Solve y=1/3(18x 12) 5x-4

Question

Function

Find the vertex

Find the axis of symmetry

Evaluate the derivative

(0, - 4)

Evaluate

y = \frac{1}{3} (18 x \times 12) \times 5 x - 4

Simplify

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Evaluate

\frac{1}{3} (18 x \times 12) \times 5 x - 4

Remove the parentheses

\frac{1}{3} \times 18 x \times 12 \times 5 x - 4

Multiply

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Evaluate

\frac{1}{3} \times 18 x \times 12 \times 5 x

Multiply the terms

360 x \times x

Multiply the terms

360 x^{2}

360 x^{2} - 4

y = 360 x^{2} - 4

Find the x -coordinate of the vertex by substituting a= 360 and b= 0 into x = - \frac{b}{2 a}

x = - \frac{0}{2 \times 360}

Solve the equation for x

x = 0

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

y = 360 \times 0^{2} - 4

Calculate

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Evaluate

360 \times 0^{2} - 4

Calculate

360 \times 0 - 4

Any expression multiplied by 0 equals 0

0 - 4

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

- 4

y = - 4

Solution

(0, - 4)

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 = \frac{1}{3} (18 x 12) 5 x - 4

Simplify the expression

y = 360 x^{2} - 4

To test if the graph of y = 360 x^{2} - 4 is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 360 (- x)^{2} - 4

Simplify

- y = 360 x^{2} - 4

Change the signs both sides

y = - 360 x^{2} + 4

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}{360} (y + 4)

Evaluate

y = \frac{1}{3} (18 x \times 12) \times 5 x - 4

Simplify

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Evaluate

\frac{1}{3} (18 x \times 12) \times 5 x - 4

Remove the parentheses

\frac{1}{3} \times 18 x \times 12 \times 5 x - 4

Multiply

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Evaluate

\frac{1}{3} \times 18 x \times 12 \times 5 x

Multiply the terms

360 x \times x

Multiply the terms

360 x^{2}

360 x^{2} - 4

y = 360 x^{2} - 4

Swap the sides of the equation

360 x^{2} - 4 = y

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

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

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

360 x^{2} = y + 4

Multiply both sides of the equation by \frac{1}{360}

360 x^{2} \times \frac{1}{360} = (y + 4) \times \frac{1}{360}

Multiply the terms

x^{2} = (y + 4) \times \frac{1}{360}

Multiply the terms

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Evaluate

(y + 4) \times \frac{1}{360}

Apply the distributive property

y \times \frac{1}{360} + 4 \times \frac{1}{360}

Use the commutative property to reorder the terms

\frac{1}{360} y + 4 \times \frac{1}{360}

Multiply the numbers

\frac{1}{360} y + \frac{1}{90}

x^{2} = \frac{1}{360} y + \frac{1}{90}

Solution

x^{2} = \frac{1}{360} (y + 4)

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{10 y + 40}{60} x = - \frac{10 y + 40}{60}

Evaluate

y = \frac{1}{3} (18 x \times 12) \times 5 x - 4

Simplify

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Evaluate

\frac{1}{3} (18 x \times 12) \times 5 x - 4

Remove the parentheses

\frac{1}{3} \times 18 x \times 12 \times 5 x - 4

Multiply

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Evaluate

\frac{1}{3} \times 18 x \times 12 \times 5 x

Multiply the terms

360 x \times x

Multiply the terms

360 x^{2}

360 x^{2} - 4

y = 360 x^{2} - 4

Swap the sides of the equation

360 x^{2} - 4 = y

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

360 x^{2} = y + 4

Divide both sides

\frac{360 x ^{2}}{360} = \frac{y + 4}{360}

Divide the numbers

x^{2} = \frac{y + 4}{360}

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

x = \pm \frac{y + 4}{360}

Simplify the expression

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Evaluate

\frac{y + 4}{360}

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

\frac{y + 4}{360}

Simplify the radical expression

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Evaluate

360

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

36 \times 10

Write the number in exponential form with the base of 6

6^{2} \times 10

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

6^{2} \times 10

Reduce the index of the radical and exponent with 2

610

\frac{y + 4}{6 10}

Multiply by the Conjugate

\frac{y + 4 \times 10}{6 10 \times 10}

Calculate

\frac{y + 4 \times 10}{6 \times 10}

Calculate

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Evaluate

y + 4 \times 10

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

(y + 4) \times 10

Calculate the product

10 y + 40

\frac{10 y + 40}{6 \times 10}

Calculate

\frac{10 y + 40}{60}

x = \pm \frac{10 y + 40}{60}

Solution

x = \frac{10 y + 40}{60} x = - \frac{10 y + 40}{60}

Show Solution

Rewrite the equation

r = \frac{sin ( θ ) - 5759 cos ^{2} ( θ ) + 5761}{720 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 5759 cos ^{2} ( θ ) + 5761}{720 cos ^{2} ( θ )}

Evaluate

y = \frac{1}{3} (18 x \times 12) \times 5 x - 4

Simplify

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Evaluate

\frac{1}{3} (18 x \times 12) \times 5 x - 4

Remove the parentheses

\frac{1}{3} \times 18 x \times 12 \times 5 x - 4

Multiply

More Steps

Evaluate

\frac{1}{3} \times 18 x \times 12 \times 5 x

Multiply the terms

360 x \times x

Multiply the terms

360 x^{2}

360 x^{2} - 4

y = 360 x^{2} - 4

Move the expression to the left side

y - 360 x^{2} = - 4

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

sin (θ) \times r - 360 (cos (θ) \times r)^{2} = - 4

Factor the expression

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

Subtract the terms

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

Evaluate

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

Solve using the quadratic formula

r = \frac{- sin ( θ ) \pm sin ^{2} ( θ ) - 4 ( - 360 cos ^{2} ( θ ) ) \times 4}{- 720 cos ^{2} ( θ )}

Simplify

r = \frac{- sin ( θ ) \pm 5759 cos ^{2} ( θ ) + 5761}{- 720 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- sin ( θ ) + 5759 cos ^{2} ( θ ) + 5761}{- 720 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 5759 cos ^{2} ( θ ) + 5761}{- 720 cos ^{2} ( θ )}

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

r = \frac{sin ( θ ) - 5759 cos ^{2} ( θ ) + 5761}{720 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 5759 cos ^{2} ( θ ) + 5761}{- 720 cos ^{2} ( θ )}

Solution

r = \frac{sin ( θ ) - 5759 cos ^{2} ( θ ) + 5761}{720 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 5759 cos ^{2} ( θ ) + 5761}{720 cos ^{2} ( θ )}

Show Solution

Function

Testing for symmetry

Identify the conic

Solve the equation

Rewrite the equation

Graph