Graph ((x-i)(2x+i))/(x-i)

$\frac{(x - i) (2 x + i)}{(x - i)}$

Step 1

Find the standard form of the hyperbola.

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

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

$y - 2 x = i$

Reorder $y$ and $- 2 x$ .

$- 2 x + y = i$

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

$- \frac{2 x}{i} + \frac{y}{i} = \frac{i}{i}$

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 $1$ .

$\frac{y}{\frac{1}{i}} - \frac{x}{\frac{1}{2 i}} = 1$

Step 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 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, $a$ .

$a = i \sqrt{i}$

$b = \frac{i \sqrt{2 i}}{2}$

$k = 0$

$h = 0$

Step 4

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

$y = \pm (- \frac{i \sqrt{2 i^{2}}}{2} x) + 0$

Step 5

Simplify $- \frac{i \sqrt{2 i^{2}}}{2} x + 0$ .

Add $- \frac{i \sqrt{2 i^{2}}}{2} x$ and $0$ .

$y = - \frac{i \sqrt{2 i^{2}}}{2} x$

Simplify the numerator.

Rewrite $i^{2}$ as $- 1$ .

$y = - \frac{i \sqrt{2 \cdot - 1}}{2} x$

Multiply $2$ by $- 1$ .

$y = - \frac{i \sqrt{- 2}}{2} x$

Rewrite $- 2$ as $- 1 (2)$ .

$y = - \frac{i \sqrt{- 1 (2)}}{2} x$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$y = - \frac{i (\sqrt{- 1} \cdot \sqrt{2})}{2} x$

Rewrite $\sqrt{- 1}$ as $i$ .

$y = - \frac{i (i \cdot \sqrt{2})}{2} x$

Remove unnecessary parentheses.

$y = - \frac{i i \sqrt{2}}{2} x$

Combine exponents.

Raise $i$ to the power of $1$ .

$y = - \frac{i^{1} i \sqrt{2}}{2} x$

Raise $i$ to the power of $1$ .

$y = - \frac{i^{1} i^{1} \sqrt{2}}{2} x$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - \frac{i^{1 + 1} \sqrt{2}}{2} x$

Add $1$ and $1$ .

$y = - \frac{i^{2} \sqrt{2}}{2} x$

Rewrite $i^{2}$ as $- 1$ .

$y = - \frac{- 1 \sqrt{2}}{2} x$

Move the negative in front of the fraction.

$y = - - \frac{\sqrt{2}}{2} x$

Multiply $- - \frac{\sqrt{2}}{2}$ .

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{\sqrt{2}}{2} x$

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$y = \frac{\sqrt{2}}{2} x$

Combine $\frac{\sqrt{2}}{2}$ and $x$ .

$y = \frac{\sqrt{2} x}{2}$

Step 6

This hyperbola has two asymptotes.

$y = \frac{\sqrt{2} x}{2}, y = \frac{\sqrt{2} x}{2}$

Step 7

The asymptotes are $y = \frac{\sqrt{2} x}{2}$ and $y = \frac{\sqrt{2} x}{2}$ .

Asymptotes: $y = \frac{\sqrt{2} x}{2}, y = \frac{\sqrt{2} x}{2}$

Step 8

Graph ((x-i)(2x+i))/(x-i)

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