r(t)=2t2-24t+81

Rewrite the equation in vertex form.

Complete the square for 2×2-24x+81.

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

a=2,b=-24,c=81

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=-242(2)

Simplify the right side.

Cancel the common factor of 24 and 2.

Factor 2 out of 24.

d=-2⋅122⋅2

Cancel the common factors.

Factor 2 out of 2⋅2.

d=-2⋅122(2)

Cancel the common factor.

d=-2⋅122⋅2

Rewrite the expression.

d=-122

d=-122

d=-122

Cancel the common factor of 12 and 2.

Factor 2 out of 12.

d=-2⋅62

Cancel the common factors.

Factor 2 out of 2.

d=-2⋅62(1)

Cancel the common factor.

d=-2⋅62⋅1

Rewrite the expression.

d=-61

Divide 6 by 1.

d=-1⋅6

d=-1⋅6

d=-1⋅6

Multiply -1 by 6.

d=-6

d=-6

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

Simplify each term.

Raise -24 to the power of 2.

e=81-5764⋅2

Multiply 4 by 2.

e=81-5768

Divide 576 by 8.

e=81-1⋅72

Multiply -1 by 72.

e=81-72

e=81-72

Subtract 72 from 81.

e=9

e=9

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

2(x-6)2+9

2(x-6)2+9

Set y equal to the new right side.

y=2(x-6)2+9

y=2(x-6)2+9

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

a=2

h=6

k=9

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

Opens Up

Find the vertex (h,k).

(6,9)

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⋅2

Multiply 4 by 2.

18

18

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.

(6,738)

(6,738)

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

x=6

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=718

y=718

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

Direction: Opens Up

Vertex: (6,9)

Focus: (6,738)

Axis of Symmetry: x=6

Directrix: y=718

Direction: Opens Up

Vertex: (6,9)

Focus: (6,738)

Axis of Symmetry: x=6

Directrix: y=718

Replace the variable x with 5 in the expression.

f(5)=2(5)2-24⋅5+81

Simplify the result.

Simplify each term.

Raise 5 to the power of 2.

f(5)=2⋅25-24⋅5+81

Multiply 2 by 25.

f(5)=50-24⋅5+81

Multiply -24 by 5.

f(5)=50-120+81

f(5)=50-120+81

Simplify by adding and subtracting.

Subtract 120 from 50.

f(5)=-70+81

Add -70 and 81.

f(5)=11

f(5)=11

The final answer is 11.

11

11

The y value at x=5 is 11.

y=11

Replace the variable x with 4 in the expression.

f(4)=2(4)2-24⋅4+81

Simplify the result.

Simplify each term.

Raise 4 to the power of 2.

f(4)=2⋅16-24⋅4+81

Multiply 2 by 16.

f(4)=32-24⋅4+81

Multiply -24 by 4.

f(4)=32-96+81

f(4)=32-96+81

Simplify by adding and subtracting.

Subtract 96 from 32.

f(4)=-64+81

Add -64 and 81.

f(4)=17

f(4)=17

The final answer is 17.

17

17

The y value at x=4 is 17.

y=17

Replace the variable x with 7 in the expression.

f(7)=2(7)2-24⋅7+81

Simplify the result.

Simplify each term.

Raise 7 to the power of 2.

f(7)=2⋅49-24⋅7+81

Multiply 2 by 49.

f(7)=98-24⋅7+81

Multiply -24 by 7.

f(7)=98-168+81

f(7)=98-168+81

Simplify by adding and subtracting.

Subtract 168 from 98.

f(7)=-70+81

Add -70 and 81.

f(7)=11

f(7)=11

The final answer is 11.

11

11

The y value at x=7 is 11.

y=11

Replace the variable x with 8 in the expression.

f(8)=2(8)2-24⋅8+81

Simplify the result.

Simplify each term.

Raise 8 to the power of 2.

f(8)=2⋅64-24⋅8+81

Multiply 2 by 64.

f(8)=128-24⋅8+81

Multiply -24 by 8.

f(8)=128-192+81

f(8)=128-192+81

Simplify by adding and subtracting.

Subtract 192 from 128.

f(8)=-64+81

Add -64 and 81.

f(8)=17

f(8)=17

The final answer is 17.

17

17

The y value at x=8 is 17.

y=17

Graph the parabola using its properties and the selected points.

xy41751169711817

xy41751169711817

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: (6,9)

Focus: (6,738)

Axis of Symmetry: x=6

Directrix: y=718

xy41751169711817

Graph r(t)=2t^2-24t+81