Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
−7x=16y
To find the x-intercept,set y=0
−7x=16×0
Any expression multiplied by 0 equals 0
−7x=0
Change the signs on both sides of the equation
7x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=−716y
Evaluate
−7x=16y
Change the signs on both sides of the equation
7x=−16y
Divide both sides
77x=7−16y
Divide the numbers
x=7−16y
Solution
x=−716y
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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
−7x=16y
To test if the graph of −7x=16y is symmetry with respect to the origin,substitute -x for x and -y for y
−7(−x)=16(−y)
Evaluate
7x=16(−y)
Evaluate
7x=−16y
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(−167)+kπ,k∈Z
Evaluate
−7x=16y
Move the expression to the left side
−7x−16y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
−7cos(θ)×r−16sin(θ)×r=0
Factor the expression
(−7cos(θ)−16sin(θ))r=0
Separate into possible cases
r=0−7cos(θ)−16sin(θ)=0
Solution
More Steps

Evaluate
−7cos(θ)−16sin(θ)=0
Move the expression to the right side
−16sin(θ)=0−(−7cos(θ))
Subtract the terms
−16sin(θ)=7cos(θ)
Divide both sides
cos(θ)−16sin(θ)=7
Divide the terms
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Evaluate
cos(θ)−16sin(θ)
Use b−a=−ba=−ba to rewrite the fraction
−cos(θ)16sin(θ)
Rewrite the expression
−16cos−1(θ)sin(θ)
Rewrite the expression
−16tan(θ)
−16tan(θ)=7
Multiply both sides of the equation by −161
−16tan(θ)(−161)=7(−161)
Calculate
tan(θ)=7(−161)
Calculate
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Evaluate
7(−161)
Multiplying or dividing an odd number of negative terms equals a negative
−7×161
Multiply the numbers
−167
tan(θ)=−167
Use the inverse trigonometric function
θ=arctan(−167)
Add the period of kπ,k∈Z to find all solutions
θ=arctan(−167)+kπ,k∈Z
r=0θ=arctan(−167)+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=−167
Calculate
−7x=16y
Take the derivative of both sides
dxd(−7x)=dxd(16y)
Calculate the derivative
More Steps

Evaluate
dxd(−7x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−7×dxd(x)
Use dxdxn=nxn−1 to find derivative
−7×1
Any expression multiplied by 1 remains the same
−7
−7=dxd(16y)
Calculate the derivative
More Steps

Evaluate
dxd(16y)
Use differentiation rules
dyd(16y)×dxdy
Evaluate the derivative
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Evaluate
dyd(16y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
16×dyd(y)
Use dxdxn=nxn−1 to find derivative
16×1
Any expression multiplied by 1 remains the same
16
16dxdy
−7=16dxdy
Swap the sides of the equation
16dxdy=−7
Divide both sides
1616dxdy=16−7
Divide the numbers
dxdy=16−7
Solution
dxdy=−167
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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
−7x=16y
Take the derivative of both sides
dxd(−7x)=dxd(16y)
Calculate the derivative
More Steps

Evaluate
dxd(−7x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−7×dxd(x)
Use dxdxn=nxn−1 to find derivative
−7×1
Any expression multiplied by 1 remains the same
−7
−7=dxd(16y)
Calculate the derivative
More Steps

Evaluate
dxd(16y)
Use differentiation rules
dyd(16y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(16y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
16×dyd(y)
Use dxdxn=nxn−1 to find derivative
16×1
Any expression multiplied by 1 remains the same
16
16dxdy
−7=16dxdy
Swap the sides of the equation
16dxdy=−7
Divide both sides
1616dxdy=16−7
Divide the numbers
dxdy=16−7
Use b−a=−ba=−ba to rewrite the fraction
dxdy=−167
Take the derivative of both sides
dxd(dxdy)=dxd(−167)
Calculate the derivative
dx2d2y=dxd(−167)
Solution
dx2d2y=0
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