Question

Graph the equation: \(y^{2}=x^{2}-1\)

Short answer

The hyperbola centered at (0,0) opens upward and downward, with vertices at (0,1) and (0,-1), and asymptotes y=x and y=-x.

Step-by-step solution

- Identify the type of equation

The given equation is \(y^{2} = x^{2} -1\). This can be rewritten in the form \(y^{2} - x^{2} = -1\), which is a hyperbola.

- Rewrite it in standard form

The standard form for a hyperbola is \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) or \( \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\). Rewrite the equation \(y^{2} - x^{2} = -1\) as \( \frac{y^{2}}{1} - \frac{x^{2}}{1} = -1\). Now it is in standard form for a hyperbola that opens along the y-axis.

- Identify the center, vertices, and slopes of the asymptotes

For the hyperbola \( \frac{y^{2}}{1} - \frac{x^{2}}{1} = -1\), the center is at the origin \((0, 0)\). The vertices will be at the points where \( y = \frac{1}{2} \) and \( y = -\frac{1}{2} \) since \( a^2 = 1 \). The slopes of the asymptotes are \( \frac{\text{rise}}{\text{run}} = \frac{a}{b}\). Since \( a = b = 1 \), the slopes of the asymptotes are \( 1 \) and \( -1 \).

- Draw the asymptotes

Draw the asymptotes passing through the origin with slopes \( 1 \) and \( -1 \). These are the lines \( y = x \) and \( y = -x \).

- Draw the hyperbola

Sketch the hyperbola by drawing curves approaching but never touching the asymptotes, passing through the vertices \((0, 1)\) and \((0, -1)\). The hyperbola will be opening upwards and downwards.

Key concepts

Understanding Hyperbolas

A hyperbola is a type of conic section that forms two open, mirror-image curves. It looks like two open arms that extend infinitely in opposite directions. Hyperbolas occur when the plane intersects both halves of a double cone.

Unlike ellipses, hyperbolas have two branches. These branches are symmetric with respect to the center of the hyperbola. The center is the midpoint of the segment that connects the vertices.

Hyperbolas can open horizontally (left and right) or vertically (up and down). This depends on the orientation of the intersecting plane with the cone.

Asymptotes of a Hyperbola

Asymptotes are lines that a hyperbola approaches but never touches. These lines are crucial because they guide the shape and direction of the hyperbola’s branches. Asymptotes cross at the hyperbola's center and set the boundary for the curves.

For the given example with equation \[ y^2 = x^2 - 1 \], we reframe it as \[ y^2 - x^2 = -1 \]. From this form, it’s evident that the hyperbola’s asymptotes are the diagonal lines \[ y = x \] and \[ y = -x \].

These asymptotes illustrate the slanting paths the open curves will follow. In graphing, it’s crucial to draw these lines first to better visualize where the hyperbola’s branches will be plotted.

Remember, the curves get closer and closer to these asymptotes as they extend, but they never actually intersect.

Standard Form of a Hyperbola

The standard form of the hyperbola depends on its orientation: whether it opens horizontally or vertically.

For a hyperbola opening left/right (horizontally), the standard form is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \]

For a hyperbola opening up/down (vertically), the standard form is: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1. \]

In the example \[ y^2 - x^2 = -1 \], rewriting it gives \[ \frac{y^2}{1} - \frac{x^2}{1} = -1. \] This tells us that the hyperbola opens vertically because \[ y^2 \] is positive.

Here, \[ a^2 = 1 \] and \[ b^2 = 1 \], so \[ a = 1 \] and \[ b = 1. \] This information is used to determine the distance from the center to the vertices and the slopes of the asymptotes.

Vertices of a Hyperbola

The vertices of a hyperbola are the points where the branches make their closest approach to each other. Vertices are crucial because they help define the overall shape and location of the hyperbola.

In our example equation \[ y^2 = x^2 - 1 \], rewritten as \[ \frac{y^2}{1} - \frac{x^2}{1} = -1 \], the center is at the origin \[ (0, 0) \].

The vertices are located a distance \[ a \] away from the center along the direction the hyperbola opens. Since \[ a = 1 \], the vertices are at \[ y = 1 \] and \[ y = -1 \]; so, the points are \[ (0, 1) \] and \[ (0, -1) \]. These illustrate where each branch of the hyperbola is closest to the center before it curves outwards.