Shifted Hyperbolas

Just like we did with parabolas and ellipses, we now want to consider hyperbolas that are centered some point \((h,k)\) rather than at the origin. We will see the same relationships where expression of \((x-h)\) respresent a horizontal shift and expressions of \((y-k)\) represent a vertical shift. Let's look at the standard formula, the key characteristics, and then a few examples of shifted ellipses.

The standard conic equations of a hyperbola with center at the point \((h,k)\) have the following form.

Horizontal Transverse Axis

Vertical Transverse Axis

Graph of a hyperbola centered at the point (h,k) with horziontal transverse axis. The vertices are located at the intersection of the transverse axis with the hyperbola. The foci are equal distance from the center along the line of the transverse axis.

\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]

Assuming that both \(a,b > 0\), the vertices are located at \((h \pm a,k)\) and the foci are at \((h \pm c,k)\). The transverse axis has a length of \(2a\). The distance between the foci is \(2c\). The rectangular box passing through the vertices with corner points on the asymptotes will have a width of \(2a\) and a height of \(2b\). The slopes of the asymptotes are \(m = \pm \frac{b}{a}\).

Graph of a hyperbola centered at the point (h,k) with vertical transverse axis. The vertices are located at the intersection of the transverse axis with the hyperbola. The foci are equal distance from the center along the line of the transverse axis.

\[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]

Assuming that both \(a,b > 0\), the vertices are located at \((h,k \pm a)\) and the foci are at \((h,k \pm c)\). The transverse axis has a length of \(2a\). The distance between the foci is \(2c\). The rectangular box passing through the vertices with corner points on the asymptotes will have a width of \(2b\) and a height of \(2a\). The slopes of the asymptotes are \(m = \pm \frac{a}{b}\).

When dealing with shifted hyperbolas, the overall shape and distances remain the same. That means that the values of \(a\) and \(b\) sill represent the dimensions of the rectangular box, and \(c\) represents the distance from the center to each focus point. The variable contained in the positive fraction still determines the orientation of the hyperbola, we still have the equation \(c^2 = a^2 + b^2\), and the asymptote slopes are still \(m = \pm \frac{b}{a}\) for a horziontal hyperbola or \(m = \pm \frac{a}{b}\) for a vertical hyperbola.

Let's look at some examples of hyperbolas that have been shifted away from the origin.