y=12×2+2x-3

Combine 12 and x2.

y=x22+2x-3

Rewrite the equation in vertex form.

Complete the square for x22+2x-3.

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

a=12,b=2,c=-3

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=22(12)

Simplify the right side.

Cancel the common factor of 2.

Cancel the common factor.

d=22(12)

Rewrite the expression.

d=112

d=112

Multiply the numerator by the reciprocal of the denominator.

d=1⋅2

Multiply 2 by 1.

d=2

d=2

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

Simplify each term.

Raise 2 to the power of 2.

e=-3-44(12)

Combine 4 and 12.

e=-3-442

Divide 4 by 2.

e=-3-42

Divide 4 by 2.

e=-3-1⋅2

Multiply -1 by 2.

e=-3-2

e=-3-2

Subtract 2 from -3.

e=-5

e=-5

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

12⋅(x+2)2-5

12⋅(x+2)2-5

Set y equal to the new right side.

y=12⋅(x+2)2-5

y=12⋅(x+2)2-5

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

a=12

h=-2

k=-5

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

Opens Up

Find the vertex (h,k).

(-2,-5)

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.

(-2,-92)

(-2,-92)

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

x=-2

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=-112

y=-112

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

Direction: Opens Up

Vertex: (-2,-5)

Focus: (-2,-92)

Axis of Symmetry: x=-2

Directrix: y=-112

Direction: Opens Up

Vertex: (-2,-5)

Focus: (-2,-92)

Axis of Symmetry: x=-2

Directrix: y=-112

Replace the variable x with -4 in the expression.

f(-4)=(-4)22+2(-4)-3

Simplify the result.

Simplify each term.

Raise -4 to the power of 2.

f(-4)=162+2(-4)-3

Divide 16 by 2.

f(-4)=8+2(-4)-3

Multiply 2 by -4.

f(-4)=8-8-3

f(-4)=8-8-3

Simplify by subtracting numbers.

Subtract 8 from 8.

f(-4)=0-3

Subtract 3 from 0.

f(-4)=-3

f(-4)=-3

The final answer is -3.

-3

-3

The y value at x=-4 is -3.

y=-3

Replace the variable x with -3 in the expression.

f(-3)=(-3)22+2(-3)-3

Simplify the result.

Find the common denominator.

Write 2(-3) as a fraction with denominator 1.

f(-3)=(-3)22+2(-3)1-3

Multiply 2(-3)1 by 22.

f(-3)=(-3)22+2(-3)1⋅22-3

Multiply 2(-3)1 and 22.

f(-3)=(-3)22+2(-3)⋅22-3

Write -3 as a fraction with denominator 1.

f(-3)=(-3)22+2(-3)⋅22+-31

Multiply -31 by 22.

f(-3)=(-3)22+2(-3)⋅22+-31⋅22

Multiply -31 and 22.

f(-3)=(-3)22+2(-3)⋅22+-3⋅22

f(-3)=(-3)22+2(-3)⋅22+-3⋅22

Combine fractions.

Combine fractions with similar denominators.

f(-3)=(-3)2+2(-3)⋅2-3⋅22

Multiply.

Multiply 2 by -3.

f(-3)=(-3)2-6⋅2-3⋅22

Multiply -6 by 2.

f(-3)=(-3)2-12-3⋅22

Multiply -3 by 2.

f(-3)=(-3)2-12-62

f(-3)=(-3)2-12-62

f(-3)=(-3)2-12-62

Simplify the numerator.

Raise -3 to the power of 2.

f(-3)=9-12-62

Subtract 12 from 9.

f(-3)=-3-62

Subtract 6 from -3.

f(-3)=-92

f(-3)=-92

Move the negative in front of the fraction.

f(-3)=-92

The final answer is -92.

-92

-92

The y value at x=-3 is -92.

y=-92

Replace the variable x with 0 in the expression.

f(0)=(0)22+2(0)-3

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=02+2(0)-3

Divide 0 by 2.

f(0)=0+2(0)-3

Multiply 2 by 0.

f(0)=0+0-3

f(0)=0+0-3

Simplify by adding zeros.

Add 0 and 0.

f(0)=0-3

Subtract 3 from 0.

f(0)=-3

f(0)=-3

The final answer is -3.

-3

-3

The y value at x=0 is -3.

y=-3

Replace the variable x with -1 in the expression.

f(-1)=(-1)22+2(-1)-3

Simplify the result.

Simplify each term.

Raise -1 to the power of 2.

f(-1)=12+2(-1)-3

Multiply 2 by -1.

f(-1)=12-2-3

f(-1)=12-2-3

Find the common denominator.

Write -2 as a fraction with denominator 1.

f(-1)=12+-21-3

Multiply -21 by 22.

f(-1)=12+-21⋅22-3

Multiply -21 and 22.

f(-1)=12+-2⋅22-3

Write -3 as a fraction with denominator 1.

f(-1)=12+-2⋅22+-31

Multiply -31 by 22.

f(-1)=12+-2⋅22+-31⋅22

Multiply -31 and 22.

f(-1)=12+-2⋅22+-3⋅22

f(-1)=12+-2⋅22+-3⋅22

Combine fractions.

Combine fractions with similar denominators.

f(-1)=1-2⋅2-3⋅22

Multiply.

Multiply -2 by 2.

f(-1)=1-4-3⋅22

Multiply -3 by 2.

f(-1)=1-4-62

f(-1)=1-4-62

f(-1)=1-4-62

Simplify the numerator.

Subtract 4 from 1.

f(-1)=-3-62

Subtract 6 from -3.

f(-1)=-92

f(-1)=-92

Move the negative in front of the fraction.

f(-1)=-92

The final answer is -92.

-92

-92

The y value at x=-1 is -92.

y=-92

Graph the parabola using its properties and the selected points.

xy-4-3-3-92-2-5-1-920-3

xy-4-3-3-92-2-5-1-920-3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-2,-5)

Focus: (-2,-92)

Axis of Symmetry: x=-2

Directrix: y=-112

xy-4-3-3-92-2-5-1-920-3

Graph y=1/2x^2+2x-3