Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
3y=24x
To find the x-intercept,set y=0
3×0=24x
Any expression multiplied by 0 equals 0
0=24x
Swap the sides of the equation
24x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=8y
Evaluate
3y=24x
Swap the sides of the equation
24x=3y
Divide both sides
2424x=243y
Divide the numbers
x=243y
Solution
x=8y
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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=24x
To test if the graph of 3y=24x is symmetry with respect to the origin,substitute -x for x and -y for y
3(−y)=24(−x)
Evaluate
−3y=24(−x)
Evaluate
−3y=−24x
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(81)+kπ,k∈Z
Evaluate
3y=24x
Move the expression to the left side
3y−24x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
3sin(θ)×r−24cos(θ)×r=0
Factor the expression
(3sin(θ)−24cos(θ))r=0
Separate into possible cases
r=03sin(θ)−24cos(θ)=0
Solution
More Steps

Evaluate
3sin(θ)−24cos(θ)=0
Move the expression to the right side
−24cos(θ)=0−3sin(θ)
Subtract the terms
−24cos(θ)=−3sin(θ)
Divide both sides
sin(θ)−24cos(θ)=−3
Divide the terms
More Steps

Evaluate
sin(θ)−24cos(θ)
Use b−a=−ba=−ba to rewrite the fraction
−sin(θ)24cos(θ)
Rewrite the expression
−24sin−1(θ)cos(θ)
Rewrite the expression
−24cot(θ)
−24cot(θ)=−3
Multiply both sides of the equation by −241
−24cot(θ)(−241)=−3(−241)
Calculate
cot(θ)=−3(−241)
Calculate
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Evaluate
−3(−241)
Multiplying or dividing an even number of negative terms equals a positive
3×241
Reduce the numbers
1×81
Multiply the numbers
81
cot(θ)=81
Use the inverse trigonometric function
θ=arccot(81)
Add the period of kπ,k∈Z to find all solutions
θ=arccot(81)+kπ,k∈Z
r=0θ=arccot(81)+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=8
Calculate
3y=24x
Take the derivative of both sides
dxd(3y)=dxd(24x)
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
3dxdy=dxd(24x)
Calculate the derivative
More Steps

Evaluate
dxd(24x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
24×dxd(x)
Use dxdxn=nxn−1 to find derivative
24×1
Any expression multiplied by 1 remains the same
24
3dxdy=24
Divide both sides
33dxdy=324
Divide the numbers
dxdy=324
Solution
More Steps

Evaluate
324
Reduce the numbers
18
Calculate
8
dxdy=8
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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
3y=24x
Take the derivative of both sides
dxd(3y)=dxd(24x)
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(24x)
Calculate the derivative
More Steps

Evaluate
dxd(24x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
24×dxd(x)
Use dxdxn=nxn−1 to find derivative
24×1
Any expression multiplied by 1 remains the same
24
3dxdy=24
Divide both sides
33dxdy=324
Divide the numbers
dxdy=324
Divide the numbers
More Steps

Evaluate
324
Reduce the numbers
18
Calculate
8
dxdy=8
Take the derivative of both sides
dxd(dxdy)=dxd(8)
Calculate the derivative
dx2d2y=dxd(8)
Solution
dx2d2y=0
Show Solution
