In Exercises 51 and 52, write an equation for the hyperbola. Graph cannot copy

Name: Problem 52, Chapter 8: Precalculus: Graphical, Numerical, Algebraic
Uploaded: 2019-07-19T21:53:03-08:00
Duration: 3 min 32 s
Channel: Margaret Hubacher
Description: In Exercises 51 and 52, write an equation for the hyperbola. Graph cannot copy

In Exercises 51 and 52, write an equation for the hyperbola....

Key Concepts

Asymptotes

Asymptotes are straight lines that the branches of a hyperbola approach but never meet. They pass through the center of the hyperbola and help define its shape; the slopes of these lines are determined by the values of 'a' and 'b' in the hyperbola’s equation and are essential in guiding the precise graphing of the curve.

Vertices

Vertices are the points where the hyperbola intersects its transverse axis, and they are located a fixed distance, denoted as 'a', from the center. Identifying the vertices on a graph allows for the determination of the value of 'a' and also helps in establishing the overall structure of the hyperbola.

Orientation

The orientation of a hyperbola indicates whether it opens horizontally or vertically. This distinction is determined by which variable is associated with the positive term in the standard equation. The orientation is crucial when writing the correct equation, as it affects the location of the vertices and the direction of the asymptotes.

Standard Equation of a Hyperbola

The equation of a hyperbola can be written in a standard form that depends on its orientation. For a hyperbola opening horizontally, the standard form is (x-h)²/a² - (y-k)²/b² = 1, and for one opening vertically, it is (y-k)²/a² - (x-h)²/b² = 1, where (h, k) is the center of the hyperbola, and a and b are distances that relate to the vertices and the 'steepness' of the curves respectively.

Hyperbola

A hyperbola is a type of conic section formed by intersecting a double-napped cone with a plane in such a way that the intersection produces two separate curves, or branches. It is defined geometrically as the set of all points in the plane such that the absolute difference of the distances from any point on the hyperbola to two fixed points (the foci) is constant.

Center

The center of a hyperbola is the midpoint between its two branches and serves as the point of symmetry for the graph. In the standard equation, the center is represented by the coordinates (h, k), and it is the reference point from which other key parameters like vertices and asymptotes are determined.

Transcript

00:01 So we have here, we have a graph of a hyperbola, and we want to find the equation for this hyperbola from the graph.

00:09 So the first thing that we know is that our vertices of the graph are at the points 0 plus or minus the square root of 2.

00:16 We also know that this graph passes through this point here, which is the point 2 -2, and we also know that the focal axis of the graph is the y -axis.

00:26 So we're going to be looking for some equation that looks like this.

00:29 We're going to be looking for an equation that is y squared over a squared minus x squared over b squared is equal to 1.

00:36 So we just have to find what a and b are equal to.

00:40 So first we can use the vertices of the graph to find our a because our vertices are just found at the points 0 plus or minus a, which means that a is just equal to the square root of 2.

00:53 So a is equal to the square root of 2.

00:56 And now that we have a value for a and we know a point on the graph, we can see that.

01:00 Solve for the value of b by plugging in negative 2 for y and 2 for x into this equation over here and root 2 for a.

01:10 So we'll write all of this out again.

01:12 So we'll have, so in the place of y, we'll put negative 2.

01:17 We'll have negative 2 squared, negative 2 squared, and we'll put this over a squared and root 2 squared is just 2.

01:28 So we have negative 2 squared over 2, minus, and we'll put this over a squared, and root 2 squared, and our x is 2, so we have 2 squared, 2 squared over b squared is equal to 1...

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