The graph of a hyperbola is given. Match each graph to its equation. (A) (x^2)/(4)-y^2=1 (B) x^2

step 1 With this hyperbola, there's a couple things to pay attention to. It has vertices at zero positi

The graph of a hyperbola is given. Match each graph to its...

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Key Concepts

Standard Form of a Hyperbola

A hyperbola’s equation is often given in a standard form that clearly shows its orientation and key features. For a horizontal hyperbola, the form is (x – h)²/a² – (y – k)²/b² = 1, and for a vertical hyperbola it is (y – k)²/a² – (x – h)²/b² = 1. This representation provides the center (h, k), the vertices, and helps in determining the directions in which the hyperbola opens.

Orientation of a Hyperbola

The orientation of a hyperbola is determined by which squared term has the positive coefficient in its standard equation. If the x-term is positive, this indicates a horizontal hyperbola that opens to the left and right; if the y-term is positive, it indicates a vertical hyperbola that opens upwards and downwards. Understanding this concept is key to matching a given graph to its corresponding equation.

Asymptotes of a Hyperbola

Hyperbolas have asymptotes—lines that the branches of the hyperbola approach as they extend toward infinity. The equations of the asymptotes can be derived from the standard form and are given by y = k ± (b/a)(x – h) for horizontal hyperbolas, and y = k ± (a/b)(x – h) for vertical hyperbolas. These asymptotes guide the overall shape of the graph and are essential for correctly identifying which hyperbola corresponds to a given equation.

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