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x=3y
Question
x=3y
Function
x=0
Evaluate
x=3y
\text{To find the }x\text{-intercept,set }y\text{=0}
x=3\times 0
Solution
x=0
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\text{Find the }x\text{-intercept/zero}
Find the y-intercept
Find the slope
Solve the equation
y=\frac{x}{3}
Evaluate
x=3y
Swap the sides of the equation
3y=x
\text{Multiply both sides of the equation by }\frac{1}{3}
3y\times \frac{1}{3}=x\times \frac{1}{3}
Calculate the product
3y\times \frac{1}{3}=\frac{x}{3}
Solution
y=\frac{x}{3}
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Testing for symmetry
\textrm{Symmetry with respect to the origin}
Evaluate
x=3y
\text{To test if the graph of }x=3y\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
-x=3\left(-y\right)
Evaluate
-x=-3y
Solution
\textrm{Symmetry with respect to the origin}
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Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Rewrite the equation
\begin{align}&r=0\\&\theta =\arctan\left(\frac{1}{3}\right)+k\pi ,k \in \mathbb{Z}\end{align}
Evaluate
x=3y
Move the expression to the left side
x-3y=0
\text{To convert the equation to polar coordinates,substitute }x\text{ for }r\cos\left(\theta \right)\text{ and }y\text{ for }r\sin\left(\theta \right)
\cos\left(\theta \right)\times r-3\sin\left(\theta \right)\times r=0
Factor the expression
\left(\cos\left(\theta \right)-3\sin\left(\theta \right)\right)r=0
Separate into possible cases
\begin{align}&r=0\\&\cos\left(\theta \right)-3\sin\left(\theta \right)=0\end{align}
Solution
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Evaluate
\cos\left(\theta \right)-3\sin\left(\theta \right)=0
Move the expression to the right side
-3\sin\left(\theta \right)=0-\cos\left(\theta \right)
Subtract the terms
-3\sin\left(\theta \right)=-\cos\left(\theta \right)
Divide both sides
\frac{-3\sin\left(\theta \right)}{\cos\left(\theta \right)}=-1
Divide the terms
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Evaluate
\frac{-3\sin\left(\theta \right)}{\cos\left(\theta \right)}
Rewrite the expression
-\frac{3\sin\left(\theta \right)}{\cos\left(\theta \right)}
Rewrite the expression
-3\cos^{-1}\left(\theta \right)\sin\left(\theta \right)
Rewrite the expression
-3\tan\left(\theta \right)
-3\tan\left(\theta \right)=-1
\text{Multiply both sides of the equation by }-\frac{1}{3}
-3\tan\left(\theta \right)\left(-\frac{1}{3}\right)=-\left(-\frac{1}{3}\right)
Calculate
\tan\left(\theta \right)=-\left(-\frac{1}{3}\right)
Multiplying or dividing an even number of negative terms equals a positive
\tan\left(\theta \right)=\frac{1}{3}
Use the inverse trigonometric function
\theta =\arctan\left(\frac{1}{3}\right)
\text{Add the period of }k\pi ,k \in \mathbb{Z}\text{ to find all solutions}
\theta =\arctan\left(\frac{1}{3}\right)+k\pi ,k \in \mathbb{Z}
\begin{align}&r=0\\&\theta =\arctan\left(\frac{1}{3}\right)+k\pi ,k \in \mathbb{Z}\end{align}
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Rewrite in polar form
Rewrite in standard form
Rewrite in slope-intercept form
Find the first derivative
\frac{dy}{dx}=\frac{1}{3}
Calculate
x=3y
Take the derivative of both sides
\frac{d}{dx}\left(x\right)=\frac{d}{dx}\left(3y\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
1=\frac{d}{dx}\left(3y\right)
Calculate the derivative
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Evaluate
\frac{d}{dx}\left(3y\right)
Use differentiation rules
\frac{d}{dy}\left(3y\right)\times \frac{dy}{dx}
Evaluate the derivative
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Evaluate
\frac{d}{dy}\left(3y\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
3\times \frac{d}{dy}\left(y\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
3\times 1
Any expression multiplied by 1 remains the same
3
3\frac{dy}{dx}
1=3\frac{dy}{dx}
Swap the sides of the equation
3\frac{dy}{dx}=1
\text{Multiply both sides of the equation by }\frac{1}{3}
3\frac{dy}{dx}\times \frac{1}{3}=1\times \frac{1}{3}
Any expression multiplied by 1 remains the same
3\frac{dy}{dx}\times \frac{1}{3}=\frac{1}{3}
Solution
\frac{dy}{dx}=\frac{1}{3}
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\text{Find the derivative with respect to }x
\text{Find the derivative with respect to }y
Find the second derivative
\frac{d^2y}{dx^2}=0
Calculate
x=3y
Take the derivative of both sides
\frac{d}{dx}\left(x\right)=\frac{d}{dx}\left(3y\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
1=\frac{d}{dx}\left(3y\right)
Calculate the derivative
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Evaluate
\frac{d}{dx}\left(3y\right)
Use differentiation rules
\frac{d}{dy}\left(3y\right)\times \frac{dy}{dx}
Evaluate the derivative
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Evaluate
\frac{d}{dy}\left(3y\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
3\times \frac{d}{dy}\left(y\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
3\times 1
Any expression multiplied by 1 remains the same
3
3\frac{dy}{dx}
1=3\frac{dy}{dx}
Swap the sides of the equation
3\frac{dy}{dx}=1
\text{Multiply both sides of the equation by }\frac{1}{3}
3\frac{dy}{dx}\times \frac{1}{3}=1\times \frac{1}{3}
Any expression multiplied by 1 remains the same
3\frac{dy}{dx}\times \frac{1}{3}=\frac{1}{3}
\text{Cancel out the greatest common factor }3
\frac{dy}{dx}=\frac{1}{3}
Take the derivative of both sides
\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{1}{3}\right)
Calculate the derivative
\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{1}{3}\right)
Solution
\frac{d^2y}{dx^2}=0
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\text{Find the derivative with respect to }x
\text{Find the derivative with respect to }y
Graph
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