Learn How to Graph a Hyperbola | Graphing Hyperbola with Center at the Origin

Prof D
27 Apr 202108:12
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, the instructor demonstrates how to graph a hyperbola centered at the origin. The process involves rewriting the equation into standard form, identifying the hyperbola's orientation as vertical, and calculating key parameters such as 'a', 'b', and 'c'. The video then explains how to determine the vertices and the length of the transverse axis. Furthermore, it provides the equations for the asymptotes and guides viewers through sketching the hyperbola, including the use of the asymptotes to form the graph. The video concludes with a call for questions and clarifications, inviting engagement from the audience.

Takeaways
  • 📚 The video is a tutorial on graphing a hyperbola with its center at the origin.
  • 🔍 The process begins by rewriting the equation of the hyperbola into standard form.
  • 📐 The standard form of a hyperbola is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
  • 📈 The hyperbola in the video opens vertically, indicated by a negative term in the standard equation.
  • 📝 The values of 'a', 'b', and 'c' are identified from the standard form equation.
  • 📌 'a' is determined to be 2 by taking the square root of the denominator under 'y^2'.
  • 🧭 'b' is found to be 1 by recognizing that 'b^2' is the denominator under 'x^2'.
  • 📉 'c' is calculated using the formula \( c^2 = a^2 + b^2 \), resulting in \( c = \sqrt{5} \).
  • 📍 The vertices of the hyperbola are at (0, ±2), derived from the 'a' value.
  • 📏 The length of the transverse axis is 4 units, which is 2 times 'a'.
  • 📈 The equations of the asymptotes are \( y = ±2x \), based on the values of 'a' and 'b'.
  • 🎨 The graph of the hyperbola is sketched with the center at the origin, vertices, transverse axis, and asymptotes.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is to demonstrate how to graph a hyperbola with its center at the origin.

  • What is the first step in graphing a hyperbola as described in the video?

    -The first step is to rewrite the equation of the hyperbola into its standard form by dividing both sides by four and simplifying it to y^2/4 - x^2 = 1.

  • How does the video determine that the hyperbola opens vertically?

    -The video determines that the hyperbola opens vertically because the standard form equation has y^2 in the numerator and x^2 in the denominator.

  • What is the standard form equation for a hyperbola?

    -The standard form equation for a hyperbola is y^2/a^2 - x^2/b^2 = 1 for a vertically opening hyperbola, or x^2/a^2 - y^2/b^2 = 1 for a horizontally opening hyperbola.

  • What is the value of 'a' in the hyperbola equation provided in the video?

    -The value of 'a' is 2, as determined by taking the square root of the denominator in the y^2 term of the standard form equation.

  • How is the value of 'b' found in the video?

    -The value of 'b' is found by recognizing that b^2 is the denominator of the x^2 term in the standard form equation, which is 1, so b equals 1.

  • What is the value of 'c' in the context of the hyperbola graphing in the video?

    -The value of 'c' is the square root of 5, found by using the formula c^2 = a^2 + b^2.

  • What are the coordinates of the vertices of the hyperbola in the video?

    -The vertices of the hyperbola are at (0, 2) and (0, -2), since the hyperbola opens vertically and 'a' is 2.

  • How long is the transverse axis of the hyperbola described in the video?

    -The length of the transverse axis is 4 units, which is 2 times the value of 'a'.

  • What are the equations of the asymptotes for the hyperbola graphed in the video?

    -The equations of the asymptotes are y = 2x and y = -2x, derived from the formula y = ±a/b*x, with 'a' being 2 and 'b' being 1.

  • How does the video guide viewers to sketch the graph of the hyperbola?

    -The video guides viewers to sketch the graph by first plotting the vertices, then drawing the asymptotes, and finally using the asymptotes to outline the hyperbola shape from the vertex.

Outlines
00:00
📚 Introduction to Graphing a Hyperbola

The video begins with a brief introduction to the task at hand: graphing a hyperbola with its center at the origin. The presenter outlines the steps required to find the vertices, the length of the transverse axis, and the equations of the asymptotes, emphasizing the importance of rewriting the equation into standard form. The standard form for a hyperbola is given as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), where \( a \) and \( b \) are constants that determine the shape of the hyperbola. The presenter then proceeds to simplify the equation by dividing both sides by four, resulting in \( \frac{y^2}{4} - x^2 = 1 \), which indicates that the hyperbola opens vertically.

05:01
📐 Identifying Key Features of the Hyperbola

In this section, the presenter moves on to identify the key features of the hyperbola, such as the vertices and the length of the transverse axis. The vertices are found using the formula \( (0, \pm a) \), leading to the vertices being at (0, 2) and (0, -2). The length of the transverse axis is calculated as \( 2a \), which in this case is 4 units. The presenter also explains how to derive the equations of the asymptotes for a vertically opening hyperbola, which are \( y = \pm \frac{a}{b}x \). Substituting the values of \( a \) and \( b \), the asymptotes are found to be \( y = \pm 2x \).

🎨 Graphing the Hyperbola

The final part of the script focuses on the actual graphing process. The presenter describes how to plot the hyperbola with its center at the origin, starting 2 units upwards from the origin to locate one vertex and 2 units downwards for the other. The transverse axis is marked accordingly. The presenter also explains how to draw the asymptotes, which are lines that the hyperbola approaches but never touches. The video concludes with a reminder to viewers to ask questions or seek clarifications in the comments section and a sign-off from the presenter.

Mindmap
Keywords
💡Hyperbola
A hyperbola is a type of conic section that resembles two mirror-imaged parabolas. It is defined as the set of all points in a plane, the difference of whose distances from two fixed points (foci) is constant. In the video, the theme revolves around graphing a hyperbola with its center at the origin, indicating the focus on understanding and visualizing this mathematical shape.
💡Vertices
Vertices in the context of a hyperbola refer to the points on the curve that are closest to the center. They are significant as they represent the endpoints of the transverse axis. The script mentions finding the vertices to be at (0, ±2), which helps in sketching the basic shape of the hyperbola.
💡Transverse Axis
The transverse axis of a hyperbola is the segment that passes through the center and the vertices. It is the longest diameter of the hyperbola and is perpendicular to the conjugate axis. The video script discusses finding the length of the transverse axis, which is crucial for accurately graphing the hyperbola.
💡Asymptotes
Asymptotes are lines that a hyperbola approaches but never intersects, regardless of how far it extends. They provide a guide for sketching the hyperbola and are essential for understanding its behavior. The script explains finding the equations of the asymptotes as y = ±2x, which are critical for graphing the hyperbola.
💡Standard Form
The standard form of a hyperbola equation is a specific way of writing the equation that makes it easier to identify its properties. The script describes rewriting the equation to standard form, which is necessary to identify the hyperbola's orientation and other characteristics.
💡Center at the Origin
When the center of a hyperbola is at the origin, it means that the geometric center of the hyperbola is located at the coordinate (0,0). This is a common scenario for teaching purposes and simplifies the process of graphing. The video script uses this as a starting point for graphing the hyperbola.
💡Equation of Hyperbola
The equation of a hyperbola is a mathematical representation that defines the set of points that form the hyperbola. In the script, the equation is manipulated and simplified to standard form, which is essential for identifying key properties and graphing the hyperbola.
💡Conjugate Axis
The conjugate axis of a hyperbola is perpendicular to the transverse axis and passes through the center. It is used in conjunction with the transverse axis to define the hyperbola's shape. The script mentions using the conjugate axis to determine the points for graphing the hyperbola.
💡Simplifying Equations
Simplifying equations is a mathematical process used to make complex expressions easier to understand and work with. In the script, the equation of the hyperbola is simplified to standard form, which helps in identifying the hyperbola's properties and graphing it.
💡Graphing
Graphing in mathematics involves visually representing data or equations on a graph. The video script is focused on teaching how to graph a hyperbola by finding its vertices, asymptotes, and other properties, which is essential for understanding the shape and behavior of the hyperbola.
Highlights

Introduction to graphing a hyperbola with the center at the origin.

Objective to find vertices, length of the transverse axis, and equations of asymptotes.

Rewriting the equation to standard form by dividing both sides by four.

Simplification leads to the standard form of a hyperbola equation.

Identification of the hyperbola opening vertically due to the negative term.

Explanation of the hyperbola formula y^2/a^2 - x^2/b^2 = 1.

Solving for the value of 'a' using the formula.

Determining 'b' value by adding b^2 to the denominator.

Calculating 'c' using the relationship c^2 = a^2 + b^2.

Identifying vertices of the hyperbola using the formula (0, ±a).

Calculating the length of the transverse axis as 2a.

Deriving the equations of the asymptotes for a vertically opening hyperbola.

Graphing the hyperbola by plotting the vertices and transverse axis.

Drawing the asymptotes using the calculated equations.

Final graphing steps to complete the hyperbola sketch.

Conclusion of the video with an invitation for questions and comments.

Sign-off from Prof D with a promise to catch viewers on the flip side.

Transcripts

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Hyperbola GraphingVerticesTransverse AxisAsymptotesMath TutorialEducational VideoGraphing BasicsAlgebraic GeometryStandard FormMathematics Channel