Hyperbolas: Videos & Practice Problems

Hyperbolas are conic sections with two curved branches that open away from each other. A key way to recognize a hyperbola is the minus sign in its standard form, unlike an ellipse which uses a plus sign. The basic equations are \( \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) and \( \frac{y^2}{b^2}-\frac{x^2}{a^2}=1 \) . The leading squared term determines whether the transverse axis is horizontal or vertical.

To graph a hyperbola, identify the center, find \(a\) and \(b\) from the denominators, and build the fundamental rectangle by moving \(a\) units left and right and \(b\) units up and down from the center. The vertices lie on the transverse axis, and the asymptotes pass through the diagonals of the rectangle, guiding the branches without touching them. If the equation is shifted, as in \((x-h)\) and \((y-k)\), then the center is \((h,k)\) and the same graphing strategy still applies.

Intro To Hyperbolas

A hyperbola is a unique conic section characterized by two distinct curved branches that open away from each other. These branches can be oriented horizontally or vertically, forming what are known as horizontal and vertical hyperbolas, respectively. In a horizontal hyperbola, the branches open left and right along the x-axis, while in a vertical hyperbola, they open up and down along the y-axis. A key feature of hyperbolas is the presence of asymptotes—lines that the branches approach but never touch. These asymptotes guide the shape and steepness of the hyperbola's branches, providing a framework for sketching the graph.

The hyperbola also includes vertices, which are the points where each branch crosses the transverse axis. For a horizontal hyperbola, the transverse axis aligns with the x-axis, and for a vertical hyperbola, it aligns with the y-axis. Additionally, hyperbolas have two fixed points called foci (singular: focus). The defining property of a hyperbola is that for any point on the curve, the absolute difference of the distances to the two foci remains constant. This geometric definition is fundamental to understanding the shape and behavior of hyperbolas, even though detailed calculations involving the foci are often not required in basic courses.

The standard equation of a hyperbola depends on its orientation. For a horizontal hyperbola centered at the origin, the equation is given by:

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

Here, a represents the distance from the center to each vertex along the x-axis, and b relates to the distance that helps define the asymptotes. For a vertical hyperbola, the equation switches the positions of x and y:

\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]

The minus sign in the equation distinguishes hyperbolas from ellipses, which have a plus sign between the terms. The order of the terms is crucial because it determines the hyperbola’s orientation.

To find the equation of a hyperbola from its graph, one typically identifies the fundamental rectangle formed by the distances a and b. The vertices lie at ±a along the transverse axis, and the rectangle’s dimensions help determine the slopes of the asymptotes. For example, if the vertices are at ±3 on the x-axis and the corresponding b value is 4, the equation of a horizontal hyperbola would be:

\[\frac{x^2}{3^2} - \frac{y^2}{4^2} = 1 \quad \Rightarrow \quad \frac{x^2}{9} - \frac{y^2}{16} = 1\]

Similarly, for a vertical hyperbola with vertices at ±4 on the y-axis and b equal to 3, the equation becomes:

\[\frac{y^2}{4^2} - \frac{x^2}{3^2} = 1 \quad \Rightarrow \quad \frac{y^2}{16} - \frac{x^2}{9} = 1\]

Understanding these fundamental properties and equations allows for the recognition and graphing of hyperbolas. The asymptotes, vertices, and foci provide a comprehensive framework for analyzing these curves. With practice, identifying the orientation and writing the equation of a hyperbola becomes more intuitive, making this complex shape more approachable in algebra and precalculus studies.

Determine if the transverse axis is horizontal or vertical for the following hyperbolas.
$\frac{x^2}{12}-\frac{y^2}{16}=1$

• A. Horizontal
• B. Vertical
• C. Cannot be determined.

Answer: (A)

Determine if the transverse axis is horizontal or vertical for the following hyperbolas.
$\frac{y^2}{9}-\frac{x^2}{9}=1$

• A. Horizontal
• B. Vertical
• C. Cannot be determined

Answer: (B)

Determine if the transverse axis is horizontal or vertical for the following hyperbolas.
$3x^2-y^2=18$

• A. Horizontal
• B. Vertical
• C. Cannot be determined

Answer: (A)

Identify whether the equation is of an ellipse or hyperbola.
$\frac{x^2}{4}-\frac{y^2}{9}=1$

• A. Ellipse
• B. Hyperbola
• C. Neither of the above

Answer: (B)

Identify whether the equation is of an ellipse or hyperbola.
$\frac{x^2}{16}+\frac{y^2}{9}=1$

• A. Ellipse
• B. Hyperbola
• C. Neither of the above

Answer: (A)

Identify whether the equation is of an ellipse or hyperbola.
$y^2-\frac{x^2}{16}=1$

• A. Ellipse
• B. Hyperbola
• C. Neither of the above

Answer: (B)

Graphing Hyperbolas

A hyperbola is a type of conic section characterized by its distinct shape and equation form, typically involving a subtraction between squared terms. When graphing a hyperbola, the equation often appears in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for a horizontal hyperbola or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for a vertical hyperbola. The presence of the minus sign between the terms confirms the hyperbola, and the variable with the positive term determines its orientation—horizontal if \(x^2\) is positive, vertical if \(y^2\) is positive.

The center of the hyperbola is a crucial point, often denoted as \((h, k)\). For hyperbolas centered at the origin, this is simply \((0,0)\). The values of \(a\) and \(b\) are derived from the denominators under the squared terms: \(a = \sqrt{a^2}\) and \(b = \sqrt{b^2}\). These values help construct the fundamental rectangle, which is essential for graphing. Starting at the center, move \(a\) units left and right and \(b\) units up and down to form this rectangle. For example, with the equation \(\frac{x^2}{9} - \frac{y^2}{4} = 1\), \(a = 3\) and \(b = 2\(, so the rectangle extends 3 units horizontally and 2 units vertically from the center.

Once the fundamental rectangle is drawn, the next step is to sketch the asymptotes. These are straight lines passing through the diagonals of the rectangle and serve as guides for the hyperbola's branches. The asymptotes can be expressed by the equations:

For a hyperbola centered at the origin with a horizontal transverse axis:

\[y = \pm \frac{b}{a} x\]

For a hyperbola centered at \)(h, k)\(, the asymptotes shift accordingly:

\[y - k = \pm \frac{b}{a} (x - h)\]

The vertices of the hyperbola lie along the transverse axis, located \)a\) units from the center. In the horizontal case, these are at \((h \pm a, k)\), and in the vertical case, at \((h, k \pm a)\). The branches of the hyperbola curve outward from these vertices, approaching but never touching the asymptotes.

Graphing any hyperbola involves identifying the center, calculating \(a\) and \(b\) from the equation, drawing the fundamental rectangle, sketching the asymptotes through its diagonals, and finally drawing the hyperbola’s branches through the vertices, approaching the asymptotes. This method applies universally, whether the hyperbola is horizontal, vertical, or shifted away from the origin.

Graphing Hyperbolas Example 1

When graphing hyperbolas, the first step is to identify the center and determine whether the hyperbola opens horizontally or vertically. For hyperbolas in the standard form x²/a² − y²/b² = 1, the center is at the origin (0,0), and the hyperbola opens horizontally. Conversely, for the form y²/a² − x²/b² = 1, the center is also at (0,0), but the hyperbola opens vertically.

To find the values of a and b, take the square roots of the denominators under the squared terms. For example, in the equation x²/16 − y²/25 = 1, a = √16 = 4 and b = √25 = 5. These values help define the fundamental rectangle used to sketch the hyperbola. Starting at the center, move a units left and right, and b units up and down to mark the rectangle’s vertices.

The asymptotes of the hyperbola pass through the diagonals of this fundamental rectangle. Drawing straight lines along these diagonals provides the asymptotes, which guide the shape of the hyperbola’s branches. The branches themselves pass through the vertices located at (±a, 0) for horizontal hyperbolas or (0, ±a) for vertical hyperbolas, curving outward and approaching the asymptotes without touching them.

For instance, if the equation is y²/9 − x²/1 = 1, rewriting it as y²/9 − x²/1 = 1 clarifies the form. Here, a = √9 = 3 and b = √1 = 1, indicating a vertical hyperbola centered at (0,0). The fundamental rectangle extends 1 unit left and right and 3 units up and down from the center. The asymptotes are drawn through the rectangle’s diagonals, and the branches curve through the vertices at (0, ±3), approaching the asymptotes.

In summary, graphing hyperbolas involves recognizing the standard form to determine the center and orientation, calculating a and b from the denominators, constructing the fundamental rectangle, drawing asymptotes along its diagonals, and sketching the hyperbola’s branches through the vertices. With practice, this process becomes intuitive, enabling accurate and efficient graphing of both horizontal and vertical hyperbolas.

Graphing Hyperbolas Example 2

When graphing hyperbolas, it is essential to first identify the center, orientation, and key parameters such as a and b. The general form of a hyperbola centered at (h, k) is either
\(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\) for a horizontal hyperbola, or
\(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\) for a vertical hyperbola.

For example, consider the equation \(\frac{(x - 2)^2}{4} - \frac{(y - 1)^2}{9} = 1\). Here, the center is at (2, 1), derived from the values subtracted inside the parentheses. Since the x-term appears first and is positive, this hyperbola opens horizontally. The values of a and b are found by taking the square roots of the denominators: \(a = \sqrt{4} = 2\) and \(b = \sqrt{9} = 3\).

To graph this hyperbola, start by plotting the center at (2, 1). From the center, move a units left and right (2 units) to locate the vertices at (0, 1) and (4, 1). Then, move b units up and down (3 units) to form a fundamental rectangle. Drawing the rectangle helps in sketching the asymptotes, which are the diagonals of this rectangle. The hyperbola’s branches approach these asymptotes but never touch them, passing through the vertices and curving outward horizontally.

In contrast, consider a vertical hyperbola such as \(\frac{y^2}{16} - \frac{(x + 2)^2}{1} = 1\). This can be rewritten as \(\frac{(y - 0)^2}{16} - \frac{(x - (-2))^2}{1} = 1\) to match the standard form, revealing the center at (-2, 0). Since the y-term is positive and first, the hyperbola opens vertically. Here, \(a = \sqrt{16} = 4\) and \(b = \sqrt{1} = 1\).

Plotting the center at (-2, 0), move a units up and down (4 units) to find the vertices at (-2, 4) and (-2, -4). Then, move b units left and right (1 unit) to complete the fundamental rectangle. The asymptotes are drawn along the diagonals of this rectangle, guiding the shape of the hyperbola’s branches, which open vertically and approach these asymptotes without intersecting them.

Understanding how to identify the center, orientation, and parameters a and b allows for accurate graphing of hyperbolas, even when they are not centered at the origin. The fundamental rectangle and asymptotes are crucial tools in sketching the hyperbola’s shape, ensuring the branches are correctly positioned relative to the vertices and asymptotes. Mastery of these concepts enables confident graphing of both horizontal and vertical hyperbolas in various positions on the coordinate plane.

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola.
$5x^2+5y^2=180$

• A. A circle
• B. An ellipse
• C. A hyperbola
• D. A parabola

Answer: (A)

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola.
$\frac{x^2}{100}+y^2=1$

• A. A circle
• B. An ellipse
• C. A hyperbola
• D. A parabola

Answer: (B)

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola.
$4x^2-9y^2=36$

• A. A circle
• B. An ellipse
• C. A hyperbola
• D. A parabola

Answer: (C)

Graphing Hyperbolas Example 3

When graphing hyperbolas, it is essential to identify whether the hyperbola is horizontal or vertical, as this determines the orientation of its branches. A vertical hyperbola typically has the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), while a horizontal hyperbola is expressed as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). The center of the hyperbola is found by examining the equation; if there are no horizontal or vertical shifts (no terms like \((x-h)\) or \((y-k)\)), the center is at the origin \((0,0)\).

To graph a vertical hyperbola, first determine the values of \(a\) and \(b\) by taking the square roots of the denominators under \(y^2\) and \(x^2\), respectively. For example, if \(a^2 = 9\) and \(b^2 = 4\), then \(a = 3\) and \(b = 2\). These values help construct the fundamental rectangle, which is crucial for sketching the hyperbola and its asymptotes. The rectangle extends \(a\) units vertically and \(b\( units horizontally from the center, so in this case, it spans 3 units up and down, and 2 units left and right.

The asymptotes of the hyperbola are the diagonals of this fundamental rectangle. For a vertical hyperbola centered at the origin, the equations of the asymptotes are given by:

\[y = \pm \frac{a}{b} x\]

In this example, the asymptotes are \)y = \pm \frac{3}{2} x\). These lines guide the shape of the hyperbola's branches, which approach but never touch the asymptotes.

The vertices of the vertical hyperbola lie along the \(y\)-axis at \((0, \pm a)\), so here they are at \((0, 3)\) and \((0, -3)\). The branches of the hyperbola open upward and downward, passing through these vertices and approaching the asymptotes. This contrasts with a horizontal hyperbola, where the vertices lie along the \(x\)-axis at \((\pm a, 0)\), and the branches open left and right.

Interestingly, both vertical and horizontal hyperbolas with the same \(a\) and \(b\) values share the same fundamental rectangle and asymptotes, differing only in the orientation of their branches and the location of their vertices. This means that once the fundamental rectangle and asymptotes are established, the graphing process is similar for both types of hyperbolas, with the key difference being the axis along which the vertices and branches lie.

Understanding these relationships allows for efficient graphing of hyperbolas by focusing on the center, fundamental rectangle dimensions, vertices, and asymptotes. Recognizing the orientation from the equation helps determine the correct placement of branches, ensuring an accurate and clear graph of the hyperbola.