y=13×2+4

Rewrite the equation in vertex form.

Combine 13 and x2.

y=x23+4

Complete the square for x23+4.

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

a=13,b=0,c=4

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(13)

Simplify the right side.

Cancel the common factor of 0 and 2.

Factor 2 out of 0.

d=2(0)2(13)

Cancel the common factors.

Cancel the common factor.

d=2⋅02(13)

Rewrite the expression.

d=013

d=013

d=013

Multiply the numerator by the reciprocal of the denominator.

d=0⋅3

Multiply 0 by 3.

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=4-04(13)

Combine 4 and 13.

e=4-043

Multiply the numerator by the reciprocal of the denominator.

e=4-(0(34))

Multiply 0 by 34.

e=4-0

Multiply -1 by 0.

e=4+0

e=4+0

Add 4 and 0.

e=4

e=4

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

13⋅(x+0)2+4

13⋅(x+0)2+4

Set y equal to the new right side.

y=13⋅(x+0)2+4

y=13⋅(x+0)2+4

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

a=13

h=0

k=4

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

Opens Up

Find the vertex (h,k).

(0,4)

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⋅13

Simplify.

Combine 4 and 13.

143

Multiply the numerator by the reciprocal of the denominator.

1(34)

Multiply 34 by 1.

34

34

34

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.

(0,194)

(0,194)

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=134

y=134

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

Direction: Opens Up

Vertex: (0,4)

Focus: (0,194)

Axis of Symmetry: x=0

Directrix: y=134

Direction: Opens Up

Vertex: (0,4)

Focus: (0,194)

Axis of Symmetry: x=0

Directrix: y=134

Replace the variable x with -1 in the expression.

f(-1)=(-1)23+4

Simplify the result.

Raise -1 to the power of 2.

f(-1)=13+4

To write 4 as a fraction with a common denominator, multiply by 33.

f(-1)=13+4⋅33

Combine 4 and 33.

f(-1)=13+4⋅33

Combine the numerators over the common denominator.

f(-1)=1+4⋅33

Simplify the numerator.

Multiply 4 by 3.

f(-1)=1+123

Add 1 and 12.

f(-1)=133

f(-1)=133

The final answer is 133.

133

133

The y value at x=-1 is 133.

y=133

Replace the variable x with -2 in the expression.

f(-2)=(-2)23+4

Simplify the result.

Raise -2 to the power of 2.

f(-2)=43+4

To write 4 as a fraction with a common denominator, multiply by 33.

f(-2)=43+4⋅33

Combine 4 and 33.

f(-2)=43+4⋅33

Combine the numerators over the common denominator.

f(-2)=4+4⋅33

Simplify the numerator.

Multiply 4 by 3.

f(-2)=4+123

Add 4 and 12.

f(-2)=163

f(-2)=163

The final answer is 163.

163

163

The y value at x=-2 is 163.

y=163

Replace the variable x with 3 in the expression.

f(3)=(3)23+4

Simplify the result.

Cancel the common factor of (3)2 and 3.

Factor 3 out of (3)2.

f(3)=3⋅33+4

Cancel the common factors.

Factor 3 out of 3.

f(3)=3⋅33(1)+4

Cancel the common factor.

f(3)=3⋅33⋅1+4

Rewrite the expression.

f(3)=31+4

Divide 3 by 1.

f(3)=3+4

f(3)=3+4

f(3)=3+4

Add 3 and 4.

f(3)=7

The final answer is 7.

7

7

The y value at x=3 is 7.

y=7

Replace the variable x with 1 in the expression.

f(1)=(1)23+4

Simplify the result.

One to any power is one.

f(1)=13+4

To write 4 as a fraction with a common denominator, multiply by 33.

f(1)=13+4⋅33

Combine 4 and 33.

f(1)=13+4⋅33

Combine the numerators over the common denominator.

f(1)=1+4⋅33

Simplify the numerator.

Multiply 4 by 3.

f(1)=1+123

Add 1 and 12.

f(1)=133

f(1)=133

The final answer is 133.

133

133

The y value at x=1 is 133.

y=133

Graph the parabola using its properties and the selected points.

xy-2163-113304113337

xy-2163-113304113337

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (0,4)

Focus: (0,194)

Axis of Symmetry: x=0

Directrix: y=134

xy-2163-113304113337

Graph y=1/3x^2+4