Algebra Examples

$y = 2 f (\frac{1}{2} x)$

Step 1

Find the standard form of the hyperbola.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Subtract $2 f (\frac{1}{2} x)$ from both sides of the equation.

$y - 2 f (\frac{1}{2} x) = 0$

Step 1.1.2

Simplify each term.

Tap for more steps...

Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$y - 2 \cdot \frac{1}{2} f x = 0$

Step 1.1.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.2.2.1

Factor $2$ out of $- 2$ .

$y + 2 (- 1) \cdot \frac{1}{2} f x = 0$

Step 1.1.2.2.2

Cancel the common factor.

$y + 2 \cdot - 1 \cdot \frac{1}{2} f x = 0$

Step 1.1.2.2.3

Rewrite the expression.

$y - f x = 0$

Step 1.1.3

Reorder $y$ and $- f x$ .

$- f x + y = 0$

Step 1.2

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

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

Step 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

This is the form of a hyperbola. Use this form to determine the values used to find vertices and 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, $a$ .

$a = 1$

$b = 1$

$k = 0$

$h = 0$

Step 4

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 0)$

Step 5

Find $c$ , the distance from the center to a focus.

Tap for more steps...

Step 5.1

Find the distance from the center to a focus of the hyperbola by using the following formula.

$\sqrt{a^{2} + b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(1)}^{2} + {(1)}^{2}}$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

One to any power is one.

$\sqrt{1 + {(1)}^{2}}$

Step 5.3.2

One to any power is one.

$\sqrt{1 + 1}$

Step 5.3.3

Add $1$ and $1$ .

$\sqrt{2}$

Step 6

Find the vertices.

Tap for more steps...

Step 6.1

The first vertex of a hyperbola can be found by adding $a$ to $h$ .

$(h + a, k)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(1, 0)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.4

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(- 1, 0)$

Step 6.5

The vertices of a hyperbola follow the form of $(h \pm a, k)$ . Hyperbolas have two vertices.

$(1, 0), (- 1, 0)$

Step 7

Find the foci.

Tap for more steps...

Step 7.1

The first focus of a hyperbola can be found by adding $c$ to $h$ .

$(h + c, k)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(\sqrt{2}, 0)$

Step 7.3

The second focus of a hyperbola can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.4

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(- \sqrt{2}, 0)$

Step 7.5

The foci of a hyperbola follow the form of $(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(\sqrt{2}, 0), (- \sqrt{2}, 0)$

Step 8

Find the eccentricity.

Tap for more steps...

Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} + b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(1)}^{2} + {(1)}^{2}}}{1}$

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Divide $\sqrt{{(1)}^{2} + {(1)}^{2}}$ by $1$ .

$\sqrt{{(1)}^{2} + {(1)}^{2}}$

Step 8.3.2

One to any power is one.

$\sqrt{1 + {(1)}^{2}}$

Step 8.3.3

One to any power is one.

$\sqrt{1 + 1}$

Step 8.3.4

Add $1$ and $1$ .

$\sqrt{2}$

Step 9

Find the focal parameter.

Tap for more steps...

Step 9.1

Find the value of the focal parameter of the hyperbola by using the following formula.

$\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$

Step 9.2

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{1^{2}}{\sqrt{2}}$

Step 9.3

Simplify.

Tap for more steps...

Step 9.3.1

One to any power is one.

$\frac{1}{\sqrt{2}}$

Step 9.3.2

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

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 9.3.3.1

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

$\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 9.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 9.3.3.4

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

$\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.3.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{2^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{\sqrt{2}}{2}$

Step 10

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 11

Simplify $1 \cdot x + 0$ .

Tap for more steps...

Step 11.1

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

$y = 1 \cdot x$

Step 11.2

Multiply $x$ by $1$ .

$y = x$

Step 12

Simplify $- 1 \cdot x + 0$ .

Tap for more steps...

Step 12.1

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

$y = - 1 \cdot x$

Step 12.2

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

$y = - x$

Step 13

This hyperbola has two asymptotes.

$y = x, y = - x$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(0, 0)$

Vertices: $(1, 0), (- 1, 0)$

Foci: $(\sqrt{2}, 0), (- \sqrt{2}, 0)$

Eccentricity: $\sqrt{2}$

Focal Parameter: $\frac{\sqrt{2}}{2}$

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

Step 15