Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form
for horizontal ellipses and
for vertical ellipses.
Given the standard form of an equation for an ellipse centered at
sketch the graph.
Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.
If the equation is in the form
where
then
the major axis is the
x -axis
the coordinates of the vertices are
the coordinates of the co-vertices are
the coordinates of the foci are
If the equation is in the form
where
then
the major axis is the
y -axis
the coordinates of the vertices are
the coordinates of the co-vertices are
the coordinates of the foci are
Solve for
using the equation
Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Graphing an ellipse centered at the origin
Graph the ellipse given by the equation,
Identify and label the center, vertices, co-vertices, and foci.
First, we determine the position of the major axis. Because
the major axis is on the
y -axis. Therefore, the equation is in the form
where
and
It follows that:
the center of the ellipse is
the coordinates of the vertices are
the coordinates of the co-vertices are
the coordinates of the foci are
where
Solving for
we have:
Therefore, the coordinates of the foci are
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See
[link] .
Graphing an ellipse centered at the origin from an equation not in standard form
Graph the ellipse given by the equation
Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.
First, use algebra to rewrite the equation in standard form.
Next, we determine the position of the major axis. Because
the major axis is on the
x -axis. Therefore, the equation is in the form
where
and
It follows that:
the center of the ellipse is
the coordinates of the vertices are
the coordinates of the co-vertices are
the coordinates of the foci are
where
Solving for
we have:
Therefore the coordinates of the foci are
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.
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A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon