Focus and Directrix of a Parabola 2 | Summary and Q&A

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July 14, 2009
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Khan Academy
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Focus and Directrix of a Parabola 2

TL;DR

The video explains how to find the focus and directrix of a parabola given its equation in the form y - y1 = A(x - x1)^2.

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Questions & Answers

Q: How can the equation y - y1 = A(x - x1)^2 be used to find the focus and directrix of a parabola?

By pattern matching, the x-coordinate of the focus is equal to the x-coordinate of the vertex, while the y-coordinate of the focus is determined by adding 1/(4A) to the y-coordinate of the vertex. The directrix is located 1/(4A) below the y-coordinate of the vertex.

Q: What is the importance of the vertex in finding the focus and directrix?

The vertex serves as a key point in determining the position of the focus and directrix. Its x-coordinate is the same as the x-coordinate of the focus, while its y-coordinate is used to calculate the y-coordinate of the directrix.

Q: How can the equation y - y1 = A(x - x1)^2 be graphed to visualize the focus and directrix?

The vertex, located at the point (x1, y1), represents the lowest or highest point on the parabola. The focus is located 1/(4A) units above the vertex, while the directrix is located 1/(4A) units below the vertex. The graph of the parabola will be symmetric with respect to the line of the directrix.

Q: What is the significance of the scaling factor, A, in the parabola equation?

The scaling factor, A, determines the steepness of the parabola. A larger value of A makes the parabola narrower, while a smaller value makes it wider. The reciprocal of 4A is used to calculate the distance between the vertex and the focus/directrix.

Summary & Key Takeaways

  • The video demonstrates how the equation y - y1 = A(x - x1)^2 can be used to find the focus and directrix of a parabola.

  • By pattern matching, the x-coordinate of the focus is equal to the x-coordinate of the vertex.

  • The y-coordinate of the focus is determined by adding 1/(4A) to the y-coordinate of the vertex.

  • The directrix is located 1/(4A) below the y-coordinate of the vertex.

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