How to find the equation of Hyperbola given its graph

Prof D
4 May 202108:21
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, Prof D guides viewers through the process of deriving the equation of a hyperbola from its graph. The video begins by explaining the standard equations for hyperbolas that open horizontally and vertically. Two examples are then presented, with step-by-step instructions on how to identify key parameters such as the center, vertex, and conjugate axis. The examples demonstrate how to substitute these values into the standard equations to obtain the hyperbola's equation in standard form. The video concludes with an invitation for viewers to ask questions in the comment section, ensuring an interactive learning experience.

Takeaways
  • 📚 The video provides a tutorial on how to find the equation of a hyperbola given its graph.
  • 📐 The standard equation for a horizontally opening hyperbola is \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\).
  • 📏 The standard equation for a vertically opening hyperbola is \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\).
  • 🔍 The center of the hyperbola is denoted by the coordinates (h, k).
  • 📏 The value 'a' represents the distance from the center to a vertex for a horizontally opening hyperbola.
  • 📏 The value 'b' represents the distance from the center to a co-vertex for a hyperbola.
  • 🔢 The script uses an example where the hyperbola opens horizontally with a center at (-3, 3), a vertex 4 units away, and a co-vertex 3 units away.
  • 🧩 In the example, the equation of the hyperbola is derived by substituting the values of h, k, a, and b into the standard equation.
  • 📝 The second example demonstrates a vertically opening hyperbola with a center at (-2, -1), a vertex 4 units away, and a co-vertex 2 units away.
  • 🔍 The second example also involves substituting the values of h, k, a, and b into the standard equation to find the hyperbola's equation.
  • 👋 The video concludes with an invitation for viewers to ask questions or seek clarifications in the comment section.
Q & A

  • What is the standard equation for a hyperbola that opens horizontally?

    -The standard equation for a hyperbola that opens horizontally is \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \), where \( h \) and \( k \) are the coordinates of the center, \( a \) is the distance from the center to the vertices, and \( b \) is the distance from the center to the endpoints of the conjugate axis.

  • What is the standard equation for a hyperbola that opens vertically?

    -The standard equation for a hyperbola that opens vertically is \( \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \), with the same meaning for \( h \), \( k \), \( a \), and \( b \) as in the horizontal case.

  • What does 'a' represent in the hyperbola equation?

    -'a' in the hyperbola equation represents the distance from the center of the hyperbola to its vertices along the transverse axis.

  • What does 'b' represent in the hyperbola equation?

    -'b' in the hyperbola equation represents the distance from the center of the hyperbola to its endpoints on the conjugate axis.

  • How can you determine if a hyperbola opens horizontally or vertically?

    -A hyperbola opens horizontally if the \( x \)-term is positive and the \( y \)-term is negative in the standard equation, and it opens vertically if the \( y \)-term is positive and the \( x \)-term is negative.

  • What is the center of the hyperbola in Example 1?

    -The center of the hyperbola in Example 1 is at the point \( (h, k) = (-3, 3) \).

  • What is the value of 'a' in Example 1?

    -In Example 1, the value of 'a' is 4 units, as it is the distance from the center to the vertex.

  • What is the value of 'b' in Example 1?

    -In Example 1, the value of 'b' is 3 units, as it is the distance from the center to the endpoint on the conjugate axis.

  • How is the equation of the hyperbola in Example 1 simplified?

    -The equation of the hyperbola in Example 1 is simplified by substituting the values of \( h \), \( k \), \( a \), and \( b \) into the standard equation for a horizontally opening hyperbola and simplifying the resulting expression.

  • What is the center of the hyperbola in Example 2?

    -The center of the hyperbola in Example 2 is at the point \( (h, k) = (-2, -1) \).

  • What is the value of 'a' in Example 2?

    -In Example 2, the value of 'a' is 4 units, as it is the distance from the center to the vertex along the vertical axis.

  • What is the value of 'b' in Example 2?

    -In Example 2, the value of 'b' is 2 units, as it is the distance from the center to the endpoint on the conjugate axis along the horizontal axis.

  • How is the equation of the hyperbola in Example 2 simplified?

    -The equation of the hyperbola in Example 2 is simplified by substituting the values of \( h \), \( k \), \( a \), and \( b \) into the standard equation for a vertically opening hyperbola and simplifying the resulting expression.

Outlines
00:00
📚 Introduction to Finding the Hyperbola Equation

This paragraph introduces the video's objective to teach viewers how to derive the equation of a hyperbola from its graph. It explains the standard equations for hyperbolas that open horizontally and vertically, highlighting the importance of identifying the center, vertices, and the distance between the center and the vertex (a), as well as the distance between the center and the endpoint of the conjugate axis (b). The video promises to guide through examples to illustrate these concepts.

05:02
📐 Example 1: Horizontal Hyperbola

The first example provided in the script involves a hyperbola that opens horizontally. The instructor begins by identifying the center (h, k) and the values of a and b, which are derived from the distances between the center and the vertex, and the center and the endpoint of the conjugate axis, respectively. The formula for a horizontal hyperbola is given, and the instructor proceeds to substitute the identified values into the formula, simplifying it to find the equation of the hyperbola in standard form.

📉 Example 2: Vertical Hyperbola

In the second example, the script discusses a hyperbola that opens vertically. Similar to the first example, the instructor identifies the center (h, k) and the values of a and b. The standard equation for a vertical hyperbola is presented, and the instructor substitutes the values of h, k, a, and b into this equation. After simplification, the equation of the vertical hyperbola in standard form is obtained. The video concludes with an invitation for viewers to ask questions or seek clarifications in the comment section and a farewell from the instructor.

Mindmap
Keywords
💡Hyperbola
A hyperbola is a type of conic section that resembles two mirror-image parabolas opening away from each other. In the context of the video, it is the main geometric figure being discussed. The script explains how to derive the equation of a hyperbola given its graph, which is central to the video's educational theme.
💡Equation
In mathematics, an equation is a statement that asserts the equality of two expressions. In the script, the term 'equation' specifically refers to the mathematical formula used to describe the shape and position of a hyperbola. The video's purpose is to show viewers how to find this equation based on the hyperbola's graph.
💡Graph
A graph in the context of the video refers to the visual representation of the hyperbola on a coordinate plane. It is the starting point for determining the hyperbola's equation. The script uses the graph to identify key features of the hyperbola, such as its vertices and asymptotes.
💡Center
The center of a hyperbola is the point from which the hyperbola is symmetrically distributed. In the script, the center is identified as a crucial point for determining the hyperbola's equation, as it helps in locating the vertices and the foci.
💡Vertex
A vertex of a hyperbola is one of the two points on the curve that is closest to the center. The script explains that the distance from the center to the vertex is denoted by 'a' and is used in the hyperbola's standard equation.
💡Asymptote
An asymptote of a hyperbola is a line that the hyperbola approaches but never intersects. In the script, the asymptotes are mentioned in relation to the conjugate axis and are important for understanding the orientation and shape of the hyperbola.
💡Conjugate Axis
The conjugate axis of a hyperbola is the axis perpendicular to the transverse axis and passes through the center of the hyperbola. The script uses the distance from the center to the endpoints on the conjugate axis, denoted by 'b', to form part of the hyperbola's equation.
💡Horizontally Opening Hyperbola
This term refers to a hyperbola that opens in the horizontal direction, with its branches extending left and right on the coordinate plane. The script provides the standard equation for such a hyperbola and uses it as an example to demonstrate the process of finding the equation.
💡Vertically Opening Hyperbola
A vertically opening hyperbola has branches that extend upwards and downwards from the center. The script also provides the standard equation for this type of hyperbola and uses it in another example to illustrate the equation derivation process.
💡Standard Equation
The standard equation of a hyperbola is a specific form of the equation that allows for easy identification of the hyperbola's key features, such as its center, vertices, and asymptotes. The script explains two forms of the standard equation, one for horizontally opening and one for vertically opening hyperbolas.
💡Transverse Axis
The transverse axis is the line segment that passes through the vertices and is perpendicular to the conjugate axis. In the script, the orientation of the hyperbola (whether it opens horizontally or vertically) is determined by the position of the transverse axis.
Highlights

Introduction to finding the equation of a hyperbola given its graph.

Explanation of the standard equation for a horizontally opening hyperbola.

Explanation of the standard equation for a vertically opening hyperbola.

Identification of the center, vertex, and foci for a horizontally opening hyperbola.

Calculation of 'a' as the distance between the center and the vertex.

Calculation of 'b' as the distance between the center and the endpoint on the conjugate axis.

Substitution of 'h', 'k', 'a', and 'b' into the hyperbola formula for a horizontally opening hyperbola.

Simplification of the hyperbola equation into standard form for a horizontally opening hyperbola.

Introduction to example number two with a vertically opening hyperbola.

Identification of the center and vertex for a vertically opening hyperbola.

Calculation of 'a' and 'b' for a vertically opening hyperbola.

Substitution of 'h', 'k', 'a', and 'b' into the hyperbola formula for a vertically opening hyperbola.

Simplification of the hyperbola equation into standard form for a vertically opening hyperbola.

End of the video with an invitation for questions and clarifications in the comments.

Closing remarks and sign-off from Prof D.

Transcripts

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Hyperbola EquationMathematics TutorialGraph AnalysisStandard FormEducational VideoGeometry LessonsEquation DerivationConjugate AxisVertex DistanceAsymptote