y+2=(x+1)2

Subtract 2 from both sides of the equation.

y=(x+1)2-2

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

a=1

h=-1

k=-2

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

Opens Up

Find the vertex (h,k).

(-1,-2)

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.

(-1,-74)

(-1,-74)

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

x=-1

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=-94

y=-94

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

Direction: Opens Up

Vertex: (-1,-2)

Focus: (-1,-74)

Axis of Symmetry: x=-1

Directrix: y=-94

Direction: Opens Up

Vertex: (-1,-2)

Focus: (-1,-74)

Axis of Symmetry: x=-1

Directrix: y=-94

Replace the variable x with -2 in the expression.

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

Simplify the result.

Simplify each term.

Raise -2 to the power of 2.

f(-2)=4+2(-2)-1

Multiply 2 by -2.

f(-2)=4-4-1

f(-2)=4-4-1

Simplify by subtracting numbers.

Subtract 4 from 4.

f(-2)=0-1

Subtract 1 from 0.

f(-2)=-1

f(-2)=-1

The final answer is -1.

-1

-1

The y value at x=-2 is -1.

y=-1

Replace the variable x with -3 in the expression.

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

Simplify the result.

Simplify each term.

Raise -3 to the power of 2.

f(-3)=9+2(-3)-1

Multiply 2 by -3.

f(-3)=9-6-1

f(-3)=9-6-1

Simplify by subtracting numbers.

Subtract 6 from 9.

f(-3)=3-1

Subtract 1 from 3.

f(-3)=2

f(-3)=2

The final answer is 2.

2

2

The y value at x=-3 is 2.

y=2

Replace the variable x with 0 in the expression.

f(0)=(0)2+2(0)-1

Simplify the result.

Simplify each term.

Raising 0 to any positive power yields 0.

f(0)=0+2(0)-1

Multiply 2 by 0.

f(0)=0+0-1

f(0)=0+0-1

Simplify by adding zeros.

Add 0 and 0.

f(0)=0-1

Subtract 1 from 0.

f(0)=-1

f(0)=-1

The final answer is -1.

-1

-1

The y value at x=0 is -1.

y=-1

Replace the variable x with 1 in the expression.

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

Simplify the result.

Simplify each term.

One to any power is one.

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

Multiply 2 by 1.

f(1)=1+2-1

f(1)=1+2-1

Simplify by adding and subtracting.

Add 1 and 2.

f(1)=3-1

Subtract 1 from 3.

f(1)=2

f(1)=2

The final answer is 2.

2

2

The y value at x=1 is 2.

y=2

Graph the parabola using its properties and the selected points.

xy-32-2-1-1-20-112

xy-32-2-1-1-20-112

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (-1,-2)

Focus: (-1,-74)

Axis of Symmetry: x=-1

Directrix: y=-94

xy-32-2-1-1-20-112

Graph y+2=(x+1)^2