Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
35x=75y
To find the x-intercept,set y=0
35x=75×0
Any expression multiplied by 0 equals 0
35x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=715y
Evaluate
35x=75y
Divide both sides
3535x=3575y
Divide the numbers
x=3575y
Solution
x=715y
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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
35x=75y
To test if the graph of 35x=75y is symmetry with respect to the origin,substitute -x for x and -y for y
35(−x)=75(−y)
Evaluate
−35x=75(−y)
Evaluate
−35x=−75y
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(157)+kπ,k∈Z
Evaluate
35x=75y
Move the expression to the left side
35x−75y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
35cos(θ)×r−75sin(θ)×r=0
Factor the expression
(35cos(θ)−75sin(θ))r=0
Separate into possible cases
r=035cos(θ)−75sin(θ)=0
Solution
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Evaluate
35cos(θ)−75sin(θ)=0
Move the expression to the right side
−75sin(θ)=0−35cos(θ)
Subtract the terms
−75sin(θ)=−35cos(θ)
Divide both sides
cos(θ)−75sin(θ)=−35
Divide the terms
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Evaluate
cos(θ)−75sin(θ)
Use b−a=−ba=−ba to rewrite the fraction
−cos(θ)75sin(θ)
Rewrite the expression
−75cos−1(θ)sin(θ)
Rewrite the expression
−75tan(θ)
−75tan(θ)=−35
Multiply both sides of the equation by −751
−75tan(θ)(−751)=−35(−751)
Calculate
tan(θ)=−35(−751)
Calculate
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Evaluate
−35(−751)
Multiplying or dividing an even number of negative terms equals a positive
35×751
Reduce the numbers
7×151
Multiply the numbers
157
tan(θ)=157
Use the inverse trigonometric function
θ=arctan(157)
Add the period of kπ,k∈Z to find all solutions
θ=arctan(157)+kπ,k∈Z
r=0θ=arctan(157)+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=157
Calculate
35x=75y
Take the derivative of both sides
dxd(35x)=dxd(75y)
Calculate the derivative
More Steps

Evaluate
dxd(35x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
35×dxd(x)
Use dxdxn=nxn−1 to find derivative
35×1
Any expression multiplied by 1 remains the same
35
35=dxd(75y)
Calculate the derivative
More Steps

Evaluate
dxd(75y)
Use differentiation rules
dyd(75y)×dxdy
Evaluate the derivative
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Evaluate
dyd(75y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
75×dyd(y)
Use dxdxn=nxn−1 to find derivative
75×1
Any expression multiplied by 1 remains the same
75
75dxdy
35=75dxdy
Swap the sides of the equation
75dxdy=35
Divide both sides
7575dxdy=7535
Divide the numbers
dxdy=7535
Solution
dxdy=157
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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
35x=75y
Take the derivative of both sides
dxd(35x)=dxd(75y)
Calculate the derivative
More Steps

Evaluate
dxd(35x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
35×dxd(x)
Use dxdxn=nxn−1 to find derivative
35×1
Any expression multiplied by 1 remains the same
35
35=dxd(75y)
Calculate the derivative
More Steps

Evaluate
dxd(75y)
Use differentiation rules
dyd(75y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(75y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
75×dyd(y)
Use dxdxn=nxn−1 to find derivative
75×1
Any expression multiplied by 1 remains the same
75
75dxdy
35=75dxdy
Swap the sides of the equation
75dxdy=35
Divide both sides
7575dxdy=7535
Divide the numbers
dxdy=7535
Cancel out the common factor 5
dxdy=157
Take the derivative of both sides
dxd(dxdy)=dxd(157)
Calculate the derivative
dx2d2y=dxd(157)
Solution
dx2d2y=0
Show Solution
