Question

find an equation for the hyperbola whose graph is below.

Explanation:

Step1: Identify center and orientation

Center at (4,0), horizontal transverse axis.

Step2: Find a and c

Vertices at (0,0) and (8,0): $a = 4$. Foci at (-2,0) and (10,0): $c = 6$.

Step3: Calculate b²

Use $c² = a² + b²$: $b² = 6² - 4² = 20$.

Step4: Write hyperbola equation

Standard form: $\frac{(x-h)²}{a²} - \frac{(y-k)²}{b²} = 1$. Substitute values: $\frac{(x-4)²}{16} - \frac{y²}{20} = 1$.

Answer:

$\frac{(x-4)^2}{16} - \frac{y^2}{20} = 1$