Solve y=-16x^2232x 97

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = - \frac{3 562 6 ^{2} x}{22504}

Evaluate

y = - 16 x^{2} \times 232 x \times 97

Simplify

More Steps

Evaluate

- 16 x^{2} \times 232 x \times 97

Multiply the terms

More Steps

Evaluate

16 \times 232 \times 97

Multiply the terms

3712 \times 97

Multiply the numbers

360064

- 360064 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 360064 x^{2 + 1}

Add the numbers

- 360064 x^{3}

y = - 360064 x^{3}

Interchange x and y

x = - 360064 y^{3}

Swap the sides of the equation

- 360064 y^{3} = x

Change the signs on both sides of the equation

360064 y^{3} = - x

Divide both sides

\frac{360064 y ^{3}}{360064} = \frac{- x}{360064}

Divide the numbers

y^{3} = \frac{- x}{360064}

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

y^{3} = - \frac{x}{360064}

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

3 y^{3} = 3 - \frac{x}{360064}

Calculate

y = 3 - \frac{x}{360064}

Simplify the root

More Steps

Evaluate

3 - \frac{x}{360064}

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

\frac{3 - x}{3 360064}

Simplify the radical expression

More Steps

Evaluate

3360064

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

3 64 \times 5626

Write the number in exponential form with the base of 4

3 4^{3} \times 5626

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

3 4^{3} \times 35626

Reduce the index of the radical and exponent with 3

435626

\frac{3 - x}{4 3 5626}

Multiply by the Conjugate

\frac{3 - x \times 3 562 6 ^{2}}{4 3 5626 \times 3 562 6 ^{2}}

Calculate

\frac{3 - x \times 3 562 6 ^{2}}{4 \times 5626}

Calculate

More Steps

Evaluate

3 - x \times 3 562 6^{2}

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

3 - x \times 562 6^{2}

Calculate the product

3 - 562 6^{2} x

An odd root of a negative radicand is always a negative

- 3 562 6^{2} x

\frac{- 3 562 6 ^{2} x}{4 \times 5626}

Calculate

\frac{- 3 562 6 ^{2} x}{22504}

Calculate

- \frac{3 562 6 ^{2} x}{22504}

y = - \frac{3 562 6 ^{2} x}{22504}

Solution

f^{- 1} (x) = - \frac{3 562 6 ^{2} x}{22504}

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 = - 16 x^{2} 232 x 97

Simplify the expression

y = - 360064 x^{3}

To test if the graph of y = - 360064 x^{3} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - 360064 (- x)^{3}

Simplify

More Steps

Evaluate

- 360064 (- x)^{3}

Rewrite the expression

- 360064 (- x^{3})

Multiply the numbers

360064 x^{3}

- y = 360064 x^{3}

Change the signs both sides

y = - 360064 x^{3}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = - \frac{3 562 6 ^{2} y}{22504}

Evaluate

y = - 16 x^{2} \times 232 x \times 97

Simplify

More Steps

Evaluate

- 16 x^{2} \times 232 x \times 97

Multiply the terms

More Steps

Evaluate

16 \times 232 \times 97

Multiply the terms

3712 \times 97

Multiply the numbers

360064

- 360064 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 360064 x^{2 + 1}

Add the numbers

- 360064 x^{3}

y = - 360064 x^{3}

Swap the sides of the equation

- 360064 x^{3} = y

Change the signs on both sides of the equation

360064 x^{3} = - y

Divide both sides

\frac{360064 x ^{3}}{360064} = \frac{- y}{360064}

Divide the numbers

x^{3} = \frac{- y}{360064}

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

x^{3} = - \frac{y}{360064}

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

3 x^{3} = 3 - \frac{y}{360064}

Calculate

x = 3 - \frac{y}{360064}

Solution

More Steps

Evaluate

3 - \frac{y}{360064}

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

\frac{3 - y}{3 360064}

Simplify the radical expression

More Steps

Evaluate

3360064

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

3 64 \times 5626

Write the number in exponential form with the base of 4

3 4^{3} \times 5626

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

3 4^{3} \times 35626

Reduce the index of the radical and exponent with 3

435626

\frac{3 - y}{4 3 5626}

Multiply by the Conjugate

\frac{3 - y \times 3 562 6 ^{2}}{4 3 5626 \times 3 562 6 ^{2}}

Calculate

\frac{3 - y \times 3 562 6 ^{2}}{4 \times 5626}

Calculate

More Steps

Evaluate

3 - y \times 3 562 6^{2}

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

3 - y \times 562 6^{2}

Calculate the product

3 - 562 6^{2} y

An odd root of a negative radicand is always a negative

- 3 562 6^{2} y

\frac{- 3 562 6 ^{2} y}{4 \times 5626}

Calculate

\frac{- 3 562 6 ^{2} y}{22504}

Calculate

- \frac{3 562 6 ^{2} y}{22504}

x = - \frac{3 562 6 ^{2} y}{22504}

Show Solution

Rewrite the equation

r = 0 r = - \frac{sin ( θ )}{360064 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{360064 cos ^{3} ( θ )}

Evaluate

y = - 16 x^{2} \times 232 x \times 97

Simplify

More Steps

Evaluate

- 16 x^{2} \times 232 x \times 97

Multiply the terms

More Steps

Evaluate

16 \times 232 \times 97

Multiply the terms

3712 \times 97

Multiply the numbers

360064

- 360064 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 360064 x^{2 + 1}

Add the numbers

- 360064 x^{3}

y = - 360064 x^{3}

Move the expression to the left side

y + 360064 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 + 360064 (cos (θ) \times r)^{3} = 0

Factor the expression

360064 cos^{3} (θ) \times r^{3} + sin (θ) \times r = 0

Factor the expression

r (360064 cos^{3} (θ) \times r^{2} + sin (θ)) = 0

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

r = 0 360064 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Solution

More Steps

Factor the expression

360064 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Subtract the terms

360064 cos^{3} (θ) \times r^{2} + sin (θ) - sin (θ) = 0 - sin (θ)

Evaluate

360064 cos^{3} (θ) \times r^{2} = - sin (θ)

Divide the terms

r^{2} = - \frac{sin ( θ )}{360064 cos ^{3} ( θ )}

Evaluate the power

r = \pm - \frac{sin ( θ )}{360064 cos ^{3} ( θ )}

Separate into possible cases

r = - \frac{sin ( θ )}{360064 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{360064 cos ^{3} ( θ )}

r = 0 r = - \frac{sin ( θ )}{360064 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{360064 cos ^{3} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph