f(x)=3×2-5x-6

Rewrite the equation in vertex form.

Complete the square for 3×2-5x-6.

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

a=3,b=-5,c=-6

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=-52(3)

Multiply 2 by 3.

d=-56

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

Simplify each term.

Raise -5 to the power of 2.

e=-6-254⋅3

Multiply 4 by 3.

e=-6-2512

e=-6-2512

To write -6 as a fraction with a common denominator, multiply by 1212.

e=-6⋅1212-2512

Combine -6 and 1212.

e=-6⋅1212-2512

Combine the numerators over the common denominator.

e=-6⋅12-2512

Simplify the numerator.

Multiply -6 by 12.

e=-72-2512

Subtract 25 from -72.

e=-9712

e=-9712

Move the negative in front of the fraction.

e=-9712

e=-9712

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

3(x-56)2-9712

3(x-56)2-9712

Set y equal to the new right side.

y=3(x-56)2-9712

y=3(x-56)2-9712

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

a=3

h=56

k=-9712

Since the value of a is positive, the parabola opens up.

Opens Up

Find the vertex (h,k).

(56,-9712)

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

Multiply 4 by 3.

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.

(56,-8)

(56,-8)

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

x=56

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=-496

y=-496

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

Direction: Opens Up

Vertex: (56,-9712)

Focus: (56,-8)

Axis of Symmetry: x=56

Directrix: y=-496

Direction: Opens Up

Vertex: (56,-9712)

Focus: (56,-8)

Axis of Symmetry: x=56

Directrix: y=-496

Replace the variable x with 0 in the expression.

f(0)=3(0)2-5⋅0-6

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=3⋅0-5⋅0-6

Multiply 3 by 0.

f(0)=0-5⋅0-6

Multiply -5 by 0.

f(0)=0+0-6

f(0)=0+0-6

Simplify by adding zeros.

Add 0 and 0.

f(0)=0-6

Subtract 6 from 0.

f(0)=-6

f(0)=-6

The final answer is -6.

-6

-6

The y value at x=0 is -6.

y=-6

Replace the variable x with -1 in the expression.

f(-1)=3(-1)2-5⋅-1-6

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=3⋅1-5⋅-1-6

Multiply 3 by 1.

f(-1)=3-5⋅-1-6

Multiply -5 by -1.

f(-1)=3+5-6

f(-1)=3+5-6

Simplify by adding and subtracting.

Add 3 and 5.

f(-1)=8-6

Subtract 6 from 8.

f(-1)=2

f(-1)=2

The final answer is 2.

2

2

The y value at x=-1 is 2.

y=2

Replace the variable x with 2 in the expression.

f(2)=3(2)2-5⋅2-6

Simplify the result.

Simplify each term.

Raise 2 to the power of 2.

f(2)=3⋅4-5⋅2-6

Multiply 3 by 4.

f(2)=12-5⋅2-6

Multiply -5 by 2.

f(2)=12-10-6

f(2)=12-10-6

Simplify by subtracting numbers.

Subtract 10 from 12.

f(2)=2-6

Subtract 6 from 2.

f(2)=-4

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)2-5⋅3-6

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)2-5⋅3-6

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

f(3)=31+2-5⋅3-6

f(3)=31+2-5⋅3-6

Add 1 and 2.

f(3)=33-5⋅3-6

f(3)=33-5⋅3-6

Raise 3 to the power of 3.

f(3)=27-5⋅3-6

Multiply -5 by 3.

f(3)=27-15-6

f(3)=27-15-6

Simplify by subtracting numbers.

Subtract 15 from 27.

f(3)=12-6

Subtract 6 from 12.

f(3)=6

f(3)=6

The final answer is 6.

6

6

The y value at x=3 is 6.

y=6

Graph the parabola using its properties and the selected points.

xy-120-656-97122-436

xy-120-656-97122-436

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (56,-9712)

Focus: (56,-8)

Axis of Symmetry: x=56

Directrix: y=-496

xy-120-656-97122-436

Graph f(x)=3x^2-5x-6