In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x²/100−y²/64=1

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1. Identify the standard form of the hyperbola equation: $\frac{x^2}{100}-\frac{y^2}{64}=\; 1$. Since the x-term is positive and comes first, this is a hyperbola that opens left and right.
2. Determine the values of $a^2$ and $b^2$ from the denominators: $a^2\; =\; 100$ and $b^2\; =\; 64$. Then find $a\; =\; \backslash ,\; ext\{sqrt\}(100)\; =\; 10$ and $b\; =\; \backslash ,\; ext\{sqrt\}(64)\; =\; 8$.
3. Locate the vertices on the x-axis at $(��\; extpm\; a,\; 0)\; =\; (\; extpm\; 10,\; 0)$ because the hyperbola opens horizontally.
4. Find the foci using the formula $c^2\; =\; a^2\; +\; b^2$. Calculate $c\; =\; \backslash ,\; ext\{sqrt\}(100\; +\; 64)\; =\; ext\{sqrt\}(164)$. The foci are at $(\; extpm\; c,\; 0)$.
5. Write the equations of the asymptotes using the slopes . The asymptotes pass through the center (0,0), so their equations are .

Key Concepts

Standard Form of a Hyperbola

A hyperbola in standard form is expressed 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), the vertices located at ±a along the transverse axis, and the relationship between a and b which determines the shape.

Foci of a Hyperbola

The foci are two fixed points located along the transverse axis of the hyperbola, found using c^2 = a^2 + b^2. They are essential in defining the hyperbola as the set of points where the difference of distances to the foci is constant. Knowing the foci helps in graphing and understanding the hyperbola's geometry.

Equations of the Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at the origin with horizontal transverse axis, the asymptotes have equations y = ±(b/a)x. These lines guide the shape and direction of the hyperbola branches and are crucial for accurate graphing.