write the equation of the hyperbola in graphing form from the given information:
Vertices at (2,0) and (-2,0); foci at (3,0) and (-3,0)
center at origin by symmetry
x^2/a^2 - y^2/b^2 = 1
transverse axis = 2 a = 2-(-2) = 4
so a = 2
now
x^2/4 -y^2/b^2 = 1 just find b from focus
center to focus = sqrt(a^2+b^2)
so 3 = sqrt(4+b^2)
9 = 4 + b^2
b = sqrt 5
so in the end
x^2/4 - y^2/5 = 1
To write the equation of a hyperbola in graphing form, we need to use the information about the vertices and foci.
First, let's analyze the given information:
- Vertices: The vertices of the hyperbola are at (2,0) and (-2,0). The vertices represent the left-most and right-most points on the transverse axis.
- Foci: The foci of the hyperbola are at (3,0) and (-3,0). The foci represent the points within the hyperbola that define its shape.
To find the equation of the hyperbola in graphing form, we can start by determining the center of the hyperbola. The center is the midpoint between the vertices, and it is located at (h, k).
- h = (2 + (-2))/2 = 0
- k = (0 + 0)/2 = 0
Thus, the center of the hyperbola is (0,0).
Next, we can find the equation of the hyperbola using the formula:
((x - h)^2/a^2) - ((y - k)^2/b^2) = 1,
where (h, k) represents the center, 'a' represents the distance from the center to the vertices, and 'b' represents the distance from the center to the foci.
Using the vertex and focus information, we can determine the values of 'a' and 'b':
- The distance between the center (0,0) and the vertices (2,0) or (-2,0) is 'a'. Therefore, a = 2.
- The distance between the center (0,0) and the foci (3,0) or (-3,0) is 'c'. The value of 'c' can be found using the formula c^2 = a^2 + b^2. In this case, c = 3, so b^2 = c^2 - a^2 = 3^2 - 2^2 = 9 - 4 = 5. Taking the square root of both sides, we find b = √5.
Now that we have the values of 'a' and 'b', we can substitute them into the equation and get the final result:
((x - 0)^2/2^2) - ((y - 0)^2/√5^2) = 1
Simplifying the equation, we have:
x^2/4 - y^2/5 = 1
Therefore, the equation of the hyperbola in graphing form is x^2/4 - y^2/5 = 1.