1Cademy - Standard Form of the Equation of a Hyperbola with Center $$(h, k)$$

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Standard Form of the Equation of a Hyperbola with Center $(h, k)$

When a hyperbola is not centered at the origin, its standard form equations use $(x - h)$ and $(y - k)$ in place of $x$ and $y$ , where $(h, k)$ is the center. If the transverse axis is horizontal, the standard form is $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ , and the hyperbola opens left and right. The vertices are located $a$ units to the left and right of the center. The central rectangle is formed by moving $a$ units left and right from the center and $b$ units above and below. If the transverse axis is vertical, the standard form is $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ , and the hyperbola opens up and down. The vertices are located $a$ units above and below the center. The central rectangle is formed by moving $a$ units above and below the center and $b$ units left and right. In both cases, the center is at $(h, k)$ .

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Updated 2026-05-25

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OpenStax Intermediate Algebra

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