Exercise 48 Page 649

We will find the desired information, and use it to draw the graph of the hyperbola.

Vertices, Foci, and Asymptotes

Let's start by recalling the equation of hyperbolas centered at (h,k). Horizontal Hyperbola (x-h)^2/a^2-(y-k)^2/b^2=1 Vertical Hyperbola (y-k)^2/a^2-(x-h)^2/b^2=1 Now we will rewrite the given equation to match one of these formats.

From the above formula, we can see that the equation represents a vertical hyperbola. Next, let's review the main characteristics of this type of hyperbola.

Using this information, we can identify that the center of the hyperbola is (4,- 1). The vertices we find using the formula (h ,k± a). (4, - 1+4)=(4,3) [0.6em] (4,- 1-4)=(4,- 5) Let's substitute h=4, k=- 1, a=4, and b=3 into the formula for the asymptotes and obtain their equations.

The asymptotes are y=± 4/3(x-4)-1. Now, let's calculate c. To do so, we will substitute a=4 and b=3 into c^2=a^2+b^2.

We can now use the formula (h ,k± c) to find the foci of the hyperbola. (4, - 1+ 5)=(4,4) [0.6em] (4,- 1- 5)=(4,- 6)

Graph

To graph the function, let's summarize all of the information that we have found.

Finally, we can graph our hyperbola with center (4,-1).