Domain and Range of Ellipses

The domain is the set of $x$x-values of the relation and the range is the set of $y$y-values of the relation. One example of a relation is an ellipse. An ellipse that is centred at the point $\left(h,k\right)$(h,k) which has a semi-major axis of length $a$a units and semi-minor axis of length $b$b units can be described by the following equation and graph.

The graph of $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2​+(y−k)2b2​=1 for $a>b$a>b.

 

The graph of the ellipse is horizontally bound by two vertical lines, one at $x=h-a$x=h−a and the other at $x=h+a$x=h+a. Because of this, the domain of the ellipse is $\left[h-a,h+a\right]$[h−a,h+a].

The ellipse bound by two vertical lines.

 

Similarly, the ellipse is bound by two vertical lines, one at $y=k-b$y=k−b and the other at $y=k+b$y=k+b and so the range of the ellipse is $\left[k-b,k+b\right]$[k−b,k+b].

The ellipse bound by two horizontal lines.

 

If we swap the length of the semi-major axis $a$a with the length of the semi-minor axis $b$b, we get the following equation and graph.

The graph of $\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$(x−h)2b2​+(y−k)2a2​=1 for $a>b$a>b.

 

And because we've swapped $a$a and $b$b, the domain of the ellipse becomes $\left[h-b,h+b\right]$[h−b,h+b] and the range becomes $\left[k-a,k+a\right]$[k−a,k+a].

In the special case that the ellipse is centred at the origin, that is $\left(h,k\right)=\left(0,0\right)$(h,k)=(0,0), then the domain and range of the ellipse only depend on the values of $a$a and $b$b.

Worked example

Find the domain and range of the ellipse $\frac{\left(x+3\right)^2}{25}+\frac{\left(x-5\right)^2}{16}=1$(x+3)225​+(x−5)216​=1.

Think: The equation of the ellipse is of the form $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$(x−h)2a2​+(y−k)2b2​=1 where $a>b$a>b. The domain of such an ellipse is given by $\left[h-a,h+a\right]$[h−a,h+a] and the range is $\left[k-b,k+b\right]$[k−b,k+b].

Do: In our case we have $h=-3$h=−3 and $a=5$a=5 so the domain of the ellipse is $\left[-3-5,-3+5\right]$[−3−5,−3+5] which simplifies to $\left[-8,2\right]$[−8,2].

We also know that $k=5$k=5 and $b=4$b=4 so the range of the ellipse is $\left[5-4,5+4\right]$[5−4,5+4] which simplifies to $\left[1,9\right]$[1,9].

Reflect: How might the domain and range differ if we swapped the semi-major and semi-minor axes?

Practice questions

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