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

Math
r(t)=2t2-24t+81
Find the properties of the given parabola.
Tap for more steps…
Rewrite the equation in vertex form.
Tap for more steps…
Complete the square for 2×2-24x+81.
Tap for more steps…
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.
Tap for more steps…
Cancel the common factor of 24 and 2.
Tap for more steps…
Factor 2 out of 24.
d=-2⋅122⋅2
Cancel the common factors.
Tap for more steps…
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.
Tap for more steps…
Factor 2 out of 12.
d=-2⋅62
Cancel the common factors.
Tap for more steps…
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.
Tap for more steps…
Simplify each term.
Tap for more steps…
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.
Tap for more steps…
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.
Tap for more steps…
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.
Tap for more steps…
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
Select a few x values, and plug them into the equation to find the corresponding y values. The x values should be selected around the vertex.
Tap for more steps…
Replace the variable x with 5 in the expression.
f(5)=2(5)2-24⋅5+81
Simplify the result.
Tap for more steps…
Simplify each term.
Tap for more steps…
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.
Tap for more steps…
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.
Tap for more steps…
Simplify each term.
Tap for more steps…
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.
Tap for more steps…
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.
Tap for more steps…
Simplify each term.
Tap for more steps…
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.
Tap for more steps…
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.
Tap for more steps…
Simplify each term.
Tap for more steps…
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.
Tap for more steps…
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

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Scroll to top