Problem 71 - Algebra and Trigonometry

Question

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\)

Step-by-Step Solution

Answer

The similarity is both hyperbolas have the same shape and orientation. The difference is their centers: the first hyperbola is centered at the origin (0,0), while the second is centered at (3,-3).

Step 1: Identify the first hyperbola characteristics

The standard form for the equation of a hyperbola centered at (h,k) is \(\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1\). The given equation \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) suggests the center of this hyperbola is at (0,0), and it opens towards the x axis since the x term comes first with a being the square root of the denominator under the x term, which here is 3, and b being the square root of the denominator under the y term, which is 1. Hence, the vertices would be at (±a, 0) = (±3, 0).

Step 2: Identify the second hyperbola characteristics

The equation \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\) is already in standard form so the center of this hyperbola is at (3,-3). It also opens towards the x axis since the x term comes first. The values of a and b remain the same as in the first hyperbola (a=3, b=1). Hence, the vertices would be at (h±a, k) = (3±3, -3).

Step 3: Identify the similarity and difference between the graphs

The similarity between the graphs is that both hyperbolas have the same shape and orientation since they share same values of a and b, and the x term comes first in both equations. The difference, however, lies in their centers. The center of the first hyperbola is at the origin (0,0) while the center of the second hyperbola is at (3,-3), indicating it has been translated 3 units to the right and 3 units down from the origin.