In Exercises 1–4, find the vertices and locate the foci of each hyperbola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). y²/4−x²/1=1

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1. Identify the standard form of the hyperbola equation given: $\frac{y^2}{4}-\frac{x^2}{1}=1$. This matches the form $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$, indicating a vertical transverse axis.
2. Determine the values of $a$ and $b$ by comparing denominators: $a^2\; =\; 4$ so $a\; =\; 2$, and $b^2\; =\; 1$ so $b\; =\; 1$.
3. Find the vertices of the hyperbola. Since the transverse axis is vertical, the vertices are located at $(0,\; \backslash ,\; \backslash \backslash pm\; a)\; =\; (0,\; \backslash ,\; \backslash \backslash pm\; 2)$.
4. Calculate the focal distance $c$ using the relationship $c^2\; =\; a^2\; +\; b^2$. Substitute the values to get $c^2\; =\; 4\; +\; 1$, so $c\; =\; \backslash \backslash sqrt\{5\}$.
5. Locate the foci at $(0,\; \backslash ,\; \backslash \backslash pm\; c)\; =\; (0,\; \backslash ,\; \backslash \backslash pm\; \backslash \backslash sqrt\{5\})$. Use this information to match the equation to the correct graph among options (a)–(d), looking for a hyperbola opening up and down with vertices at (0, ±2) and foci at (0, ±√5).

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation can be written in standard form as either (x^2/a^2) - (y^2/b^2) = 1 or (y^2/a^2) - (x^2/b^2) = 1. This form helps identify the orientation of the hyperbola (horizontal or vertical) and the values of a^2 and b^2, which are essential for locating vertices and foci.

Vertices of a Hyperbola

Vertices are the points where the hyperbola intersects its transverse axis. For the form (y^2/a^2) - (x^2/b^2) = 1, the vertices lie at (0, ±a). Knowing a allows you to find these points, which are crucial for sketching and understanding the shape of the hyperbola.

Foci of a Hyperbola

The foci are two fixed points located along the transverse axis, used to define the hyperbola. Their distance from the center is c, where c^2 = a^2 + b^2. For vertical hyperbolas, the foci are at (0, ±c). Identifying foci helps in graph matching and understanding the hyperbola's geometry.