Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
9y=3x×18
To find the x-intercept,set y=0
9×0=3x×18
Any expression multiplied by 0 equals 0
0=3x×18
Multiply the terms
0=54x
Swap the sides of the equation
54x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=6y
Evaluate
9y=3x×18
Multiply the terms
9y=54x
Swap the sides of the equation
54x=9y
Divide both sides
5454x=549y
Divide the numbers
x=549y
Solution
x=6y
Show Solution

Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Symmetry with respect to the origin
Evaluate
9y=3x18
Simplify the expression
9y=54x
To test if the graph of 9y=54x is symmetry with respect to the origin,substitute -x for x and -y for y
9(−y)=54(−x)
Evaluate
−9y=54(−x)
Evaluate
−9y=−54x
Solution
Symmetry with respect to the origin
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Rewrite the equation
Rewrite in polar form
Rewrite in standard form
Rewrite in slope-intercept form
r=0θ=arccot(61)+kπ,k∈Z
Evaluate
9y=3x×18
Evaluate
9y=54x
Move the expression to the left side
9y−54x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
9sin(θ)×r−54cos(θ)×r=0
Factor the expression
(9sin(θ)−54cos(θ))r=0
Separate into possible cases
r=09sin(θ)−54cos(θ)=0
Solution
More Steps

Evaluate
9sin(θ)−54cos(θ)=0
Move the expression to the right side
−54cos(θ)=0−9sin(θ)
Subtract the terms
−54cos(θ)=−9sin(θ)
Divide both sides
sin(θ)−54cos(θ)=−9
Divide the terms
More Steps

Evaluate
sin(θ)−54cos(θ)
Use b−a=−ba=−ba to rewrite the fraction
−sin(θ)54cos(θ)
Rewrite the expression
−54sin−1(θ)cos(θ)
Rewrite the expression
−54cot(θ)
−54cot(θ)=−9
Multiply both sides of the equation by −541
−54cot(θ)(−541)=−9(−541)
Calculate
cot(θ)=−9(−541)
Calculate
More Steps

Evaluate
−9(−541)
Multiplying or dividing an even number of negative terms equals a positive
9×541
Reduce the numbers
1×61
Multiply the numbers
61
cot(θ)=61
Use the inverse trigonometric function
θ=arccot(61)
Add the period of kπ,k∈Z to find all solutions
θ=arccot(61)+kπ,k∈Z
r=0θ=arccot(61)+kπ,k∈Z
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=6
Calculate
9y=3x18
Simplify the expression
9y=54x
Take the derivative of both sides
dxd(9y)=dxd(54x)
Calculate the derivative
More Steps

Evaluate
dxd(9y)
Use differentiation rules
dyd(9y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(9y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
9×dyd(y)
Use dxdxn=nxn−1 to find derivative
9×1
Any expression multiplied by 1 remains the same
9
9dxdy
9dxdy=dxd(54x)
Calculate the derivative
More Steps

Evaluate
dxd(54x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
54×dxd(x)
Use dxdxn=nxn−1 to find derivative
54×1
Any expression multiplied by 1 remains the same
54
9dxdy=54
Divide both sides
99dxdy=954
Divide the numbers
dxdy=954
Solution
More Steps

Evaluate
954
Reduce the numbers
16
Calculate
6
dxdy=6
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=0
Calculate
9y=3x18
Simplify the expression
9y=54x
Take the derivative of both sides
dxd(9y)=dxd(54x)
Calculate the derivative
More Steps

Evaluate
dxd(9y)
Use differentiation rules
dyd(9y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(9y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
9×dyd(y)
Use dxdxn=nxn−1 to find derivative
9×1
Any expression multiplied by 1 remains the same
9
9dxdy
9dxdy=dxd(54x)
Calculate the derivative
More Steps

Evaluate
dxd(54x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
54×dxd(x)
Use dxdxn=nxn−1 to find derivative
54×1
Any expression multiplied by 1 remains the same
54
9dxdy=54
Divide both sides
99dxdy=954
Divide the numbers
dxdy=954
Divide the numbers
More Steps

Evaluate
954
Reduce the numbers
16
Calculate
6
dxdy=6
Take the derivative of both sides
dxd(dxdy)=dxd(6)
Calculate the derivative
dx2d2y=dxd(6)
Solution
dx2d2y=0
Show Solution
