Algebra Examples

$\frac{y}{2} = f (x)$

Step 1

Solve for $y$ .

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Step 1.1

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 f (x)$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$2 \frac{y}{2} = 2 f (x)$

Step 1.2.1.2

Rewrite the expression.

$y = 2 f (x)$

Step 1.3

Multiply $2 f$ by $x$ .

$y = 2 f x$

Step 2

Find the asymptotes.

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Step 2.1

Find the standard form of the hyperbola.

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Step 2.1.1

Move all terms containing variables to the left side of the equation.

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Step 2.1.1.1

Subtract $2 f x$ from both sides of the equation.

$y - 2 f x = 0$

Step 2.1.1.2

Reorder $y$ and $- 2 f x$ .

$- 2 f x + y = 0$

Step 2.1.2

Divide each term by $0$ to make the right side equal to one.

$- \frac{2 f x}{0} + \frac{y}{0} = \frac{0}{0}$

Step 2.1.3

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$y - f x = 1$

Step 2.2

This is the form of a hyperbola. Use this form to determine the values used to find the asymptotes of the hyperbola.

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 2.3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = 1$

$b = 1$

$k = 0$

$h = 0$

Step 2.4

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm 1 \cdot x + 0$

Step 2.5

Simplify $1 \cdot x + 0$ .

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Step 2.5.1

Add $1 \cdot x$ and $0$ .

$y = 1 \cdot x$

Step 2.5.2

Multiply $x$ by $1$ .

$y = x$

Step 2.6

Simplify $- 1 \cdot x + 0$ .

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Step 2.6.1

Add $- 1 \cdot x$ and $0$ .

$y = - 1 \cdot x$

Step 2.6.2

Rewrite $- 1 x$ as $- x$ .

$y = - x$

Step 2.7

This hyperbola has two asymptotes.

$y = x, y = - x$

Step 2.8

The asymptotes are $y = x$ and $y = - x$ .

Asymptotes: $y = x, y = - x$

Step 3