Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−1)²−(y−2)²=3

Verified step by step guidance

1. Identify the center of the hyperbola by rewriting the equation in the form \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\). Here, the center is at \((h, k)\). Start by dividing both sides of the equation \( (x-1)^2 - (y-2)^2 = 3 \) by 3 to get it into standard form.
2. From the standard form, determine \(a^2\) and \(b^2\). The term with the positive sign corresponds to \(\frac{(x-h)^2}{a^2}\), and the term with the negative sign corresponds to \(\frac{(y-k)^2}{b^2}\). Identify \(a\) and \(b\) by taking square roots of \(a^2\) and \(b^2\) respectively.
3. Locate the vertices of the hyperbola. Since the \(x\)-term is positive, the transverse axis is horizontal. The vertices are located \(a\) units left and right from the center along the \(x\)-axis, at points \((h \pm a, k)\).
4. Find the foci of the hyperbola. Use the relationship \(c^2 = a^2 + b^2\) to find \(c\), where \(c\) is the distance from the center to each focus along the transverse axis. The foci are at \((h \pm c, k)\).
5. Write the equations of the asymptotes. For a hyperbola centered at \((h, k)\) with a horizontal transverse axis, the asymptotes are given by \(y - k = \pm \frac{b}{a} (x - h)\). Substitute the values of \(a\), \(b\), \(h\), and \(k\) to get the asymptote equations.

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation can be written in the form (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or vice versa, where (h, k) is the center. Recognizing and rewriting the given equation into this form helps identify the center, vertices, and orientation of the hyperbola.

Foci and Vertices of a Hyperbola

Vertices are points on the hyperbola closest to the center along the transverse axis, while foci lie further out along the same axis. The distance to the foci is found using c^2 = a^2 + b^2, where a and b come from the standard form. Locating these points is essential for graphing and understanding the hyperbola's shape.

Equations of the Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches, passing through the center. Their slopes depend on a and b from the standard form, with equations y - k = ±(b/a)(x - h) or y - k = ±(a/b)(x - h), depending on orientation. These lines guide the shape and help in sketching the hyperbola accurately.