Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
3y=4y−4x
To find the x-intercept,set y=0
3×0=4×0−4x
Any expression multiplied by 0 equals 0
3×0=0−4x
Any expression multiplied by 0 equals 0
0=0−4x
Removing 0 doesn't change the value,so remove it from the expression
0=−4x
Swap the sides of the equation
−4x=0
Change the signs on both sides of the equation
4x=0
Solution
x=0
Show Solution

Solve the equation
Solve for x
Solve for y
x=4y
Evaluate
3y=4y−4x
Swap the sides of the equation
4y−4x=3y
Move the expression to the right-hand side and change its sign
−4x=3y−4y
Subtract the terms
More Steps

Evaluate
3y−4y
Rewrite the expression
(3−4)y
Subtract the numbers
−y
−4x=−y
Change the signs on both sides of the equation
4x=y
Divide both sides
44x=4y
Solution
x=4y
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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
3y=4y−4x
To test if the graph of 3y=4y−4x is symmetry with respect to the origin,substitute -x for x and -y for y
3(−y)=4(−y)−4(−x)
Evaluate
−3y=4(−y)−4(−x)
Evaluate
More Steps

Evaluate
4(−y)−4(−x)
Multiply the numbers
−4y−4(−x)
Multiply the numbers
−4y−(−4x)
Rewrite the expression
−4y+4x
−3y=−4y+4x
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(41)+kπ,k∈Z
Evaluate
3y=4y−4x
Move the expression to the left side
−y+4x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
−sin(θ)×r+4cos(θ)×r=0
Factor the expression
(−sin(θ)+4cos(θ))r=0
Separate into possible cases
r=0−sin(θ)+4cos(θ)=0
Solution
More Steps

Evaluate
−sin(θ)+4cos(θ)=0
Move the expression to the right side
4cos(θ)=0−(−sin(θ))
Subtract the terms
4cos(θ)=sin(θ)
Divide both sides
sin(θ)4cos(θ)=1
Divide the terms
More Steps

Evaluate
sin(θ)4cos(θ)
Rewrite the expression
4sin−1(θ)cos(θ)
Rewrite the expression
4cot(θ)
4cot(θ)=1
Multiply both sides of the equation by 41
4cot(θ)×41=1×41
Calculate
cot(θ)=1×41
Any expression multiplied by 1 remains the same
cot(θ)=41
Use the inverse trigonometric function
θ=arccot(41)
Add the period of kπ,k∈Z to find all solutions
θ=arccot(41)+kπ,k∈Z
r=0θ=arccot(41)+kπ,k∈Z
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=4
Calculate
3y=4y−4x
Take the derivative of both sides
dxd(3y)=dxd(4y−4x)
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
3dxdy=dxd(4y−4x)
Calculate the derivative
More Steps

Evaluate
dxd(4y−4x)
Use differentiation rules
dxd(4y)+dxd(−4x)
Evaluate the derivative
More Steps

Evaluate
dxd(4y)
Use differentiation rules
dyd(4y)×dxdy
Evaluate the derivative
4dxdy
4dxdy+dxd(−4x)
Evaluate the derivative
More Steps

Evaluate
dxd(−4x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−4×dxd(x)
Use dxdxn=nxn−1 to find derivative
−4×1
Any expression multiplied by 1 remains the same
−4
4dxdy−4
3dxdy=4dxdy−4
Move the variable to the left side
3dxdy−4dxdy=−4
Subtract the terms
More Steps

Evaluate
3dxdy−4dxdy
Collect like terms by calculating the sum or difference of their coefficients
(3−4)dxdy
Subtract the numbers
−dxdy
−dxdy=−4
Solution
dxdy=4
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
3y=4y−4x
Take the derivative of both sides
dxd(3y)=dxd(4y−4x)
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
3dxdy=dxd(4y−4x)
Calculate the derivative
More Steps

Evaluate
dxd(4y−4x)
Use differentiation rules
dxd(4y)+dxd(−4x)
Evaluate the derivative
More Steps

Evaluate
dxd(4y)
Use differentiation rules
dyd(4y)×dxdy
Evaluate the derivative
4dxdy
4dxdy+dxd(−4x)
Evaluate the derivative
More Steps

Evaluate
dxd(−4x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−4×dxd(x)
Use dxdxn=nxn−1 to find derivative
−4×1
Any expression multiplied by 1 remains the same
−4
4dxdy−4
3dxdy=4dxdy−4
Move the variable to the left side
3dxdy−4dxdy=−4
Subtract the terms
More Steps

Evaluate
3dxdy−4dxdy
Collect like terms by calculating the sum or difference of their coefficients
(3−4)dxdy
Subtract the numbers
−dxdy
−dxdy=−4
Change the signs on both sides of the equation
dxdy=4
Take the derivative of both sides
dxd(dxdy)=dxd(4)
Calculate the derivative
dx2d2y=dxd(4)
Solution
dx2d2y=0
Show Solution
