Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
−3x×18=3y
To find the x-intercept,set y=0
−3x×18=3×0
Any expression multiplied by 0 equals 0
−3x×18=0
Multiply the terms
−54x=0
Change the signs on both sides of the equation
54x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=−18y
Evaluate
−3x×18=3y
Multiply the terms
−54x=3y
Change the signs on both sides of the equation
54x=−3y
Divide both sides
5454x=54−3y
Divide the numbers
x=54−3y
Solution
More Steps

Evaluate
54−3y
Cancel out the common factor 3
18−y
Use b−a=−ba=−ba to rewrite the fraction
−18y
x=−18y
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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
−3x18=3y
Simplify the expression
−54x=3y
To test if the graph of −54x=3y is symmetry with respect to the origin,substitute -x for x and -y for y
−54(−x)=3(−y)
Evaluate
54x=3(−y)
Evaluate
54x=−3y
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=arctan(−18)+kπ,k∈Z
Evaluate
−3x×18=3y
Evaluate
−54x=3y
Move the expression to the left side
−54x−3y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
−54cos(θ)×r−3sin(θ)×r=0
Factor the expression
(−54cos(θ)−3sin(θ))r=0
Separate into possible cases
r=0−54cos(θ)−3sin(θ)=0
Solution
More Steps

Evaluate
−54cos(θ)−3sin(θ)=0
Move the expression to the right side
−3sin(θ)=0−(−54cos(θ))
Subtract the terms
−3sin(θ)=54cos(θ)
Divide both sides
cos(θ)−3sin(θ)=54
Divide the terms
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Evaluate
cos(θ)−3sin(θ)
Use b−a=−ba=−ba to rewrite the fraction
−cos(θ)3sin(θ)
Rewrite the expression
−3cos−1(θ)sin(θ)
Rewrite the expression
−3tan(θ)
−3tan(θ)=54
Multiply both sides of the equation by −31
−3tan(θ)(−31)=54(−31)
Calculate
tan(θ)=54(−31)
Calculate
More Steps

Evaluate
54(−31)
Multiplying or dividing an odd number of negative terms equals a negative
−54×31
Reduce the numbers
−18×1
Simplify
−18
tan(θ)=−18
Use the inverse trigonometric function
θ=arctan(−18)
Add the period of kπ,k∈Z to find all solutions
θ=arctan(−18)+kπ,k∈Z
r=0θ=arctan(−18)+kπ,k∈Z
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Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−18
Calculate
−3x18=3y
Simplify the expression
−54x=3y
Take the derivative of both sides
dxd(−54x)=dxd(3y)
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
−54=dxd(3y)
Calculate the derivative
More Steps

Evaluate
dxd(3y)
Use differentiation rules
dyd(3y)×dxdy
Evaluate the derivative
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Evaluate
dyd(3y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
3×dyd(y)
Use dxdxn=nxn−1 to find derivative
3×1
Any expression multiplied by 1 remains the same
3
3dxdy
−54=3dxdy
Swap the sides of the equation
3dxdy=−54
Divide both sides
33dxdy=3−54
Divide the numbers
dxdy=3−54
Solution
More Steps

Evaluate
3−54
Reduce the numbers
1−18
Calculate
−18
dxdy=−18
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
−3x18=3y
Simplify the expression
−54x=3y
Take the derivative of both sides
dxd(−54x)=dxd(3y)
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
−54=dxd(3y)
Calculate the derivative
More Steps

Evaluate
dxd(3y)
Use differentiation rules
dyd(3y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(3y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
3×dyd(y)
Use dxdxn=nxn−1 to find derivative
3×1
Any expression multiplied by 1 remains the same
3
3dxdy
−54=3dxdy
Swap the sides of the equation
3dxdy=−54
Divide both sides
33dxdy=3−54
Divide the numbers
dxdy=3−54
Divide the numbers
More Steps

Evaluate
3−54
Reduce the numbers
1−18
Calculate
−18
dxdy=−18
Take the derivative of both sides
dxd(dxdy)=dxd(−18)
Calculate the derivative
dx2d2y=dxd(−18)
Solution
dx2d2y=0
Show Solution
