Solve y-5=-2(x 1)^2

Question

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = - \frac{10}{2}, x_{2} = \frac{10}{2}

Evaluate

y - 5 = - 2 (x \times 1)^{2}

To find the x -intercept,set y =0

0 - 5 = - 2 (x \times 1)^{2}

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

- 5 = - 2 (x \times 1)^{2}

Any expression multiplied by 1 remains the same

- 5 = - 2 x^{2}

Swap the sides of the equation

- 2 x^{2} = - 5

Change the signs on both sides of the equation

2 x^{2} = 5

Divide both sides

\frac{2 x ^{2}}{2} = \frac{5}{2}

Divide the numbers

x^{2} = \frac{5}{2}

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

x = \pm \frac{5}{2}

Simplify the expression

More Steps

Evaluate

\frac{5}{2}

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

\frac{5}{2}

Multiply by the Conjugate

\frac{5 \times 2}{2 \times 2}

Multiply the numbers

More Steps

Evaluate

5 \times 2

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

5 \times 2

Calculate the product

10

\frac{10}{2 \times 2}

When a square root of an expression is multiplied by itself,the result is that expression

\frac{10}{2}

x = \pm \frac{10}{2}

Separate the equation into 2 possible cases

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

Solution

x_{1} = - \frac{10}{2}, x_{2} = \frac{10}{2}

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{- 2 y + 10}{2} x = - \frac{- 2 y + 10}{2}

Evaluate

y - 5 = - 2 (x \times 1)^{2}

Any expression multiplied by 1 remains the same

y - 5 = - 2 x^{2}

Swap the sides of the equation

- 2 x^{2} = y - 5

Change the signs on both sides of the equation

2 x^{2} = - y + 5

Divide both sides

\frac{2 x ^{2}}{2} = \frac{- y + 5}{2}

Divide the numbers

x^{2} = \frac{- y + 5}{2}

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

x = \pm \frac{- y + 5}{2}

Simplify the expression

More Steps

Evaluate

\frac{- y + 5}{2}

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

\frac{- y + 5}{2}

Multiply by the Conjugate

\frac{- y + 5 \times 2}{2 \times 2}

Calculate

\frac{- y + 5 \times 2}{2}

Calculate

More Steps

Evaluate

- y + 5 \times 2

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

(- y + 5) \times 2

Calculate the product

- 2 y + 10

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

x = \pm \frac{- 2 y + 10}{2}

Solution

x = \frac{- 2 y + 10}{2} x = - \frac{- 2 y + 10}{2}

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 - 5 = - 2 (x 1)^{2}

Simplify the expression

y - 5 = - 2 x^{2}

To test if the graph of y - 5 = - 2 x^{2} is symmetry with respect to the origin,substitute -x for x and -y for y

- y - 5 = - 2 (- x)^{2}

Evaluate

- y - 5 = - 2 x^{2}

Solution

Not symmetry with respect to the origin

Show Solution

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}{2} (y - 5)

Evaluate

y - 5 = - 2 (x \times 1)^{2}

Any expression multiplied by 1 remains the same

y - 5 = - 2 x^{2}

Swap the sides of the equation

- 2 x^{2} = y - 5

Multiply both sides of the equation by - \frac{1}{2}

- 2 x^{2} (- \frac{1}{2}) = (y - 5) (- \frac{1}{2})

Multiply the terms

x^{2} = (y - 5) (- \frac{1}{2})

Multiply the terms

More Steps

Evaluate

(y - 5) (- \frac{1}{2})

Apply the distributive property

y (- \frac{1}{2}) - 5 (- \frac{1}{2})

Use the commutative property to reorder the terms

- \frac{1}{2} y - 5 (- \frac{1}{2})

Multiply the numbers

- \frac{1}{2} y + \frac{5}{2}

x^{2} = - \frac{1}{2} y + \frac{5}{2}

Solution

x^{2} = - \frac{1}{2} (y - 5)

Show Solution

Rewrite the equation

r = \frac{- sin ( θ ) + 39 cos ^{2} ( θ ) + 41}{4 cos ^{2} ( θ )} r = - \frac{sin ( θ ) + 39 cos ^{2} ( θ ) + 41}{4 cos ^{2} ( θ )}

Evaluate

y - 5 = - 2 (x \times 1)^{2}

Any expression multiplied by 1 remains the same

y - 5 = - 2 x^{2}

Move the expression to the left side

y - 5 + 2 x^{2} = 0

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

sin (θ) \times r - 5 + 2 (cos (θ) \times r)^{2} = 0

Factor the expression

2 cos^{2} (θ) \times r^{2} + sin (θ) \times r - 5 = 0

Solve using the quadratic formula

r = \frac{- sin ( θ ) \pm sin ^{2} ( θ ) - 4 \times 2 cos ^{2} ( θ ) ( - 5 )}{4 cos ^{2} ( θ )}

Simplify

r = \frac{- sin ( θ ) \pm 39 cos ^{2} ( θ ) + 41}{4 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- sin ( θ ) + 39 cos ^{2} ( θ ) + 41}{4 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 39 cos ^{2} ( θ ) + 41}{4 cos ^{2} ( θ )}

Solution

r = \frac{- sin ( θ ) + 39 cos ^{2} ( θ ) + 41}{4 cos ^{2} ( θ )} r = - \frac{sin ( θ ) + 39 cos ^{2} ( θ ) + 41}{4 cos ^{2} ( 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} = - 4 x

Calculate

y - 5 = - 2 (x 1)^{2}

Simplify the expression

y - 5 = - 2 x^{2}

Take the derivative of both sides

\frac{d}{d x} (y - 5) = \frac{d}{d x} (- 2 x^{2})

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (y - 5)

Use differentiation rules

\frac{d}{d x} (y) + \frac{d}{d x} (- 5)

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} (- 5)

Use \frac{d}{d x} (c) = 0 to find derivative

\frac{d y}{d x} + 0

Evaluate

\frac{d y}{d x}

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

Solution

More Steps

Evaluate

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

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

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

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

- 2 \times 2 x

Multiply the terms

- 4 x

\frac{d y}{d x} = - 4 x

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}} = - 4

Calculate

y - 5 = - 2 (x 1)^{2}

Simplify the expression

y - 5 = - 2 x^{2}

Take the derivative of both sides

\frac{d}{d x} (y - 5) = \frac{d}{d x} (- 2 x^{2})

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (y - 5)

Use differentiation rules

\frac{d}{d x} (y) + \frac{d}{d x} (- 5)

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} (- 5)

Use \frac{d}{d x} (c) = 0 to find derivative

\frac{d y}{d x} + 0

Evaluate

\frac{d y}{d x}

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

Calculate the derivative

More Steps

Evaluate

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

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

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

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

- 2 \times 2 x

Multiply the terms

- 4 x

\frac{d y}{d x} = - 4 x

Take the derivative of both sides

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

Calculate the derivative

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

Simplify

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

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = - 4 \times 1

Solution

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

Show Solution

Function

Solve the equation

Testing for symmetry

Identify the conic

Rewrite the equation

Find the first derivative

Find the second derivative

Graph