Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
5y=10x×11
To find the x-intercept,set y=0
5×0=10x×11
Any expression multiplied by 0 equals 0
0=10x×11
Multiply the terms
0=110x
Swap the sides of the equation
110x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=22y
Evaluate
5y=10x×11
Multiply the terms
5y=110x
Swap the sides of the equation
110x=5y
Divide both sides
110110x=1105y
Divide the numbers
x=1105y
Solution
x=22y
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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
5y=10x11
Simplify the expression
5y=110x
To test if the graph of 5y=110x is symmetry with respect to the origin,substitute -x for x and -y for y
5(−y)=110(−x)
Evaluate
−5y=110(−x)
Evaluate
−5y=−110x
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(221)+kπ,k∈Z
Evaluate
5y=10x×11
Evaluate
5y=110x
Move the expression to the left side
5y−110x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
5sin(θ)×r−110cos(θ)×r=0
Factor the expression
(5sin(θ)−110cos(θ))r=0
Separate into possible cases
r=05sin(θ)−110cos(θ)=0
Solution
More Steps

Evaluate
5sin(θ)−110cos(θ)=0
Move the expression to the right side
−110cos(θ)=0−5sin(θ)
Subtract the terms
−110cos(θ)=−5sin(θ)
Divide both sides
sin(θ)−110cos(θ)=−5
Divide the terms
More Steps

Evaluate
sin(θ)−110cos(θ)
Use b−a=−ba=−ba to rewrite the fraction
−sin(θ)110cos(θ)
Rewrite the expression
−110sin−1(θ)cos(θ)
Rewrite the expression
−110cot(θ)
−110cot(θ)=−5
Multiply both sides of the equation by −1101
−110cot(θ)(−1101)=−5(−1101)
Calculate
cot(θ)=−5(−1101)
Calculate
More Steps

Evaluate
−5(−1101)
Multiplying or dividing an even number of negative terms equals a positive
5×1101
Reduce the numbers
1×221
Multiply the numbers
221
cot(θ)=221
Use the inverse trigonometric function
θ=arccot(221)
Add the period of kπ,k∈Z to find all solutions
θ=arccot(221)+kπ,k∈Z
r=0θ=arccot(221)+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=22
Calculate
5y=10x11
Simplify the expression
5y=110x
Take the derivative of both sides
dxd(5y)=dxd(110x)
Calculate the derivative
More Steps

Evaluate
dxd(5y)
Use differentiation rules
dyd(5y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(5y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
5×dyd(y)
Use dxdxn=nxn−1 to find derivative
5×1
Any expression multiplied by 1 remains the same
5
5dxdy
5dxdy=dxd(110x)
Calculate the derivative
More Steps

Evaluate
dxd(110x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
110×dxd(x)
Use dxdxn=nxn−1 to find derivative
110×1
Any expression multiplied by 1 remains the same
110
5dxdy=110
Divide both sides
55dxdy=5110
Divide the numbers
dxdy=5110
Solution
More Steps

Evaluate
5110
Reduce the numbers
122
Calculate
22
dxdy=22
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
5y=10x11
Simplify the expression
5y=110x
Take the derivative of both sides
dxd(5y)=dxd(110x)
Calculate the derivative
More Steps

Evaluate
dxd(5y)
Use differentiation rules
dyd(5y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(5y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
5×dyd(y)
Use dxdxn=nxn−1 to find derivative
5×1
Any expression multiplied by 1 remains the same
5
5dxdy
5dxdy=dxd(110x)
Calculate the derivative
More Steps

Evaluate
dxd(110x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
110×dxd(x)
Use dxdxn=nxn−1 to find derivative
110×1
Any expression multiplied by 1 remains the same
110
5dxdy=110
Divide both sides
55dxdy=5110
Divide the numbers
dxdy=5110
Divide the numbers
More Steps

Evaluate
5110
Reduce the numbers
122
Calculate
22
dxdy=22
Take the derivative of both sides
dxd(dxdy)=dxd(22)
Calculate the derivative
dx2d2y=dxd(22)
Solution
dx2d2y=0
Show Solution
