Solve x=3y^2+2 - ScanMath

Question

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

y^{2} = \frac{1}{3} (x - 2)

Evaluate

x = 3 y^{2} + 2

Swap the sides of the equation

3 y^{2} + 2 = x

Move the constant to the right-hand side and change its sign

3 y^{2} = x - 2

Multiply both sides of the equation by \frac{1}{3}

3 y^{2} \times \frac{1}{3} = (x - 2) \times \frac{1}{3}

Multiply the terms

y^{2} = (x - 2) \times \frac{1}{3}

Multiply the terms

More Steps

Evaluate

(x - 2) \times \frac{1}{3}

Apply the distributive property

x \times \frac{1}{3} - 2 \times \frac{1}{3}

Use the commutative property to reorder the terms

\frac{1}{3} x - 2 \times \frac{1}{3}

Multiply the numbers

\frac{1}{3} x - \frac{2}{3}

y^{2} = \frac{1}{3} x - \frac{2}{3}

Solution

y^{2} = \frac{1}{3} (x - 2)

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Solve the equation

y = \frac{3 x - 6}{3} y = - \frac{3 x - 6}{3}

Evaluate

x = 3 y^{2} + 2

Swap the sides of the equation

3 y^{2} + 2 = x

Move the constant to the right-hand side and change its sign

3 y^{2} = x - 2

Multiply both sides of the equation by \frac{1}{3}

3 y^{2} \times \frac{1}{3} = (x - 2) \times \frac{1}{3}

Calculate the product

3 y^{2} \times \frac{1}{3} = \frac{x - 2}{3}

Cancel out the greatest common factor 3

y^{2} = \frac{x - 2}{3}

Take the root of both sides of the equation and remember to use both positive and negative roots

y = \pm \frac{x - 2}{3}

Simplify the expression

More Steps

Evaluate

\frac{x - 2}{3}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{x - 2}{3}

Multiply by the Conjugate

\frac{x - 2 \times 3}{3 \times 3}

Calculate

\frac{x - 2 \times 3}{3}

Calculate

More Steps

Evaluate

x - 2 \times 3

The product of roots with the same index is equal to the root of the product

(x - 2) \times 3

Calculate the product

3 x - 6

\frac{3 x - 6}{3}

y = \pm \frac{3 x - 6}{3}

Solution

y = \frac{3 x - 6}{3} y = - \frac{3 x - 6}{3}

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Rewrite the equation

r = \frac{cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )} r = \frac{cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )}

Evaluate

x = 3 y^{2} + 2

Move the expression to the left side

x - 3 y^{2} = 2

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

cos (θ) \times r - 3 (sin (θ) \times r)^{2} = 2

Factor the expression

- 3 sin^{2} (θ) \times r^{2} + cos (θ) \times r = 2

Subtract the terms

- 3 sin^{2} (θ) \times r^{2} + cos (θ) \times r - 2 = 2 - 2

Evaluate

- 3 sin^{2} (θ) \times r^{2} + cos (θ) \times r - 2 = 0

Solve using the quadratic formula

r = \frac{- cos ( θ ) \pm cos ^{2} ( θ ) - 4 ( - 3 sin ^{2} ( θ ) ) ( - 2 )}{- 6 sin ^{2} ( θ )}

Simplify

r = \frac{- cos ( θ ) \pm 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )} r = \frac{- cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )}

Evaluate

More Steps

Evaluate

\frac{- cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )}

Rewrite the expression

- \frac{- cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )}

Rewrite the expression

\frac{cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )}

r = \frac{cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )} r = \frac{- cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{- 6 sin ^{2} ( θ )}

Solution

r = \frac{cos ( θ ) - 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )} r = \frac{cos ( θ ) + 25 cos ^{2} ( θ ) - 24}{6 sin ^{2} ( θ )}

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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

Not symmetry with respect to the origin

Evaluate

x = 3 y^{2} + 2

To test if the graph of x = 3 y^{2} + 2 is symmetry with respect to the origin,substitute -x for x and -y for y

- x = 3 (- y)^{2} + 2

Evaluate

- x = 3 y^{2} + 2

Solution

Not symmetry with respect to the origin

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{1}{6 y}

Calculate

x = 3 y^{2} + 2

Take the derivative of both sides

\frac{d}{d x} (x) = \frac{d}{d x} (3 y^{2} + 2)

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

1 = \frac{d}{d x} (3 y^{2} + 2)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 y^{2} + 2)

Use differentiation rules

\frac{d}{d x} (3 y^{2}) + \frac{d}{d x} (2)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 y^{2})

Use differentiation rules

\frac{d}{d y} (3 y^{2}) \times \frac{d y}{d x}

Evaluate the derivative

6 y \frac{d y}{d x}

6 y \frac{d y}{d x} + \frac{d}{d x} (2)

Use \frac{d}{d x} (c) = 0 to find derivative

6 y \frac{d y}{d x} + 0

Evaluate

6 y \frac{d y}{d x}

1 = 6 y \frac{d y}{d x}

Swap the sides of the equation

6 y \frac{d y}{d x} = 1

Multiply both sides of the equation by \frac{1}{6 y}

6 y \frac{d y}{d x} \times \frac{1}{6 y} = 1 \times \frac{1}{6 y}

Any expression multiplied by 1 remains the same

6 y \frac{d y}{d x} \times \frac{1}{6 y} = \frac{1}{6 y}

Solution

\frac{d y}{d x} = \frac{1}{6 y}

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Find the second derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{36 y ^{3}}

Calculate

x = 3 y^{2} + 2

Take the derivative of both sides

\frac{d}{d x} (x) = \frac{d}{d x} (3 y^{2} + 2)

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

1 = \frac{d}{d x} (3 y^{2} + 2)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 y^{2} + 2)

Use differentiation rules

\frac{d}{d x} (3 y^{2}) + \frac{d}{d x} (2)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 y^{2})

Use differentiation rules

\frac{d}{d y} (3 y^{2}) \times \frac{d y}{d x}

Evaluate the derivative

6 y \frac{d y}{d x}

6 y \frac{d y}{d x} + \frac{d}{d x} (2)

Use \frac{d}{d x} (c) = 0 to find derivative

6 y \frac{d y}{d x} + 0

Evaluate

6 y \frac{d y}{d x}

1 = 6 y \frac{d y}{d x}

Swap the sides of the equation

6 y \frac{d y}{d x} = 1

Multiply both sides of the equation by \frac{1}{6 y}

6 y \frac{d y}{d x} \times \frac{1}{6 y} = 1 \times \frac{1}{6 y}

Any expression multiplied by 1 remains the same

6 y \frac{d y}{d x} \times \frac{1}{6 y} = \frac{1}{6 y}

Cancel out the greatest common factor 6 y

\frac{d y}{d x} = \frac{1}{6 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{1}{6 y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{1}{6 y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{6} \times \frac{d}{d x} (\frac{1}{y})

Rewrite the expression in exponential form

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{6} \times \frac{d}{d x} (y^{- 1})

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (y^{- 1})

Use differentiation rules

\frac{d}{d y} (y^{- 1}) \times \frac{d y}{d x}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

- y^{- 2} \frac{d y}{d x}

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{6} (- y^{- 2} \frac{d y}{d x})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = \frac{1}{6} (- \frac{\frac{d y}{d x}}{y ^{2}})

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x}}{6 y ^{2}}

Use equation \frac{d y}{d x} = \frac{1}{6 y} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{1}{6 y}}{6 y ^{2}}

Solution

More Steps

Calculate

- \frac{\frac{1}{6 y}}{6 y ^{2}}

Divide the terms

More Steps

Evaluate

\frac{\frac{1}{6 y}}{6 y ^{2}}

Multiply by the reciprocal

\frac{1}{6 y} \times \frac{1}{6 y ^{2}}

Multiply the terms

\frac{1}{6 y \times 6 y ^{2}}

Multiply the terms

\frac{1}{36 y ^{3}}

- \frac{1}{36 y ^{3}}

\frac{d ^{2} y}{d x ^{2}} = - \frac{1}{36 y ^{3}}

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Graph