f(x)=-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 -1 by 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=-6

e=-6

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

-3(x-1)2-6

-3(x-1)2-6

Set y equal to the new right side.

y=-3(x-1)2-6

y=-3(x-1)2-6

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

a=-3

h=1

k=-6

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

Opens Down

Find the vertex (h,k).

(1,-6)

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,-7312)

(1,-7312)

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=-7112

y=-7112

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

Direction: Opens Down

Vertex: (1,-6)

Focus: (1,-7312)

Axis of Symmetry: x=1

Directrix: y=-7112

Direction: Opens Down

Vertex: (1,-6)

Focus: (1,-7312)

Axis of Symmetry: x=1

Directrix: y=-7112

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

Subtract 9 from 0.

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.

Raise -1 to the power of 2.

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 subtracting numbers.

Subtract 6 from -3.

f(-1)=-9-9

Subtract 9 from -9.

f(-1)=-18

f(-1)=-18

The final answer is -18.

-18

-18

The y value at x=-1 is -18.

y=-18

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

Subtract 9 from 0.

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.

Raise 3 to the power of 2.

f(3)=-3⋅9+6(3)-9

Multiply -3 by 9.

f(3)=-27+6(3)-9

Multiply 6 by 3.

f(3)=-27+18-9

f(3)=-27+18-9

Simplify by adding and subtracting.

Add -27 and 18.

f(3)=-9-9

Subtract 9 from -9.

f(3)=-18

f(3)=-18

The final answer is -18.

-18

-18

The y value at x=3 is -18.

y=-18

Graph the parabola using its properties and the selected points.

xy-1-180-91-62-93-18

xy-1-180-91-62-93-18

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (1,-6)

Focus: (1,-7312)

Axis of Symmetry: x=1

Directrix: y=-7112

xy-1-180-91-62-93-18

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