Solve y-4x^(-2)=0

Question

Function

Find the x -intercept/zero

Find the y-intercept

No x -intercept

Evaluate

y - 4 x^{- 2} = 0

To find the x -intercept,set y =0

0 - 4 x^{- 2} = 0

Removing 0 doesn't change the value,so remove it from the expression

- 4 x^{- 2} = 0

Rewrite the expression

\frac{- 4}{x ^{2}} = 0

Cross multiply

- 4 = x^{2} \times 0

Simplify the equation

- 4 = 0

The statement is false for any value of x

x \in \emptyset

Solution

No x -intercept

Show Solution

Solve the equation

Solve for x

Solve for y

x = - \frac{2 y}{∣ y ∣}, y \neq = 0 x = \frac{2 y}{∣ y ∣}, y \neq = 0

Evaluate

y - 4 x^{- 2} = 0

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

- 4 x^{- 2} = 0 - y

Removing 0 doesn't change the value,so remove it from the expression

- 4 x^{- 2} = - y

Rewrite the expression

More Steps

Evaluate

- 4 x^{- 2}

Express with a positive exponent using a^{- n} = \frac{1}{a ^{n}}

- 4 \times \frac{1}{x ^{2}}

Rewrite the expression

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

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

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

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

Rewrite the expression

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

Cross multiply

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

Simplify the equation

- 4 = - y x^{2}

Evaluate

4 = y x^{2}

Swap the sides of the equation

y x^{2} = 4

Divide both sides

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

Divide the numbers

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

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

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

Simplify the expression

More Steps

Evaluate

\frac{4}{y}

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

\frac{4}{y}

Simplify the radical expression

More Steps

Evaluate

4

Write the number in exponential form with the base of 2

2^{2}

Reduce the index of the radical and exponent with 2

2

\frac{2}{y}

Multiply by the Conjugate

\frac{2 y}{y \times y}

Calculate

\frac{2 y}{∣ y ∣}

x = \pm \frac{2 y}{∣ y ∣}

Separate the equation into 2 possible cases

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

Calculate

{x = - \frac{2 y}{∣ y ∣} y \neq = 0 {x = \frac{2 y}{∣ y ∣} y \neq = 0

Solution

x = - \frac{2 y}{∣ y ∣}, y \neq = 0 x = \frac{2 y}{∣ y ∣}, y \neq = 0

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 - 4 x^{- 2} = 0

To test if the graph of y - 4 x^{- 2} = 0 is symmetry with respect to the origin,substitute -x for x and -y for y

- y - 4 (- x)^{- 2} = 0

Evaluate

- y - 4 x^{- 2} = 0

Solution

Not symmetry with respect to the origin

Show Solution

Rewrite the equation

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

Evaluate

y - 4 x^{- 2} = 0

Rewrite the expression

More Steps

Evaluate

- 4 x^{- 2}

Express with a positive exponent using a^{- n} = \frac{1}{a ^{n}}

- 4 \times \frac{1}{x ^{2}}

Rewrite the expression

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

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

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

y - \frac{4}{x ^{2}} = 0

Multiply both sides of the equation by LCD

(y - \frac{4}{x ^{2}}) x^{2} = 0 \times x^{2}

Simplify the equation

More Steps

Evaluate

(y - \frac{4}{x ^{2}}) x^{2}

Apply the distributive property

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

Simplify

y x^{2} - 4

y x^{2} - 4 = 0 \times x^{2}

Any expression multiplied by 0 equals 0

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

Factor the expression

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

Simplify the expression

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

Subtract the terms

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

Evaluate

cos^{2} (θ) sin (θ) \times r^{3} = 4

Divide the terms

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

Solution

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

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{8}{x ^{3}}

Calculate

y - 4 x^{- 2} = 0

Take the derivative of both sides

\frac{d}{d x} (y - 4 x^{- 2}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

\frac{d}{d x} (y) + \frac{d}{d x} (- 4 x^{- 2})

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (y)

Use differentiation rules

\frac{d}{d y} (y) \times \frac{d y}{d x}

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

\frac{d y}{d x}

\frac{d y}{d x} + \frac{d}{d x} (- 4 x^{- 2})

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 4 x^{- 2})

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

- 4 \times \frac{d}{d x} (x^{- 2})

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

- 4 (- 2 x^{- 3})

Multiply the terms

8 x^{- 3}

\frac{d y}{d x} + 8 x^{- 3}

\frac{d y}{d x} + 8 x^{- 3} = \frac{d}{d x} (0)

Calculate the derivative

\frac{d y}{d x} + 8 x^{- 3} = 0

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

\frac{d y}{d x} = 0 - 8 x^{- 3}

Removing 0 doesn't change the value,so remove it from the expression

\frac{d y}{d x} = - 8 x^{- 3}

Solution

More Steps

Evaluate

- 8 x^{- 3}

Express with a positive exponent using a^{- n} = \frac{1}{a ^{n}}

- 8 \times \frac{1}{x ^{3}}

Calculate the product

- \frac{8}{x ^{3}}

\frac{d y}{d x} = - \frac{8}{x ^{3}}

Show Solution

Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{24}{x ^{4}}

Calculate

y - 4 x^{- 2} = 0

Take the derivative of both sides

\frac{d}{d x} (y - 4 x^{- 2}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

\frac{d}{d x} (y) + \frac{d}{d x} (- 4 x^{- 2})

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (y)

Use differentiation rules

\frac{d}{d y} (y) \times \frac{d y}{d x}

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

\frac{d y}{d x}

\frac{d y}{d x} + \frac{d}{d x} (- 4 x^{- 2})

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 4 x^{- 2})

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

- 4 \times \frac{d}{d x} (x^{- 2})

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

- 4 (- 2 x^{- 3})

Multiply the terms

8 x^{- 3}

\frac{d y}{d x} + 8 x^{- 3}

\frac{d y}{d x} + 8 x^{- 3} = \frac{d}{d x} (0)

Calculate the derivative

\frac{d y}{d x} + 8 x^{- 3} = 0

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

\frac{d y}{d x} = 0 - 8 x^{- 3}

Removing 0 doesn't change the value,so remove it from the expression

\frac{d y}{d x} = - 8 x^{- 3}

Rewrite the expression

More Steps

Evaluate

- 8 x^{- 3}

Express with a positive exponent using a^{- n} = \frac{1}{a ^{n}}

- 8 \times \frac{1}{x ^{3}}

Calculate the product

- \frac{8}{x ^{3}}

\frac{d y}{d x} = - \frac{8}{x ^{3}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{8}{x ^{3}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{8}{x ^{3}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - 8 \times \frac{d}{d x} (\frac{1}{x ^{3}})

Rewrite the expression in exponential form

\frac{d ^{2} y}{d x ^{2}} = - 8 \times \frac{d}{d x} (x^{- 3})

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

\frac{d ^{2} y}{d x ^{2}} = - 8 (- 3 x^{- 4})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = - 8 (- \frac{3}{x ^{4}})

Solution

\frac{d ^{2} y}{d x ^{2}} = \frac{24}{x ^{4}}

Show Solution

Function

Solve the equation

Testing for symmetry

Rewrite the equation

Find the first derivative

Find the second derivative

Graph