Identify the direction of opening,the coordinates of the center, the vertices, and the foci. Find the equations of the asymptotes and sketch the graph. x^2/16-y^2/9=1
x^2/16-y^2/9=1
a^2 = 16 ---> a = 4
b^2 = 9 ----> b = 3
centre: (0,0)
vertices: (-4,0), (4,0)
asympotes: y = ±(3/4)x
You should know the basic properties of a hyperbola in its standard form like this one.
To identify the direction of opening of the given equation and find the coordinates of the center, vertices, and foci, we can rewrite the equation in the standard form of a hyperbola.
The standard form for a hyperbola with a horizontal transverse axis is:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
Where (h, k) represents the coordinates of the center, a represents the distance from the center to the vertices in the x-direction, and b represents the distance from the center to the vertices in the y-direction.
Comparing this standard form to the given equation x^2/16 - y^2/9 = 1, we can see that a^2 = 16 and b^2 = 9.
Taking the square root of a^2 and b^2, we find that a = 4 and b = 3.
Therefore, the center of the hyperbola is at (h, k) = (0, 0).
Since a > b, we can conclude that the hyperbola opens horizontally.
To find the vertices, we can take the distance of a units from the center in the x-direction. So, the vertices are (4, 0) and (-4, 0).
To find the foci, we can use the formula c^2 = a^2 + b^2, where c represents the distance from the center to the foci. In this case, a = 4 and b = 3.
c^2 = 4^2 + 3^2 = 16 + 9 = 25
Taking the square root of 25, we find that c = 5.
Therefore, the foci are located at (5, 0) and (-5, 0).
Next, let's find the equations of the asymptotes for the hyperbola.
The equations of the asymptotes for a hyperbola with a horizontal transverse axis are given by:
y = ± (b/a) * x + k
In this case, b/a = 3/4.
Thus, the equations of the asymptotes are:
y = (3/4) * x + 0 and y = -(3/4) * x + 0.
Finally, let's sketch the graph of the hyperbola based on the information we have found.
The center is at (0, 0) and the vertices are at (4, 0) and (-4, 0).
The foci are located at (5, 0) and (-5, 0).
Using this information, you can plot the points and connect them to create the hyperbola. The asymptotes will be straight lines passing through the center and extending infinitely.
The sketch should resemble two symmetrical open curves with asymptotes passing through the center.
Keep in mind that it is always helpful to use a graphing tool or software to obtain an accurate visual representation of the hyperbola.
I hope this helps you understand how to identify the direction of opening, find the coordinates of the center, vertices, and foci, determine the equations of the asymptotes, and sketch the graph of a hyperbola.