Solve y-4 = -5/2 (x^2)

Function

Find the x -intercept/zero

Find the y-intercept

x_{1} = - \frac{2 10}{5}, x_{2} = \frac{2 10}{5}

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Solve for x

Solve for y

x = \frac{- 10 y + 40}{5} x = - \frac{- 10 y + 40}{5}

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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

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Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

x^{2} = - \frac{2}{5} (y - 4)

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r = \frac{- sin ( θ ) + 1 + 39 cos ^{2} ( θ )}{5 cos ^{2} ( θ )} r = - \frac{sin ( θ ) + 1 + 39 cos ^{2} ( θ )}{5 cos ^{2} ( θ )}

Evaluate

y - 4 = - \frac{5}{2} (x^{2})

Evaluate

y - 4 = - \frac{5}{2} x^{2}

Multiply both sides of the equation by LCD

(y - 4) \times 2 = - \frac{5}{2} x^{2} \times 2

Simplify the equation

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2 y - 8 = - \frac{5}{2} x^{2} \times 2

Simplify the equation

2 y - 8 = - 5 x^{2}

Move the expression to the left side

2 y - 8 + 5 x^{2} = 0

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

2 sin (θ) \times r - 8 + 5 (cos (θ) \times r)^{2} = 0

Factor the expression

5 cos^{2} (θ) \times r^{2} + 2 sin (θ) \times r - 8 = 0

Solve using the quadratic formula

r = \frac{- 2 sin ( θ ) \pm ( 2 sin ( θ ) ) ^{2} - 4 \times 5 cos ^{2} ( θ ) ( - 8 )}{10 cos ^{2} ( θ )}

Simplify

r = \frac{- 2 sin ( θ ) \pm 4 + 156 cos ^{2} ( θ )}{10 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- 2 sin ( θ ) + 4 + 156 cos ^{2} ( θ )}{10 cos ^{2} ( θ )} r = \frac{- 2 sin ( θ ) - 4 + 156 cos ^{2} ( θ )}{10 cos ^{2} ( θ )}

Evaluate

More Steps

r = \frac{- sin ( θ ) + 1 + 39 cos ^{2} ( θ )}{5 cos ^{2} ( θ )} r = \frac{- 2 sin ( θ ) - 4 + 156 cos ^{2} ( θ )}{10 cos ^{2} ( θ )}

Solution

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r = \frac{- sin ( θ ) + 1 + 39 cos ^{2} ( θ )}{5 cos ^{2} ( θ )} r = - \frac{sin ( θ ) + 1 + 39 cos ^{2} ( θ )}{5 cos ^{2} ( θ )}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - 5 x

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Find the second derivative with respect to x

Find the second derivative with respect to y

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

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