Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
4y=8x
To find the x-intercept,set y=0
4×0=8x
Any expression multiplied by 0 equals 0
0=8x
Swap the sides of the equation
8x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=2y
Evaluate
4y=8x
Swap the sides of the equation
8x=4y
Divide both sides
88x=84y
Divide the numbers
x=84y
Solution
x=2y
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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
4y=8x
To test if the graph of 4y=8x is symmetry with respect to the origin,substitute -x for x and -y for y
4(−y)=8(−x)
Evaluate
−4y=8(−x)
Evaluate
−4y=−8x
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(21)+kπ,k∈Z
Evaluate
4y=8x
Move the expression to the left side
4y−8x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
4sin(θ)×r−8cos(θ)×r=0
Factor the expression
(4sin(θ)−8cos(θ))r=0
Separate into possible cases
r=04sin(θ)−8cos(θ)=0
Solution
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Evaluate
4sin(θ)−8cos(θ)=0
Move the expression to the right side
−8cos(θ)=0−4sin(θ)
Subtract the terms
−8cos(θ)=−4sin(θ)
Divide both sides
sin(θ)−8cos(θ)=−4
Divide the terms
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Evaluate
sin(θ)−8cos(θ)
Use b−a=−ba=−ba to rewrite the fraction
−sin(θ)8cos(θ)
Rewrite the expression
−8sin−1(θ)cos(θ)
Rewrite the expression
−8cot(θ)
−8cot(θ)=−4
Multiply both sides of the equation by −81
−8cot(θ)(−81)=−4(−81)
Calculate
cot(θ)=−4(−81)
Calculate
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Evaluate
−4(−81)
Multiplying or dividing an even number of negative terms equals a positive
4×81
Reduce the numbers
1×21
Multiply the numbers
21
cot(θ)=21
Use the inverse trigonometric function
θ=arccot(21)
Add the period of kπ,k∈Z to find all solutions
θ=arccot(21)+kπ,k∈Z
r=0θ=arccot(21)+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=2
Calculate
4y=8x
Take the derivative of both sides
dxd(4y)=dxd(8x)
Calculate the derivative
More Steps

Evaluate
dxd(4y)
Use differentiation rules
dyd(4y)×dxdy
Evaluate the derivative
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Evaluate
dyd(4y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
4×dyd(y)
Use dxdxn=nxn−1 to find derivative
4×1
Any expression multiplied by 1 remains the same
4
4dxdy
4dxdy=dxd(8x)
Calculate the derivative
More Steps

Evaluate
dxd(8x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
8×dxd(x)
Use dxdxn=nxn−1 to find derivative
8×1
Any expression multiplied by 1 remains the same
8
4dxdy=8
Divide both sides
44dxdy=48
Divide the numbers
dxdy=48
Solution
More Steps

Evaluate
48
Reduce the numbers
12
Calculate
2
dxdy=2
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Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=0
Calculate
4y=8x
Take the derivative of both sides
dxd(4y)=dxd(8x)
Calculate the derivative
More Steps

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

Evaluate
dyd(4y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
4×dyd(y)
Use dxdxn=nxn−1 to find derivative
4×1
Any expression multiplied by 1 remains the same
4
4dxdy
4dxdy=dxd(8x)
Calculate the derivative
More Steps

Evaluate
dxd(8x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
8×dxd(x)
Use dxdxn=nxn−1 to find derivative
8×1
Any expression multiplied by 1 remains the same
8
4dxdy=8
Divide both sides
44dxdy=48
Divide the numbers
dxdy=48
Divide the numbers
More Steps

Evaluate
48
Reduce the numbers
12
Calculate
2
dxdy=2
Take the derivative of both sides
dxd(dxdy)=dxd(2)
Calculate the derivative
dx2d2y=dxd(2)
Solution
dx2d2y=0
Show Solution
