12. Conic Sections & Systems of Nonlinear Equations

Hyperbolas

12. Conic Sections & Systems of Nonlinear Equations

Hyperbolas: Videos & Practice Problems

Topic summary

Hyperbolas are conic sections with two curved branches that open away from each other, either horizontally or vertically. A key feature is the pair of asymptotes, lines the branches approach without touching, and the vertices, which lie on the transverse axis. The graph is built from a fundamental rectangle, whose diagonals give the asymptotes and help control the shape of the branches. The foci are fixed points related to the constant difference in distances from any point on the hyperbola.

A hyperbola is recognized by the minus sign in its standard form: $ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 $ or $ \frac{y^2}{b^2}-\frac{x^2}{a^2}=1 $ . The term that appears first determines whether the hyperbola is horizontal or vertical. If the center is not at the origin, the form shifts to $ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 $ or $ \frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1 $ .

Concept

Intro To Hyperbolas

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Intro To Hyperbolas Video Summary

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 visual framework for graphing.

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 graphing.

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

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

For a vertical hyperbola, the equation switches the positions of the variables:

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

Here, a and b are positive real numbers representing distances related to the hyperbola’s geometry. The value a corresponds to the distance from the center to each vertex along the transverse axis, while b relates to the distance that helps define the slopes of the asymptotes.

To find the equation of a hyperbola from its graph, one can use the fundamental rectangle formed by the distances a and b. This rectangle is drawn using the vertices and the corresponding vertical or horizontal distances, although it does not appear on the actual graph of the hyperbola. For example, if the vertices lie at ±3 on the x-axis and the corresponding vertical distance is 4, then a = 3 and b = 4. Substituting these into the horizontal hyperbola equation yields:

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

Similarly, for a vertical hyperbola with vertices at ±4 on the y-axis and a horizontal distance of 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 equations and their components allows for the recognition and graphing of hyperbolas. The minus sign in the equation distinguishes hyperbolas from ellipses, which have a plus sign between the squared terms. The order of the variables in the equation determines the orientation of the hyperbola, making it essential to identify which term comes first.

Mastering hyperbolas involves recognizing their defining features—two branches, asymptotes, vertices, and foci—and understanding how these relate to the standard equations. With practice, graphing hyperbolas and interpreting their equations becomes more intuitive, providing a solid foundation for exploring more complex conic sections and their applications in mathematics and science.

Problem

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

Horizontal

Vertical

Cannot be determined.

Problem

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

Horizontal

Vertical

Cannot be determined

Problem

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

Horizontal

Vertical

Cannot be determined

Problem

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

Ellipse

Hyperbola

Neither of the above

Problem

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

Ellipse

Hyperbola

Neither of the above

Problem

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

Ellipse

Hyperbola

Neither of the above

Concept

Graphing Hyperbolas

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Graphing Hyperbolas Video Summary

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 general equation takes 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 leading squared term determines its orientation—horizontal if $x^2$ leads, vertical if $y^2$ leads.

The center of the hyperbola, often denoted as $(h, k)$, is the midpoint around which the hyperbola is symmetric. For the standard form centered at the origin, the center is at $(0,0)$. When the hyperbola is shifted, the equation adjusts to $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$, reflecting the new center coordinates.

To graph a hyperbola effectively, begin by identifying the values of $a$ and $b$ from the denominators of the squared terms. These values represent distances from the center: $a$ units along the transverse axis (the axis along which the hyperbola opens) and $b$ units along the conjugate axis (perpendicular to the transverse axis). For example, if the equation is $\frac{x^2}{9} - \frac{y^2}{4} = 1$, then $a = \sqrt{9} = 3$ and $b = \sqrt{4} = 2$.

Using $a$ and $b$, construct the fundamental rectangle by moving $a$ units left and right from the center and $b$ units up and down. This rectangle serves as a guide for drawing the asymptotes, which are straight lines passing through the center and the rectangle's opposite corners. The equations of the asymptotes for a hyperbola centered at $(h, k)$ are:

$y - k = \pm \frac{b}{a} (x - h)$ for a horizontal hyperbola, and
$y - k = \pm \frac{a}{b} (x - h)$ for a vertical hyperbola.

These asymptotes act as boundaries that the hyperbola's branches approach but never touch. The vertices of the hyperbola lie along the transverse axis at a distance $a$ from the center, located at $(h \pm a, k)$ for horizontal hyperbolas or $(h, k \pm a)$ for vertical hyperbolas. The branches of the hyperbola curve outward from these vertices, approaching the asymptotes as they extend further from the center.

Graphing a hyperbola involves plotting the center, drawing the fundamental rectangle using $a$ and $b$, sketching the asymptotes through the rectangle's diagonals, marking the vertices on the transverse axis, and finally drawing the two branches that open away from the center along the transverse axis. This method applies universally, regardless of the hyperbola's orientation or center location, making it a reliable strategy for mastering hyperbola graphs.

Example

Graphing Hyperbolas Example 1

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Graphing Hyperbolas Example 1 Video Summary

When graphing hyperbolas, the first step is to identify the center and determine whether the hyperbola opens horizontally or vertically. For hyperbolas in standard form, such as $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$, the center is typically at the origin $(0,0)$ if there are no horizontal or vertical shifts in the equation.

In the case of a horizontal hyperbola, the equation takes the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Here, the transverse axis is along the x-axis, meaning the hyperbola opens left and right. The values of $a$ and $b$ are found by taking the square roots of the denominators under $x^2$ and $y^2$, respectively. For example, if the equation is $\frac{x^2}{16} - \frac{y^2}{25} = 1$, then $a = \sqrt{16} = 4$ and $b = \sqrt{25} = 5$. The center is at $(0,0)$, and the vertices are located at $(\pm a, 0)$, or $(\pm 4, 0)$ in this case.

To graph the hyperbola, first draw the fundamental rectangle centered at the origin. This rectangle extends $a$ units left and right along the x-axis and $b$ units up and down along the y-axis. For the example above, the rectangle spans from $-4$ to $4$ on the x-axis and approximately from $-5$ to \$5\( on the y-axis. The asymptotes of the hyperbola are the diagonals of this rectangle, and their equations can be expressed as:

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

In this example, the asymptotes are \)y = \pm \frac{5}{4} x\(. The hyperbola branches pass through the vertices and approach these asymptotes without touching them, creating the characteristic open curves.

For a vertical hyperbola, the equation is of the form $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$, where the transverse axis is along the y-axis, and the hyperbola opens upward and downward. For instance, if the equation is $\frac{y^2}{9} - x^2 = 1$, it can be rewritten as $\frac{y^2}{9} - \frac{x^2}{1} = 1$, revealing \)a = 1$ and $b = 3$. The center remains at $(0,0)$, and the vertices are at \((0, \pm b)$, or $(0, \pm 3)$ here.

Similar to the horizontal case, the fundamental rectangle is drawn by moving \)a$ units left and right and $b\( units up and down from the center. The asymptotes are the diagonals of this rectangle, with equations:

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

For this vertical hyperbola, the asymptotes are \)y = \pm 3x$. The branches of the hyperbola pass through the vertices at \((0, \pm 3)$ and approach the asymptotes as they extend outward.

Understanding the orientation of the hyperbola depends on which variable's squared term is positive and appears first in the equation. The center is found by identifying any horizontal or vertical shifts; if none exist, it is at the origin. The values of \)a$ and $b\( are derived from the denominators under the squared terms, and these values help construct the fundamental rectangle. Drawing the asymptotes as the diagonals of this rectangle guides the shape of the hyperbola's branches, which pass through the vertices and approach the asymptotes without intersecting them.

Mastering these steps—identifying the center, determining the orientation, calculating \)a$ and $b$, sketching the fundamental rectangle, drawing asymptotes, and finally sketching the branches—enables efficient and accurate graphing of hyperbolas. With practice, recognizing these patterns becomes intuitive, making it easier to handle a variety of hyperbola graphing problems.

Example

Graphing Hyperbolas Example 2

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Graphing Hyperbolas Example 2 Video Summary

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 given by 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. The values h and k represent the coordinates of the hyperbola's center, which may not always be at the origin.

For example, consider the hyperbola defined by

\[\frac{(x - 2)^2}{4} - \frac{(y - 1)^2}{9} = 1.\]

Here, the center is at (2, 1). 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 = 2 and b = 3. These values help construct the fundamental rectangle used to sketch the hyperbola. From the center, move 2 units left and right to mark the vertices at (0, 1) and (4, 1), and 3 units up and down to define the rectangle's height. The asymptotes are drawn as the diagonals of this rectangle, guiding the shape of the hyperbola's branches, which approach but never touch these lines.

In contrast, a vertical hyperbola example might be

\[\frac{(y - 0)^2}{16} - \frac{(x + 2)^2}{1} = 1,\]

which can be rewritten as

\[\frac{(y - 0)^2}{16} - \frac{(x - (-2))^2}{1} = 1.\]

This reveals the center at (-2, 0). Since the y-term is positive and leads, the hyperbola opens vertically. Here, a = 4 and b = 1. From the center, move 4 units up and down to locate the vertices at (-2, 4) and (-2, -4), and 1 unit left and right to complete the fundamental rectangle. The asymptotes, drawn along the rectangle's diagonals, guide the hyperbola's branches, which curve outward vertically, approaching the asymptotes.

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 this process, providing a framework to sketch the hyperbola's shape precisely. Mastery of these concepts ensures confidence in handling various hyperbola equations and their graphs.

Problem

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

A circle

An ellipse

A hyperbola

A parabola

Problem

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 circle

An ellipse

A hyperbola

A parabola

Problem

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

A circle

An ellipse

A hyperbola

A parabola

Example

Graphing Hyperbolas Example 3

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Graphing Hyperbolas Example 3 Video Summary

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 where the $y^2$ term is positive and comes first, while a horizontal hyperbola has the $x^2$ term positive and first. The center of the hyperbola is found by examining the equation; if there are no shifts inside the squared terms (no values subtracted from $x$ or $y$), the center is at the origin $(0,0)$.

To graph a vertical hyperbola, start by determining the values of $a$ and $b$ from the equation. These values come from the denominators under the squared terms, where $a^2$ and $b^2$ are given. For example, if $a^2 = 9$, then $a = \sqrt{9} = 3$, and if $b^2 = 4$, then $b = \sqrt{4} = 2$. These values help construct the fundamental rectangle, which guides the shape and asymptotes of the hyperbola.

The fundamental rectangle is drawn by moving $a$ units left and right from the center along the $x$-axis and $b$ units up and down along the $y$-axis. In this case, you would move 3 units left and right and 2 units up and down, forming a rectangle centered at the origin. The asymptotes of the hyperbola are the diagonals of this rectangle, and the hyperbola’s branches approach these asymptotes without touching them.

For a vertical hyperbola, the vertices lie along the $y$-axis at points $(0, a)$ and $(0, -a)$, which in this example are $(0, 3)$ and $(0, -3)$. The branches curve outward from these vertices, approaching the asymptotes defined by the diagonals of the fundamental rectangle. This contrasts with a horizontal hyperbola, where the vertices lie along the $x$-axis at $(a, 0)$ and $(-a, 0)$, but both share the same fundamental rectangle and asymptotes.

Understanding that both horizontal and vertical hyperbolas share the same fundamental rectangle and asymptotes allows for a streamlined graphing process. The key difference lies in the orientation of the vertices and the direction in which the branches open. This approach simplifies sketching hyperbolas by focusing on the center, values of $a$ and $b$, the fundamental rectangle, and the asymptotes, ensuring accurate and efficient graphing of both types of hyperbolas.

Here's what students ask on this topic:

The standard form of a hyperbola's equation is either $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ for a horizontal hyperbola or $\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$ for a vertical hyperbola. The orientation depends on which variable term appears first. If the $x^{2}$ term is positive and comes first, the hyperbola opens horizontally. If the $y^{2}$ term is positive and comes first, it opens vertically. The minus sign between the two fractions is key to identifying the shape as a hyperbola, distinguishing it from ellipses which have a plus sign.

To graph a hyperbola, first identify the center, usually at $(0,0)$ or $(h,k)$ if shifted. Then find $a$ and $b$ by taking the square roots of the denominators under $x^2$ and $y^2$ . Use these to draw the fundamental rectangle centered at the hyperbola's center, extending $a$ units horizontally and $b$ units vertically. Next, draw the asymptotes by sketching lines through the diagonals of this rectangle. The vertices lie on the transverse axis at a distance $a$ from the center. Finally, sketch the two branches of the hyperbola opening away from each other, approaching but never touching the asymptotes. This method works for both horizontal and vertical hyperbolas.

Asymptotes are straight lines that the branches of a hyperbola approach but never touch. They are important because they guide the shape and direction of the hyperbola's branches. When graphing, the asymptotes are drawn through the diagonals of the fundamental rectangle, which is constructed using the values of $a$ and $b$ . The branches get closer to these lines as they extend infinitely, helping to define the hyperbola's steepness and orientation. Although asymptotes do not appear on the graph of the hyperbola itself, they are essential for accurately sketching the curve.

The foci (plural of focus) are two fixed points located inside each branch of the hyperbola. The defining property of a hyperbola is that for any point on the curve, the absolute difference of the distances to the two foci is constant. This constant difference helps define the shape of the hyperbola. While calculations involving the foci are less common in basic algebra courses, understanding their role is important for grasping the geometric nature of hyperbolas and their applications in fields like physics and engineering.

The center of a hyperbola is the midpoint between its vertices and foci. When the center is at the origin, the equation is in the form $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ or its vertical counterpart. If the center is shifted to $(h,k)$ , the equation becomes $\frac{{(x−h)}^{2}}{a^{2}} - \frac{{(y−k)}^{2}}{b^{2}} = 1$ for a horizontal hyperbola. The center shifts the entire graph horizontally and vertically, changing the location of vertices, foci, and the fundamental rectangle, but the shape and orientation remain the same.

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Hyperbolas: Videos & Practice Problems