g(x)=x2-x-6

Rewrite the equation in vertex form.

Complete the square for x2-x-6.

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

a=1,b=-1,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=-12(1)

Cancel the common factor of 1.

Cancel the common factor.

d=-12⋅1

Rewrite the expression.

d=-12

d=-12

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

Simplify each term.

Raise -1 to the power of 2.

e=-6-14⋅1

Multiply 4 by 1.

e=-6-14

e=-6-14

To write -6 as a fraction with a common denominator, multiply by 44.

e=-6⋅44-14

Combine -6 and 44.

e=-6⋅44-14

Combine the numerators over the common denominator.

e=-6⋅4-14

Simplify the numerator.

Multiply -6 by 4.

e=-24-14

Subtract 1 from -24.

e=-254

e=-254

Move the negative in front of the fraction.

e=-254

e=-254

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

(x-12)2-254

(x-12)2-254

Set y equal to the new right side.

y=(x-12)2-254

y=(x-12)2-254

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

a=1

h=12

k=-254

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

Opens Up

Find the vertex (h,k).

(12,-254)

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⋅1

Cancel the common factor of 1.

Cancel the common factor.

14⋅1

Rewrite the expression.

14

14

14

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.

(12,-6)

(12,-6)

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

x=12

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=-132

y=-132

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

Direction: Opens Up

Vertex: (12,-254)

Focus: (12,-6)

Axis of Symmetry: x=12

Directrix: y=-132

Direction: Opens Up

Vertex: (12,-254)

Focus: (12,-6)

Axis of Symmetry: x=12

Directrix: y=-132

Replace the variable x with -1 in the expression.

f(-1)=(-1)2-(-1)-6

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=1-(-1)-6

Multiply -1 by -1.

f(-1)=1+1-6

f(-1)=1+1-6

Simplify by adding and subtracting.

Add 1 and 1.

f(-1)=2-6

Subtract 6 from 2.

f(-1)=-4

f(-1)=-4

The final answer is -4.

-4

-4

The y value at x=-1 is -4.

y=-4

Replace the variable x with -2 in the expression.

f(-2)=(-2)2-(-2)-6

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=4-(-2)-6

Multiply -1 by -2.

f(-2)=4+2-6

f(-2)=4+2-6

Simplify by adding and subtracting.

Add 4 and 2.

f(-2)=6-6

Subtract 6 from 6.

f(-2)=0

f(-2)=0

The final answer is 0.

0

0

The y value at x=-2 is 0.

y=0

Replace the variable x with 1 in the expression.

f(1)=(1)2-(1)-6

Simplify the result.

Simplify each term.

One to any power is one.

f(1)=1-(1)-6

Multiply -1 by 1.

f(1)=1-1-6

f(1)=1-1-6

Simplify by subtracting numbers.

Subtract 1 from 1.

f(1)=0-6

Subtract 6 from 0.

f(1)=-6

f(1)=-6

The final answer is -6.

-6

-6

The y value at x=1 is -6.

y=-6

Replace the variable x with 2 in the expression.

f(2)=(2)2-(2)-6

Simplify the result.

Simplify each term.

Raise 2 to the power of 2.

f(2)=4-(2)-6

Multiply -1 by 2.

f(2)=4-2-6

f(2)=4-2-6

Simplify by subtracting numbers.

Subtract 2 from 4.

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

Graph the parabola using its properties and the selected points.

xy-20-1-412-2541-62-4

xy-20-1-412-2541-62-4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (12,-254)

Focus: (12,-6)

Axis of Symmetry: x=12

Directrix: y=-132

xy-20-1-412-2541-62-4

Graph g(x)=x^2-x-6