Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
3x×8=4y×7
To find the x-intercept,set y=0
3x×8=4×0×7
Any expression multiplied by 0 equals 0
3x×8=0
Multiply the terms
24x=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=67y
Evaluate
3x×8=4y×7
Multiply the terms
24x=4y×7
Multiply the terms
24x=28y
Divide both sides
2424x=2428y
Divide the numbers
x=2428y
Solution
x=67y
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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
3x8=4y7
Simplify the expression
24x=28y
To test if the graph of 24x=28y is symmetry with respect to the origin,substitute -x for x and -y for y
24(−x)=28(−y)
Evaluate
−24x=28(−y)
Evaluate
−24x=−28y
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(76)+kπ,k∈Z
Evaluate
3x×8=4y×7
Evaluate
24x=4y×7
Evaluate
24x=28y
Move the expression to the left side
24x−28y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
24cos(θ)×r−28sin(θ)×r=0
Factor the expression
(24cos(θ)−28sin(θ))r=0
Separate into possible cases
r=024cos(θ)−28sin(θ)=0
Solution
More Steps

Evaluate
24cos(θ)−28sin(θ)=0
Move the expression to the right side
−28sin(θ)=0−24cos(θ)
Subtract the terms
−28sin(θ)=−24cos(θ)
Divide both sides
cos(θ)−28sin(θ)=−24
Divide the terms
More Steps

Evaluate
cos(θ)−28sin(θ)
Use b−a=−ba=−ba to rewrite the fraction
−cos(θ)28sin(θ)
Rewrite the expression
−28cos−1(θ)sin(θ)
Rewrite the expression
−28tan(θ)
−28tan(θ)=−24
Multiply both sides of the equation by −281
−28tan(θ)(−281)=−24(−281)
Calculate
tan(θ)=−24(−281)
Calculate
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Evaluate
−24(−281)
Multiplying or dividing an even number of negative terms equals a positive
24×281
Reduce the numbers
6×71
Multiply the numbers
76
tan(θ)=76
Use the inverse trigonometric function
θ=arctan(76)
Add the period of kπ,k∈Z to find all solutions
θ=arctan(76)+kπ,k∈Z
r=0θ=arctan(76)+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=76
Calculate
3x8=4y7
Simplify the expression
24x=28y
Take the derivative of both sides
dxd(24x)=dxd(28y)
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
24=dxd(28y)
Calculate the derivative
More Steps

Evaluate
dxd(28y)
Use differentiation rules
dyd(28y)×dxdy
Evaluate the derivative
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Evaluate
dyd(28y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
28×dyd(y)
Use dxdxn=nxn−1 to find derivative
28×1
Any expression multiplied by 1 remains the same
28
28dxdy
24=28dxdy
Swap the sides of the equation
28dxdy=24
Divide both sides
2828dxdy=2824
Divide the numbers
dxdy=2824
Solution
dxdy=76
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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
3x8=4y7
Simplify the expression
24x=28y
Take the derivative of both sides
dxd(24x)=dxd(28y)
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
24=dxd(28y)
Calculate the derivative
More Steps

Evaluate
dxd(28y)
Use differentiation rules
dyd(28y)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(28y)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
28×dyd(y)
Use dxdxn=nxn−1 to find derivative
28×1
Any expression multiplied by 1 remains the same
28
28dxdy
24=28dxdy
Swap the sides of the equation
28dxdy=24
Divide both sides
2828dxdy=2824
Divide the numbers
dxdy=2824
Cancel out the common factor 4
dxdy=76
Take the derivative of both sides
dxd(dxdy)=dxd(76)
Calculate the derivative
dx2d2y=dxd(76)
Solution
dx2d2y=0
Show Solution
