y=12×2-9

Rewrite the equation in vertex form.

Combine 12 and x2.

y=x22-9

Complete the square for x22-9.

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

a=12,b=0,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=02(12)

Simplify the right side.

Cancel the common factor of 0 and 2.

Factor 2 out of 0.

d=2(0)2(12)

Cancel the common factors.

Cancel the common factor.

d=2⋅02(12)

Rewrite the expression.

d=012

d=012

d=012

Multiply the numerator by the reciprocal of the denominator.

d=0⋅2

Multiply 0 by 2.

d=0

d=0

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

Simplify each term.

Raising 0 to any positive power yields 0.

e=-9-04(12)

Combine 4 and 12.

e=-9-042

Divide 4 by 2.

e=-9-02

Divide 0 by 2.

e=-9-0

Multiply -1 by 0.

e=-9+0

e=-9+0

Add -9 and 0.

e=-9

e=-9

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

12⋅(x+0)2-9

12⋅(x+0)2-9

Set y equal to the new right side.

y=12⋅(x+0)2-9

y=12⋅(x+0)2-9

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

a=12

h=0

k=-9

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

Opens Up

Find the vertex (h,k).

(0,-9)

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⋅12

Simplify.

Combine 4 and 12.

142

Divide 4 by 2.

12

12

12

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.

(0,-172)

(0,-172)

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

x=0

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=-192

y=-192

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

Direction: Opens Up

Vertex: (0,-9)

Focus: (0,-172)

Axis of Symmetry: x=0

Directrix: y=-192

Direction: Opens Up

Vertex: (0,-9)

Focus: (0,-172)

Axis of Symmetry: x=0

Directrix: y=-192

Replace the variable x with -2 in the expression.

f(-2)=(-2)22-9

Simplify the result.

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

Rewrite -2 as -1(2).

f(-2)=(-1⋅2)22-9

Apply the product rule to -1(2).

f(-2)=(-1)2⋅222-9

Raise -1 to the power of 2.

f(-2)=1⋅222-9

Multiply 22 by 1.

f(-2)=222-9

Factor 2 out of 22.

f(-2)=2⋅22-9

Cancel the common factors.

Factor 2 out of 2.

f(-2)=2⋅22(1)-9

Cancel the common factor.

f(-2)=2⋅22⋅1-9

Rewrite the expression.

f(-2)=21-9

Divide 2 by 1.

f(-2)=2-9

f(-2)=2-9

f(-2)=2-9

Subtract 9 from 2.

f(-2)=-7

The final answer is -7.

-7

-7

The y value at x=-2 is -7.

y=-7

Replace the variable x with -1 in the expression.

f(-1)=(-1)22-9

Simplify the result.

Raise -1 to the power of 2.

f(-1)=12-9

To write -9 as a fraction with a common denominator, multiply by 22.

f(-1)=12-9⋅22

Combine -9 and 22.

f(-1)=12+-9⋅22

Combine the numerators over the common denominator.

f(-1)=1-9⋅22

Simplify the numerator.

Multiply -9 by 2.

f(-1)=1-182

Subtract 18 from 1.

f(-1)=-172

f(-1)=-172

Move the negative in front of the fraction.

f(-1)=-172

The final answer is -172.

-172

-172

The y value at x=-1 is -172.

y=-172

Replace the variable x with 2 in the expression.

f(2)=(2)22-9

Simplify the result.

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

Factor 2 out of (2)2.

f(2)=2⋅22-9

Cancel the common factors.

Factor 2 out of 2.

f(2)=2⋅22(1)-9

Cancel the common factor.

f(2)=2⋅22⋅1-9

Rewrite the expression.

f(2)=21-9

Divide 2 by 1.

f(2)=2-9

f(2)=2-9

f(2)=2-9

Subtract 9 from 2.

f(2)=-7

The final answer is -7.

-7

-7

The y value at x=2 is -7.

y=-7

Replace the variable x with 1 in the expression.

f(1)=(1)22-9

Simplify the result.

One to any power is one.

f(1)=12-9

To write -9 as a fraction with a common denominator, multiply by 22.

f(1)=12-9⋅22

Combine -9 and 22.

f(1)=12+-9⋅22

Combine the numerators over the common denominator.

f(1)=1-9⋅22

Simplify the numerator.

Multiply -9 by 2.

f(1)=1-182

Subtract 18 from 1.

f(1)=-172

f(1)=-172

Move the negative in front of the fraction.

f(1)=-172

The final answer is -172.

-172

-172

The y value at x=1 is -172.

y=-172

Graph the parabola using its properties and the selected points.

xy-2-7-1-1720-91-1722-7

xy-2-7-1-1720-91-1722-7

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (0,-9)

Focus: (0,-172)

Axis of Symmetry: x=0

Directrix: y=-192

xy-2-7-1-1720-91-1722-7

Graph y=1/2x^2-9