Solve y=-2x^2 40x

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = - \frac{3 100 x}{20}

Evaluate

y = - 2 x^{2} \times 40 x

Simplify

More Steps

Evaluate

- 2 x^{2} \times 40 x

Multiply the terms

- 80 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 80 x^{2 + 1}

Add the numbers

- 80 x^{3}

y = - 80 x^{3}

Interchange x and y

x = - 80 y^{3}

Swap the sides of the equation

- 80 y^{3} = x

Change the signs on both sides of the equation

80 y^{3} = - x

Divide both sides

\frac{80 y ^{3}}{80} = \frac{- x}{80}

Divide the numbers

y^{3} = \frac{- x}{80}

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

y^{3} = - \frac{x}{80}

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

3 y^{3} = 3 - \frac{x}{80}

Calculate

y = 3 - \frac{x}{80}

Simplify the root

More Steps

Evaluate

3 - \frac{x}{80}

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

\frac{3 - x}{3 80}

Simplify the radical expression

More Steps

Evaluate

380

Write the expression as a product where the root of one of the factors can be evaluated

3 8 \times 10

Write the number in exponential form with the base of 2

3 2^{3} \times 10

The root of a product is equal to the product of the roots of each factor

3 2^{3} \times 310

Reduce the index of the radical and exponent with 3

2310

\frac{3 - x}{2 3 10}

Multiply by the Conjugate

\frac{3 - x \times 3 1 0 ^{2}}{2 3 10 \times 3 1 0 ^{2}}

Calculate

\frac{3 - x \times 3 1 0 ^{2}}{2 \times 10}

Calculate

More Steps

Evaluate

3 - x \times 3 1 0^{2}

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

3 - x \times 1 0^{2}

Calculate the product

3 - 1 0^{2} x

An odd root of a negative radicand is always a negative

- 3 1 0^{2} x

\frac{- 3 1 0 ^{2} x}{2 \times 10}

Calculate

\frac{- 3 1 0 ^{2} x}{20}

Calculate

- \frac{3 1 0 ^{2} x}{20}

Calculate

- \frac{3 100 x}{20}

y = - \frac{3 100 x}{20}

Solution

f^{- 1} (x) = - \frac{3 100 x}{20}

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 = - 2 x^{2} 40 x

Simplify the expression

y = - 80 x^{3}

To test if the graph of y = - 80 x^{3} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - 80 (- x)^{3}

Simplify

More Steps

Evaluate

- 80 (- x)^{3}

Rewrite the expression

- 80 (- x^{3})

Multiply the numbers

80 x^{3}

- y = 80 x^{3}

Change the signs both sides

y = - 80 x^{3}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = - \frac{3 100 y}{20}

Evaluate

y = - 2 x^{2} \times 40 x

Simplify

More Steps

Evaluate

- 2 x^{2} \times 40 x

Multiply the terms

- 80 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 80 x^{2 + 1}

Add the numbers

- 80 x^{3}

y = - 80 x^{3}

Swap the sides of the equation

- 80 x^{3} = y

Change the signs on both sides of the equation

80 x^{3} = - y

Divide both sides

\frac{80 x ^{3}}{80} = \frac{- y}{80}

Divide the numbers

x^{3} = \frac{- y}{80}

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

x^{3} = - \frac{y}{80}

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

3 x^{3} = 3 - \frac{y}{80}

Calculate

x = 3 - \frac{y}{80}

Solution

More Steps

Evaluate

3 - \frac{y}{80}

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

\frac{3 - y}{3 80}

Simplify the radical expression

More Steps

Evaluate

380

Write the expression as a product where the root of one of the factors can be evaluated

3 8 \times 10

Write the number in exponential form with the base of 2

3 2^{3} \times 10

The root of a product is equal to the product of the roots of each factor

3 2^{3} \times 310

Reduce the index of the radical and exponent with 3

2310

\frac{3 - y}{2 3 10}

Multiply by the Conjugate

\frac{3 - y \times 3 1 0 ^{2}}{2 3 10 \times 3 1 0 ^{2}}

Calculate

\frac{3 - y \times 3 1 0 ^{2}}{2 \times 10}

Calculate

More Steps

Evaluate

3 - y \times 3 1 0^{2}

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

3 - y \times 1 0^{2}

Calculate the product

3 - 1 0^{2} y

An odd root of a negative radicand is always a negative

- 3 1 0^{2} y

\frac{- 3 1 0 ^{2} y}{2 \times 10}

Calculate

\frac{- 3 1 0 ^{2} y}{20}

Calculate

- \frac{3 1 0 ^{2} y}{20}

Calculate

- \frac{3 100 y}{20}

x = - \frac{3 100 y}{20}

Show Solution

Rewrite the equation

r = 0 r = - \frac{sin ( θ )}{80 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{80 cos ^{3} ( θ )}

Evaluate

y = - 2 x^{2} \times 40 x

Simplify

More Steps

Evaluate

- 2 x^{2} \times 40 x

Multiply the terms

- 80 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 80 x^{2 + 1}

Add the numbers

- 80 x^{3}

y = - 80 x^{3}

Move the expression to the left side

y + 80 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 + 80 (cos (θ) \times r)^{3} = 0

Factor the expression

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

Factor the expression

r (80 cos^{3} (θ) \times r^{2} + sin (θ)) = 0

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

r = 0 80 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Solution

More Steps

Factor the expression

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

Subtract the terms

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

Evaluate

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

Divide the terms

r^{2} = - \frac{sin ( θ )}{80 cos ^{3} ( θ )}

Evaluate the power

r = \pm - \frac{sin ( θ )}{80 cos ^{3} ( θ )}

Separate into possible cases

r = - \frac{sin ( θ )}{80 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{80 cos ^{3} ( θ )}

r = 0 r = - \frac{sin ( θ )}{80 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{80 cos ^{3} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph