 by 29.

d=0

d=0

d=0

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

Simplify each term.

Raising 0 to any positive power yields 0.

e=193-04⋅(-1(92))

Simplify the denominator.

Multiply 4 by -1.

e=193-0-4(92)

Combine -4 and 92.

e=193-0-4⋅92

e=193-0-4⋅92

Multiply -4 by 9.

e=193-0-362

Divide -36 by 2.

e=193-0-18

Divide 0 by -18.

e=193-0

Multiply -1 by 0.

e=193+0

e=193+0

Add 193 and 0.

e=193

e=193

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

-92⋅(x+0)2+193

-92⋅(x+0)2+193

Set y equal to the new right side.

y=-92⋅(x+0)2+193

y=-92⋅(x+0)2+193

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

a=-92

h=0

k=193

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

Opens Down

Find the vertex (h,k).

(0,193)

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⋅(-92)

Simplify.

Cancel the common factor of 1 and -1.

Rewrite 1 as -1(-1).

-1(-1)4⋅(-92)

Move the negative in front of the fraction.

-14(92)

-14(92)

Combine 4 and 92.

-14⋅92

Simplify the expression.

Multiply 4 by 9.

-1362

Divide 36 by 2.

-118

-118

-118

-118

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.

(0,11318)

(0,11318)

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

x=0

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=11518

y=11518

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

Direction: Opens Down

Vertex: (0,193)

Focus: (0,11318)

Axis of Symmetry: x=0

Directrix: y=11518

Direction: Opens Down

Vertex: (0,193)

Focus: (0,11318)

Axis of Symmetry: x=0

Directrix: y=11518

Replace the variable x with -1 in the expression.

f(-1)=-9(-1)22+193

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=-9⋅12+193

Multiply 9 by 1.

f(-1)=-92+193

f(-1)=-92+193

To write -92 as a fraction with a common denominator, multiply by 33.

f(-1)=-92⋅33+193

To write 193 as a fraction with a common denominator, multiply by 22.

f(-1)=-92⋅33+193⋅22

Write each expression with a common denominator of 6, by multiplying each by an appropriate factor of 1.

Multiply 92 and 33.

f(-1)=-9⋅32⋅3+193⋅22

Multiply 2 by 3.

f(-1)=-9⋅36+193⋅22

Multiply 193 and 22.

f(-1)=-9⋅36+19⋅23⋅2

Multiply 3 by 2.

f(-1)=-9⋅36+19⋅26

f(-1)=-9⋅36+19⋅26

Combine the numerators over the common denominator.

f(-1)=-9⋅3+19⋅26

Simplify the numerator.

Multiply -9 by 3.

f(-1)=-27+19⋅26

Multiply 19 by 2.

f(-1)=-27+386

Add -27 and 38.

f(-1)=116

f(-1)=116

The final answer is 116.

116

116

The y value at x=-1 is 116.

y=116

Replace the variable x with -2 in the expression.

f(-2)=-9(-2)22+193

Simplify the result.

Simplify each term.

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

Rewrite -2 as -1(2).

f(-2)=-9(-1⋅2)22+193

Apply the product rule to -1(2).

f(-2)=-9((-1)2⋅22)2+193

Raise -1 to the power of 2.

f(-2)=-9(1⋅22)2+193

Multiply 22 by 1.

f(-2)=-9⋅222+193

Factor 2 out of 9⋅22.

f(-2)=-2(9⋅2)2+193

Cancel the common factors.

Factor 2 out of 2.

f(-2)=-2(9⋅2)2(1)+193

Cancel the common factor.

f(-2)=-2(9⋅2)2⋅1+193

Rewrite the expression.

f(-2)=-9⋅21+193

Divide 9⋅2 by 1.

f(-2)=-(9⋅2)+193

f(-2)=-(9⋅2)+193

f(-2)=-(9⋅2)+193

Multiply 9 by 2.

f(-2)=-1⋅18+193

Multiply -1 by 18.

f(-2)=-18+193

f(-2)=-18+193

To write -18 as a fraction with a common denominator, multiply by 33.

f(-2)=-18⋅33+193

Combine -18 and 33.

f(-2)=-18⋅33+193

Combine the numerators over the common denominator.

f(-2)=-18⋅3+193

Simplify the numerator.

Multiply -18 by 3.

f(-2)=-54+193

Add -54 and 19.

f(-2)=-353

f(-2)=-353

Move the negative in front of the fraction.

f(-2)=-353

The final answer is -353.

-353

-353

The y value at x=-2 is -353.

y=-353

Replace the variable x with 1 in the expression.

f(1)=-9(1)22+193

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=-9⋅12+193

Multiply 9 by 1.

f(1)=-92+193

f(1)=-92+193

To write -92 as a fraction with a common denominator, multiply by 33.

f(1)=-92⋅33+193

To write 193 as a fraction with a common denominator, multiply by 22.

f(1)=-92⋅33+193⋅22

Write each expression with a common denominator of 6, by multiplying each by an appropriate factor of 1.

Multiply 92 and 33.

f(1)=-9⋅32⋅3+193⋅22

Multiply 2 by 3.

f(1)=-9⋅36+193⋅22

Multiply 193 and 22.

f(1)=-9⋅36+19⋅23⋅2

Multiply 3 by 2.

f(1)=-9⋅36+19⋅26

f(1)=-9⋅36+19⋅26

Combine the numerators over the common denominator.

f(1)=-9⋅3+19⋅26

Simplify the numerator.

Multiply -9 by 3.

f(1)=-27+19⋅26

Multiply 19 by 2.

f(1)=-27+386

Add -27 and 38.

f(1)=116

f(1)=116

The final answer is 116.

116

116

The y value at x=1 is 116.

y=116

Replace the variable x with 2 in the expression.

f(2)=-9(2)22+193

Simplify the result.

Simplify each term.

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

Factor 2 out of 9(2)2.

f(2)=-2(9⋅2)2+193

Cancel the common factors.

Factor 2 out of 2.

f(2)=-2(9⋅2)2(1)+193

Cancel the common factor.

f(2)=-2(9⋅2)2⋅1+193

Rewrite the expression.

f(2)=-9⋅21+193

Divide 9⋅2 by 1.

f(2)=-(9⋅2)+193

f(2)=-(9⋅2)+193

f(2)=-(9⋅2)+193

Multiply 9 by 2.

f(2)=-1⋅18+193

Multiply -1 by 18.

f(2)=-18+193

f(2)=-18+193

To write -18 as a fraction with a common denominator, multiply by 33.

f(2)=-18⋅33+193

Combine -18 and 33.

f(2)=-18⋅33+193

Combine the numerators over the common denominator.

f(2)=-18⋅3+193

Simplify the numerator.

Multiply -18 by 3.

f(2)=-54+193

Add -54 and 19.

f(2)=-353

f(2)=-353

Move the negative in front of the fraction.

f(2)=-353

The final answer is -353.

-353

-353

The y value at x=2 is -353.

y=-353

Graph the parabola using its properties and the selected points.

xy-2-353-1116019311162-353

xy-2-353-1116019311162-353

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (0,193)

Focus: (0,11318)

Axis of Symmetry: x=0

Directrix: y=11518

xy-2-353-1116019311162-353

Graph f(x)=-9/2x^2+1/3+6