Find the standard equation of an ellipse with the given characteristics and sketch the graph.

Vertices at (0,-5), and (0,5); Foci at (0,-4)and (0,4).

The ellipse is centred at the origin (0,0).

The standard formula for the ellipse is then
x²/a²+y²/b²=1
where
2a=distance between vertices on the major axis
=(5-(-5)=10, so
a=5

Distance between foci
2c=(4-(-4))=8
so
c=4

To find b, we use the relation
a²=b²+c²
or
5²=b²+4²
b=sqrt(25-16)=3

So the equation for the given ellipse is
x²/5²+y²/3²=1

To find the standard equation of an ellipse, we need to determine its major and minor axes, as well as the location of the center. The given characteristics provide the information we need to determine these values.

1. The vertices of the ellipse are located at (0, -5) and (0, 5). These points represent the endpoints of the major axis, which is the vertical line connecting them. Therefore, the length of the major axis (2a) is 10 units.

2. The foci of the ellipse are at (0, -4) and (0, 4). These points represent the focal points of the ellipse, which lie on the major axis. The distance between the center and each focus point is called c. Therefore, c = 4 units.

3. The distance between the center and each vertex is called b. To find its value, we can use the relationship between a, b, and c in an ellipse: a^2 = b^2 + c^2. Substitute the given values: 10/2^2 = b^2 + 4^2. Simplify: 25 = b^2 + 16. Subtract 16 from both sides: b^2 = 9. Taking the square root: b = 3.

Now, we have the necessary information to write the standard equation of the ellipse.

The standard equation of an ellipse with a vertical major axis is given by:
(x - h)^2/b^2 + (y - k)^2/a^2 = 1,

where (h, k) represents the center of the ellipse.

Since the center is at (0, 0), the equation becomes:
x^2/3^2 + y^2/5^2 = 1.

Simplifying further, we obtain the standard equation of the ellipse:
x^2/9 + y^2/25 = 1.

To sketch the graph, plot the center at (0, 0), then mark the vertices at (0, -5) and (0, 5). Also, plot the focal points at (0, -4) and (0, 4). Finally, draw the ellipse by using the length of the major and minor axes as guides, making sure the ellipse passes through both vertices and the focal points.