y=-3×2-6x+9

Rewrite the equation in vertex form.

Complete the square for -3×2-6x+9.

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

a=-3,b=-6,c=9

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(-3)

Simplify the right side.

Cancel the common factor of 6 and 2.

Factor 2 out of 6.

d=-2⋅32⋅-3

Cancel the common factors.

Factor 2 out of 2⋅-3.

d=-2⋅32(-3)

Cancel the common factor.

d=-2⋅32⋅-3

Rewrite the expression.

d=-3-3

d=-3-3

d=-3-3

Cancel the common factor of 3 and -3.

Factor 3 out of 3.

d=-3(1)-3

Move the negative one from the denominator of 1-1.

d=-(-1⋅1)

d=-(-1⋅1)

Multiply.

Multiply -1 by 1.

d=1

Multiply -1 by -1.

d=1

d=1

d=1

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

Simplify each term.

Raise -6 to the power of 2.

e=9-364⋅-3

Multiply 4 by -3.

e=9-36-12

Divide 36 by -12.

e=9+3

Multiply -1 by -3.

e=9+3

e=9+3

Add 9 and 3.

e=12

e=12

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

-3(x+1)2+12

-3(x+1)2+12

Set y equal to the new right side.

y=-3(x+1)2+12

y=-3(x+1)2+12

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

a=-3

h=-1

k=12

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

Opens Down

Find the vertex (h,k).

(-1,12)

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⋅-3

Simplify.

Multiply 4 by -3.

1-12

Move the negative in front of the fraction.

-112

-112

-112

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.

(-1,14312)

(-1,14312)

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

x=-1

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=14512

y=14512

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

Direction: Opens Down

Vertex: (-1,12)

Focus: (-1,14312)

Axis of Symmetry: x=-1

Directrix: y=14512

Direction: Opens Down

Vertex: (-1,12)

Focus: (-1,14312)

Axis of Symmetry: x=-1

Directrix: y=14512

Replace the variable x with -2 in the expression.

f(-2)=-3(-2)2-6⋅-2+9

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=-3⋅4-6⋅-2+9

Multiply -3 by 4.

f(-2)=-12-6⋅-2+9

Multiply -6 by -2.

f(-2)=-12+12+9

f(-2)=-12+12+9

Simplify by adding numbers.

Add -12 and 12.

f(-2)=0+9

Add 0 and 9.

f(-2)=9

f(-2)=9

The final answer is 9.

9

9

The y value at x=-2 is 9.

y=9

Replace the variable x with -3 in the expression.

f(-3)=-3(-3)2-6⋅-3+9

Simplify the result.

Simplify each term.

Multiply -3 by (-3)2 by adding the exponents.

Multiply -3 by (-3)2.

Raise -3 to the power of 1.

f(-3)=(-3)(-3)2-6⋅-3+9

Use the power rule aman=am+n to combine exponents.

f(-3)=(-3)1+2-6⋅-3+9

f(-3)=(-3)1+2-6⋅-3+9

Add 1 and 2.

f(-3)=(-3)3-6⋅-3+9

f(-3)=(-3)3-6⋅-3+9

Raise -3 to the power of 3.

f(-3)=-27-6⋅-3+9

Multiply -6 by -3.

f(-3)=-27+18+9

f(-3)=-27+18+9

Simplify by adding numbers.

Add -27 and 18.

f(-3)=-9+9

Add -9 and 9.

f(-3)=0

f(-3)=0

The final answer is 0.

0

0

The y value at x=-3 is 0.

y=0

Replace the variable x with 0 in the expression.

f(0)=-3(0)2-6⋅0+9

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=-3⋅0-6⋅0+9

Multiply -3 by 0.

f(0)=0-6⋅0+9

Multiply -6 by 0.

f(0)=0+0+9

f(0)=0+0+9

Simplify by adding zeros.

Add 0 and 0.

f(0)=0+9

Add 0 and 9.

f(0)=9

f(0)=9

The final answer is 9.

9

9

The y value at x=0 is 9.

y=9

Replace the variable x with 1 in the expression.

f(1)=-3(1)2-6⋅1+9

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=-3⋅1-6⋅1+9

Multiply -3 by 1.

f(1)=-3-6⋅1+9

Multiply -6 by 1.

f(1)=-3-6+9

f(1)=-3-6+9

Simplify by adding and subtracting.

Subtract 6 from -3.

f(1)=-9+9

Add -9 and 9.

f(1)=0

f(1)=0

The final answer is 0.

0

0

The y value at x=1 is 0.

y=0

Graph the parabola using its properties and the selected points.

xy-30-29-1120910

xy-30-29-1120910

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (-1,12)

Focus: (-1,14312)

Axis of Symmetry: x=-1

Directrix: y=14512

xy-30-29-1120910

Graph y=-3x^2-6x+9