Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
y×7=2(x×7)
To find the x-intercept,set y=0
0×7=2(x×7)
Any expression multiplied by 0 equals 0
0=2(x×7)
Remove the parentheses
0=2x×7
Multiply the terms
0=14x
Swap the sides of the equation
14x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=2y
Evaluate
y×7=2(x×7)
Remove the parentheses
y×7=2x×7
Simplify
y=2x
Swap the sides of the equation
2x=y
Divide both sides
22x=2y
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
y7=2(x7)
Simplify the expression
y=2x
To test if the graph of y=2x is symmetry with respect to the origin,substitute -x for x and -y for y
−y=2(−x)
Evaluate
−y=−2x
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
y×7=2(x×7)
Use the commutative property to reorder the terms
7y=2(x×7)
Evaluate
More Steps

Evaluate
2(x×7)
Remove the parentheses
2x×7
Multiply the terms
14x
7y=14x
Move the expression to the left side
7y−14x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
7sin(θ)×r−14cos(θ)×r=0
Factor the expression
(7sin(θ)−14cos(θ))r=0
Separate into possible cases
r=07sin(θ)−14cos(θ)=0
Solution
More Steps

Evaluate
7sin(θ)−14cos(θ)=0
Move the expression to the right side
−14cos(θ)=0−7sin(θ)
Subtract the terms
−14cos(θ)=−7sin(θ)
Divide both sides
sin(θ)−14cos(θ)=−7
Divide the terms
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Evaluate
sin(θ)−14cos(θ)
Use b−a=−ba=−ba to rewrite the fraction
−sin(θ)14cos(θ)
Rewrite the expression
−14sin−1(θ)cos(θ)
Rewrite the expression
−14cot(θ)
−14cot(θ)=−7
Multiply both sides of the equation by −141
−14cot(θ)(−141)=−7(−141)
Calculate
cot(θ)=−7(−141)
Calculate
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Evaluate
−7(−141)
Multiplying or dividing an even number of negative terms equals a positive
7×141
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
y7=2(x7)
Simplify the expression
7y=14x
Take the derivative of both sides
dxd(7y)=dxd(14x)
Calculate the derivative
More Steps

Evaluate
dxd(7y)
Use differentiation rules
dyd(7y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(7y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
7×dyd(y)
Use dxdxn=nxn−1 to find derivative
7×1
Any expression multiplied by 1 remains the same
7
7dxdy
7dxdy=dxd(14x)
Calculate the derivative
More Steps

Evaluate
dxd(14x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
14×dxd(x)
Use dxdxn=nxn−1 to find derivative
14×1
Any expression multiplied by 1 remains the same
14
7dxdy=14
Divide both sides
77dxdy=714
Divide the numbers
dxdy=714
Solution
More Steps

Evaluate
714
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
y7=2(x7)
Simplify the expression
7y=14x
Take the derivative of both sides
dxd(7y)=dxd(14x)
Calculate the derivative
More Steps

Evaluate
dxd(7y)
Use differentiation rules
dyd(7y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(7y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
7×dyd(y)
Use dxdxn=nxn−1 to find derivative
7×1
Any expression multiplied by 1 remains the same
7
7dxdy
7dxdy=dxd(14x)
Calculate the derivative
More Steps

Evaluate
dxd(14x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
14×dxd(x)
Use dxdxn=nxn−1 to find derivative
14×1
Any expression multiplied by 1 remains the same
14
7dxdy=14
Divide both sides
77dxdy=714
Divide the numbers
dxdy=714
Divide the numbers
More Steps

Evaluate
714
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
