f(x)=-3×2-4x

Rewrite the equation in vertex form.

Complete the square for -3×2-4x.

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

a=-3,b=-4,c=0

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=-42(-3)

Simplify the right side.

Cancel the common factor of 4 and 2.

Factor 2 out of 4.

d=-2⋅22⋅-3

Cancel the common factors.

Factor 2 out of 2⋅-3.

d=-2⋅22(-3)

Cancel the common factor.

d=-2⋅22⋅-3

Rewrite the expression.

d=-2-3

d=-2-3

d=-2-3

Move the negative in front of the fraction.

d=23

Multiply –23.

Multiply -1 by -1.

d=1(23)

Multiply 23 by 1.

d=23

d=23

d=23

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

Simplify each term.

Cancel the common factor of (-4)2 and 4.

Rewrite -4 as -1(4).

e=0-(-1⋅4)24(-3)

Apply the product rule to -1(4).

e=0-(-1)2⋅424(-3)

Raise -1 to the power of 2.

e=0-1⋅424(-3)

Multiply 42 by 1.

e=0-424(-3)

Factor 4 out of 42.

e=0-4⋅44⋅-3

Cancel the common factors.

Factor 4 out of 4⋅-3.

e=0-4⋅44(-3)

Cancel the common factor.

e=0-4⋅44⋅-3

Rewrite the expression.

e=0-4-3

e=0-4-3

e=0-4-3

Move the negative in front of the fraction.

e=0+43

Multiply –43.

Multiply -1 by -1.

e=0+1(43)

Multiply 43 by 1.

e=0+43

e=0+43

e=0+43

Add 0 and 43.

e=43

e=43

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

-3(x+23)2+43

-3(x+23)2+43

Set y equal to the new right side.

y=-3(x+23)2+43

y=-3(x+23)2+43

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

a=-3

h=-23

k=43

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

Opens Down

Find the vertex (h,k).

(-23,43)

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.

(-23,54)

(-23,54)

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

x=-23

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=1712

y=1712

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

Direction: Opens Down

Vertex: (-23,43)

Focus: (-23,54)

Axis of Symmetry: x=-23

Directrix: y=1712

Direction: Opens Down

Vertex: (-23,43)

Focus: (-23,54)

Axis of Symmetry: x=-23

Directrix: y=1712

Replace the variable x with -2 in the expression.

f(-2)=-3(-2)2-4⋅-2

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=-3⋅4-4⋅-2

Multiply -3 by 4.

f(-2)=-12-4⋅-2

Multiply -4 by -2.

f(-2)=-12+8

f(-2)=-12+8

Add -12 and 8.

f(-2)=-4

The final answer is -4.

-4

-4

The y value at x=-2 is -4.

y=-4

Replace the variable x with -3 in the expression.

f(-3)=-3(-3)2-4⋅-3

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)2-4⋅-3

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

f(-3)=(-3)1+2-4⋅-3

f(-3)=(-3)1+2-4⋅-3

Add 1 and 2.

f(-3)=(-3)3-4⋅-3

f(-3)=(-3)3-4⋅-3

Raise -3 to the power of 3.

f(-3)=-27-4⋅-3

Multiply -4 by -3.

f(-3)=-27+12

f(-3)=-27+12

Add -27 and 12.

f(-3)=-15

The final answer is -15.

-15

-15

The y value at x=-3 is -15.

y=-15

Replace the variable x with 0 in the expression.

f(0)=-3(0)2-4⋅0

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=-3⋅0-4⋅0

Multiply -3 by 0.

f(0)=0-4⋅0

Multiply -4 by 0.

f(0)=0+0

f(0)=0+0

Add 0 and 0.

f(0)=0

The final answer is 0.

0

0

The y value at x=0 is 0.

y=0

Replace the variable x with 1 in the expression.

f(1)=-3(1)2-4⋅1

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=-3⋅1-4⋅1

Multiply -3 by 1.

f(1)=-3-4⋅1

Multiply -4 by 1.

f(1)=-3-4

f(1)=-3-4

Subtract 4 from -3.

f(1)=-7

The final answer is -7.

-7

-7

The y value at x=1 is -7.

y=-7

Graph the parabola using its properties and the selected points.

xy-3-15-2-4-2343001-7

xy-3-15-2-4-2343001-7

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (-23,43)

Focus: (-23,54)

Axis of Symmetry: x=-23

Directrix: y=1712

xy-3-15-2-4-2343001-7

Graph f(x)=-3x^2-4x