Graph t^2+2/3t+1/9

t2+23t+19
Combine 23 and x.
y=x2+2×3+19
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for x2+2×3+19.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=23,c=19
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=232(1)
Simplify the right side.
Multiply the numerator by the reciprocal of the denominator.
d=23⋅12⋅1
Cancel the common factor of 2.
Factor 2 out of 2⋅1.
d=23⋅12(1)
Cancel the common factor.
d=23⋅12⋅1
Rewrite the expression.
d=13
d=13
d=13
Find the value of e using the formula e=c-b24a.
Simplify each term.
Simplify the numerator.
Apply the product rule to 23.
e=19-22324⋅1
Raise 2 to the power of 2.
e=19-4324⋅1
Raise 3 to the power of 2.
e=19-494⋅1
e=19-494⋅1
Multiply 4 by 1.
e=19-494
Multiply the numerator by the reciprocal of the denominator.
e=19-(49⋅14)
Cancel the common factor of 4.
Cancel the common factor.
e=19-(49⋅14)
Rewrite the expression.
e=19-19
e=19-19
e=19-19
Combine the numerators over the common denominator.
e=1-19
Subtract 1 from 1.
e=09
Divide 0 by 9.
e=0
e=0
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x+13)2+0
(x+13)2+0
Set y equal to the new right side.
y=(x+13)2+0
y=(x+13)2+0
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=1
h=-13
k=0
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-13,0)
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.
(-13,14)
(-13,14)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-13
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=-14
y=-14
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-13,0)
Focus: (-13,14)
Axis of Symmetry: x=-13
Directrix: y=-14
Direction: Opens Up
Vertex: (-13,0)
Focus: (-13,14)
Axis of Symmetry: x=-13
Directrix: y=-14
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.
Replace the variable x with -1 in the expression.
f(-1)=(-1)2+2(-1)3+19
Simplify the result.
Multiply 2 by -1.
f(-1)=(-1)2+-23+19
Simplify each term.
Raise -1 to the power of 2.
f(-1)=1+-23+19
Move the negative in front of the fraction.
f(-1)=1-23+19
f(-1)=1-23+19
Simplify the expression.
Write 1 as a fraction with a common denominator.
f(-1)=33-23+19
Combine the numerators over the common denominator.
f(-1)=3-23+19
f(-1)=3-23+19
Subtract 2 from 3.
f(-1)=13+19
To write 13 as a fraction with a common denominator, multiply by 33.
f(-1)=13⋅33+19
Write each expression with a common denominator of 9, by multiplying each by an appropriate factor of 1.
Multiply 13 and 33.
f(-1)=33⋅3+19
Multiply 3 by 3.
f(-1)=39+19
f(-1)=39+19
Combine the numerators over the common denominator.
f(-1)=3+19
f(-1)=49
49
49
The y value at x=-1 is 49.
y=49
Replace the variable x with -2 in the expression.
f(-2)=(-2)2+2(-2)3+19
Simplify the result.
Multiply 2 by -2.
f(-2)=(-2)2+-43+19
Simplify each term.
Raise -2 to the power of 2.
f(-2)=4+-43+19
Move the negative in front of the fraction.
f(-2)=4-43+19
f(-2)=4-43+19
To write 4 as a fraction with a common denominator, multiply by 33.
f(-2)=4⋅33-43+19
Combine 4 and 33.
f(-2)=4⋅33-43+19
Combine the numerators over the common denominator.
f(-2)=4⋅3-43+19
Simplify the numerator.
Multiply 4 by 3.
f(-2)=12-43+19
Subtract 4 from 12.
f(-2)=83+19
f(-2)=83+19
To write 83 as a fraction with a common denominator, multiply by 33.
f(-2)=83⋅33+19
Write each expression with a common denominator of 9, by multiplying each by an appropriate factor of 1.
Multiply 83 and 33.
f(-2)=8⋅33⋅3+19
Multiply 3 by 3.
f(-2)=8⋅39+19
f(-2)=8⋅39+19
Combine the numerators over the common denominator.
f(-2)=8⋅3+19
Simplify the numerator.
Multiply 8 by 3.
f(-2)=24+19
f(-2)=259
f(-2)=259
259
259
The y value at x=-2 is 259.
y=259
Replace the variable x with 1 in the expression.
f(1)=(1)2+2(1)3+19
Simplify the result.
Multiply 2 by 1.
f(1)=(1)2+23+19
One to any power is one.
f(1)=1+23+19
Find the common denominator.
Write 1 as a fraction with denominator 1.
f(1)=11+23+19
Multiply 11 by 99.
f(1)=11⋅99+23+19
Multiply 11 and 99.
f(1)=99+23+19
Multiply 23 by 33.
f(1)=99+23⋅33+19
Combine.
f(1)=99+2⋅33⋅3+19
Multiply 3 by 3.
f(1)=99+2⋅39+19
f(1)=99+2⋅39+19
Combine fractions.
Combine fractions with similar denominators.
f(1)=9+2⋅3+19
Multiply 2 by 3.
f(1)=9+6+19
f(1)=9+6+19
Simplify the numerator.
f(1)=15+19
f(1)=169
f(1)=169
169
169
The y value at x=1 is 169.
y=169
Replace the variable x with 2 in the expression.
f(2)=(2)2+2(2)3+19
Simplify the result.
Multiply 2 by 2.
f(2)=(2)2+43+19
Raise 2 to the power of 2.
f(2)=4+43+19
Find the common denominator.
Write 4 as a fraction with denominator 1.
f(2)=41+43+19
Multiply 41 by 99.
f(2)=41⋅99+43+19
Multiply 41 and 99.
f(2)=4⋅99+43+19
Multiply 43 by 33.
f(2)=4⋅99+43⋅33+19
Combine.
f(2)=4⋅99+4⋅33⋅3+19
Multiply 3 by 3.
f(2)=4⋅99+4⋅39+19
f(2)=4⋅99+4⋅39+19
Combine fractions.
Combine fractions with similar denominators.
f(2)=4⋅9+4⋅3+19
Multiply.
Multiply 4 by 9.
f(2)=36+4⋅3+19
Multiply 4 by 3.
f(2)=36+12+19
f(2)=36+12+19
f(2)=36+12+19
Simplify the numerator.
f(2)=48+19
f(2)=499
f(2)=499
499
499
The y value at x=2 is 499.
y=499
Graph the parabola using its properties and the selected points.
xy-2259-149-13011692499
xy-2259-149-13011692499
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-13,0)
Focus: (-13,14)
Axis of Symmetry: x=-13
Directrix: y=-14
xy-2259-149-13011692499
Graph t^2+2/3t+1/9