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). x²/4−y²/1=1

Verified step by step guidance

1. Identify the standard form of the hyperbola equation given: $\frac{x^2}{4}-\frac{y^2}{1}=1$. This matches the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, indicating a hyperbola centered at the origin with a horizontal 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 horizontal, the vertices are located at $(�pm\; a,\; 0)$, which means the vertices are at $(�pm\; 2,\; 0)$.
4. Calculate the focal distance $c$ using the relationship $c^2\; =\; a^2\; +\; b^2$. Substitute the known values to get $c^2\; =\; 4\; +\; 1$, so $c\; =\; �sqrt\{5\}$.
5. Locate the foci of the hyperbola at $(�pm\; c,\; 0)$, which means the foci are at $(�pm\; �sqrt\{5\},\; 0)$. Use this information to match the equation to the correct graph among options (a)–(d).

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation in standard form is written as (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 and b, which are essential for locating vertices and asymptotes.

Vertices of a Hyperbola

Vertices are the points where the hyperbola intersects its transverse axis. For the equation (x^2/a^2) - (y^2/b^2) = 1, the vertices are at (±a, 0). These points mark the closest approach of the hyperbola branches and are critical for graphing and understanding its shape.

Foci of a Hyperbola

Foci are two fixed points located along the transverse axis, inside each branch of the hyperbola. Their distance from the center is given by c, where c^2 = a^2 + b^2. Knowing the foci helps in graphing and understanding the reflective properties of hyperbolas.