Solve y=(x^2-2000) 500

Question

Function

Find the vertex

Find the axis of symmetry

Evaluate the derivative

(0, - 1000000)

Evaluate

y = (x^{2} - 2000) \times 500

Simplify

y = 500 (x^{2} - 2000)

Write the quadratic function in standard form

y = 500 x^{2} - 1000000

Find the x -coordinate of the vertex by substituting a= 500 and b= 0 into x = - \frac{b}{2 a}

x = - \frac{0}{2 \times 500}

Solve the equation for x

x = 0

Find the y-coordinate of the vertex by evaluating the function for x = 0

y = 500 \times 0^{2} - 1000000

Calculate

More Steps

Evaluate

500 \times 0^{2} - 1000000

Calculate

500 \times 0 - 1000000

Any expression multiplied by 0 equals 0

0 - 1000000

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

- 1000000

y = - 1000000

Solution

(0, - 1000000)

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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} - 2000) 500

Simplify the expression

y = 500 (x^{2} - 2000)

To test if the graph of y = 500 (x^{2} - 2000) is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 500 ((- x)^{2} - 2000)

Simplify

- y = 500 (x^{2} - 2000)

Change the signs both sides

y = - 500 (x^{2} - 2000)

Solution

Not symmetry with respect to the origin

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Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

x^{2} = \frac{1}{500} (y + 1000000)

Evaluate

y = (x^{2} - 2000) \times 500

Simplify

y = 500 (x^{2} - 2000)

Calculate

y = 500 x^{2} - 1000000

Swap the sides of the equation

500 x^{2} - 1000000 = y

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

500 x^{2} = y - (- 1000000)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

500 x^{2} = y + 1000000

Multiply both sides of the equation by \frac{1}{500}

500 x^{2} \times \frac{1}{500} = (y + 1000000) \times \frac{1}{500}

Multiply the terms

x^{2} = (y + 1000000) \times \frac{1}{500}

Multiply the terms

More Steps

Evaluate

(y + 1000000) \times \frac{1}{500}

Apply the distributive property

y \times \frac{1}{500} + 1000000 \times \frac{1}{500}

Use the commutative property to reorder the terms

\frac{1}{500} y + 1000000 \times \frac{1}{500}

Multiply the numbers

\frac{1}{500} y + 2000

x^{2} = \frac{1}{500} y + 2000

Solution

x^{2} = \frac{1}{500} (y + 1000000)

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{5 y + 5000000}{50} x = - \frac{5 y + 5000000}{50}

Evaluate

y = (x^{2} - 2000) \times 500

Simplify

y = 500 (x^{2} - 2000)

Swap the sides of the equation

500 (x^{2} - 2000) = y

Divide both sides

\frac{500 ( x ^{2} - 2000 )}{500} = \frac{y}{500}

Divide the numbers

x^{2} - 2000 = \frac{y}{500}

Move the constant to the right side

x^{2} = \frac{y}{500} + 2000

Add the terms

More Steps

Evaluate

\frac{y}{500} + 2000

Reduce fractions to a common denominator

\frac{y}{500} + \frac{2000 \times 500}{500}

Write all numerators above the common denominator

\frac{y + 2000 \times 500}{500}

Multiply the numbers

\frac{y + 1000000}{500}

x^{2} = \frac{y + 1000000}{500}

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

x = \pm \frac{y + 1000000}{500}

Simplify the expression

More Steps

Evaluate

\frac{y + 1000000}{500}

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

\frac{y + 1000000}{500}

Simplify the radical expression

More Steps

Evaluate

500

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

100 \times 5

Write the number in exponential form with the base of 10

1 0^{2} \times 5

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

1 0^{2} \times 5

Reduce the index of the radical and exponent with 2

105

\frac{y + 1000000}{10 5}

Multiply by the Conjugate

\frac{y + 1000000 \times 5}{10 5 \times 5}

Calculate

\frac{y + 1000000 \times 5}{10 \times 5}

Calculate

More Steps

Evaluate

y + 1000000 \times 5

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

(y + 1000000) \times 5

Calculate the product

5 y + 5000000

\frac{5 y + 5000000}{10 \times 5}

Calculate

\frac{5 y + 5000000}{50}

x = \pm \frac{5 y + 5000000}{50}

Solution

x = \frac{5 y + 5000000}{50} x = - \frac{5 y + 5000000}{50}

Show Solution

Rewrite the equation

r = \frac{sin ( θ ) - 1999999999 cos ^{2} ( θ ) + 2000000001}{1000 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 1999999999 cos ^{2} ( θ ) + 2000000001}{1000 cos ^{2} ( θ )}

Evaluate

y = (x^{2} - 2000) \times 500

Simplify

y = 500 (x^{2} - 2000)

Move the expression to the left side

y - 500 x^{2} = - 1000000

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

sin (θ) \times r - 500 (cos (θ) \times r)^{2} = - 1000000

Factor the expression

- 500 cos^{2} (θ) \times r^{2} + sin (θ) \times r = - 1000000

Subtract the terms

- 500 cos^{2} (θ) \times r^{2} + sin (θ) \times r - (- 1000000) = - 1000000 - (- 1000000)

Evaluate

- 500 cos^{2} (θ) \times r^{2} + sin (θ) \times r + 1000000 = 0

Solve using the quadratic formula

r = \frac{- sin ( θ ) \pm sin ^{2} ( θ ) - 4 ( - 500 cos ^{2} ( θ ) ) \times 1000000}{- 1000 cos ^{2} ( θ )}

Simplify

r = \frac{- sin ( θ ) \pm 1999999999 cos ^{2} ( θ ) + 2000000001}{- 1000 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- sin ( θ ) + 1999999999 cos ^{2} ( θ ) + 2000000001}{- 1000 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 1999999999 cos ^{2} ( θ ) + 2000000001}{- 1000 cos ^{2} ( θ )}

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

r = \frac{sin ( θ ) - 1999999999 cos ^{2} ( θ ) + 2000000001}{1000 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 1999999999 cos ^{2} ( θ ) + 2000000001}{- 1000 cos ^{2} ( θ )}

Solution

r = \frac{sin ( θ ) - 1999999999 cos ^{2} ( θ ) + 2000000001}{1000 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 1999999999 cos ^{2} ( θ ) + 2000000001}{1000 cos ^{2} ( θ )}

Show Solution

Function

Testing for symmetry

Identify the conic

Solve the equation

Rewrite the equation

Graph