Solve y=-x^2 4x - ScanMath

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

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

Evaluate

y = - x^{2} \times 4 x

Simplify

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Evaluate

- x^{2} \times 4 x

Multiply

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Evaluate

x^{2} \times 4 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 4

Add the numbers

x^{3} \times 4

Use the commutative property to reorder the terms

4 x^{3}

- 4 x^{3}

y = - 4 x^{3}

Interchange x and y

x = - 4 y^{3}

Swap the sides of the equation

- 4 y^{3} = x

Change the signs on both sides of the equation

4 y^{3} = - x

Divide both sides

\frac{4 y ^{3}}{4} = \frac{- x}{4}

Divide the numbers

y^{3} = \frac{- x}{4}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

y^{3} = - \frac{x}{4}

Take the 3 -th root on both sides of the equation

3 y^{3} = 3 - \frac{x}{4}

Calculate

y = 3 - \frac{x}{4}

Simplify the root

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Evaluate

3 - \frac{x}{4}

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

\frac{3 - x}{3 4}

Multiply by the Conjugate

\frac{3 - x \times 3 4 ^{2}}{3 4 \times 3 4 ^{2}}

Calculate

\frac{3 - x \times 3 4 ^{2}}{2 ^{2}}

Calculate

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Evaluate

3 - x \times 3 4^{2}

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

3 - x \times 4^{2}

Calculate the product

3 - 4^{2} x

An odd root of a negative radicand is always a negative

- 3 4^{2} x

Simplify the radical expression

- 2 3 2 x

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

Divide the terms

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Evaluate

\frac{- 2}{2 ^{2}}

Use the product rule \frac{a ^{n}}{a ^{m}} = a^{n - m} to simplify the expression

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

Subtract the terms

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

Simplify

\frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{1}{2}

\frac{- 3 2 x}{2}

Calculate

- \frac{3 2 x}{2}

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

Solution

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

Show Solution

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

y = - x^{2} 4 x

Simplify the expression

y = - 4 x^{3}

To test if the graph of y = - 4 x^{3} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - 4 (- x)^{3}

Simplify

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Evaluate

- 4 (- x)^{3}

Rewrite the expression

- 4 (- x^{3})

Multiply the numbers

4 x^{3}

- y = 4 x^{3}

Change the signs both sides

y = - 4 x^{3}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

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

Evaluate

y = - x^{2} \times 4 x

Simplify

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Evaluate

- x^{2} \times 4 x

Multiply

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Evaluate

x^{2} \times 4 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 4

Add the numbers

x^{3} \times 4

Use the commutative property to reorder the terms

4 x^{3}

- 4 x^{3}

y = - 4 x^{3}

Swap the sides of the equation

- 4 x^{3} = y

Change the signs on both sides of the equation

4 x^{3} = - y

Divide both sides

\frac{4 x ^{3}}{4} = \frac{- y}{4}

Divide the numbers

x^{3} = \frac{- y}{4}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

x^{3} = - \frac{y}{4}

Take the 3 -th root on both sides of the equation

3 x^{3} = 3 - \frac{y}{4}

Calculate

x = 3 - \frac{y}{4}

Solution

More Steps

Evaluate

3 - \frac{y}{4}

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

\frac{3 - y}{3 4}

Multiply by the Conjugate

\frac{3 - y \times 3 4 ^{2}}{3 4 \times 3 4 ^{2}}

Calculate

\frac{3 - y \times 3 4 ^{2}}{2 ^{2}}

Calculate

More Steps

Evaluate

3 - y \times 3 4^{2}

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

3 - y \times 4^{2}

Calculate the product

3 - 4^{2} y

An odd root of a negative radicand is always a negative

- 3 4^{2} y

Simplify the radical expression

- 2 3 2 y

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

Divide the terms

More Steps

Evaluate

\frac{- 2}{2 ^{2}}

Use the product rule \frac{a ^{n}}{a ^{m}} = a^{n - m} to simplify the expression

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

Subtract the terms

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

Simplify

\frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{1}{2}

\frac{- 3 2 y}{2}

Calculate

- \frac{3 2 y}{2}

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

Show Solution

Rewrite the equation

r = 0 r = - \frac{sin ( θ )}{4 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{4 cos ^{3} ( θ )}

Evaluate

y = - x^{2} \times 4 x

Simplify

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Evaluate

- x^{2} \times 4 x

Multiply

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Evaluate

x^{2} \times 4 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 4

Add the numbers

x^{3} \times 4

Use the commutative property to reorder the terms

4 x^{3}

- 4 x^{3}

y = - 4 x^{3}

Move the expression to the left side

y + 4 x^{3} = 0

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

sin (θ) \times r + 4 (cos (θ) \times r)^{3} = 0

Factor the expression

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

Factor the expression

r (4 cos^{3} (θ) \times r^{2} + sin (θ)) = 0

When the product of factors equals 0,at least one factor is 0

r = 0 4 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Solution

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Factor the expression

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

Subtract the terms

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

Evaluate

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

Divide the terms

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

Evaluate the power

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

Separate into possible cases

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

r = 0 r = - \frac{sin ( θ )}{4 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{4 cos ^{3} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph