Solve y=|x^2 2x^2|

Question

Function

Evaluate the derivative

Find the domain

Find the x -intercept/zero

y^{'} = 8 x^{3}

Evaluate

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

Simplify

More Steps

Evaluate

x^{2} \times 2 x^{2}

Multiply

More Steps

Multiply the terms

x^{2} \times 2 x^{2}

Multiply the terms with the same base by adding their exponents

x^{2 + 2} \times 2

Add the numbers

x^{4} \times 2

Use the commutative property to reorder the terms

2 x^{4}

2 x^{4}

Rewrite the expression

2 x^{4}

Simplify

2 x^{4}

y = 2 x^{4}

Take the derivative of both sides

y^{'} = \frac{d}{d x} (2 x^{4})

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

y^{'} = 2 \times \frac{d}{d x} (x^{4})

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

y^{'} = 2 \times 4 x^{3}

Solution

y^{'} = 8 x^{3}

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

Not symmetry with respect to the origin

Evaluate

y = x^{2} 2 x^{2}

Simplify the expression

y = 2 x^{4}

To test if the graph of y = 2 x^{4} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 2 (- x)^{4}

Simplify

- y = 2 x^{4}

Change the signs both sides

y = - 2 x^{4}

Solution

Not symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{4 8 y}{2} x = - \frac{4 8 y}{2}

Evaluate

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

Simplify

More Steps

Evaluate

x^{2} \times 2 x^{2}

Multiply

More Steps

Multiply the terms

x^{2} \times 2 x^{2}

Multiply the terms with the same base by adding their exponents

x^{2 + 2} \times 2

Add the numbers

x^{4} \times 2

Use the commutative property to reorder the terms

2 x^{4}

2 x^{4}

Rewrite the expression

2 x^{4}

Simplify

2 x^{4}

y = 2 x^{4}

Swap the sides of the equation

2 x^{4} = y

Divide both sides

\frac{2 x ^{4}}{2} = \frac{y}{2}

Divide the numbers

x^{4} = \frac{y}{2}

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

x = \pm 4 \frac{y}{2}

Simplify the expression

More Steps

Evaluate

4 \frac{y}{2}

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

\frac{4 y}{4 2}

Multiply by the Conjugate

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

Calculate

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

Calculate

More Steps

Evaluate

4 y \times 4 2^{3}

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

4 y \times 2^{3}

Calculate the product

4 2^{3} y

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

Calculate

\frac{4 8 y}{2}

x = \pm \frac{4 8 y}{2}

Solution

x = \frac{4 8 y}{2} x = - \frac{4 8 y}{2}

Show Solution

Rewrite the equation

r = 0 r = \frac{3 sin ( θ )}{3 2 cos ( θ ) \times cos ( θ )}

Evaluate

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

Simplify

More Steps

Evaluate

x^{2} \times 2 x^{2}

Multiply

More Steps

Multiply the terms

x^{2} \times 2 x^{2}

Multiply the terms with the same base by adding their exponents

x^{2 + 2} \times 2

Add the numbers

x^{4} \times 2

Use the commutative property to reorder the terms

2 x^{4}

2 x^{4}

Rewrite the expression

2 x^{4}

Simplify

2 x^{4}

y = 2 x^{4}

Move the expression to the left side

y - 2 x^{4} = 0

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

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

Factor the expression

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

Factor the expression

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

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

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

Solution

More Steps

Factor the expression

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

Subtract the terms

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

Evaluate

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

Divide the terms

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

Simplify the expression

More Steps

Evaluate

3 \frac{sin ( θ )}{2 cos ^{4} ( θ )}

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

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

Simplify the radical expression

\frac{3 sin ( θ )}{3 2 cos ( θ ) \times cos ( θ )}

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

r = 0 r = \frac{3 sin ( θ )}{3 2 cos ( θ ) \times cos ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph