I am trying to graph the function of an ellipse that is: $$1=\frac{x^2}{49}+\frac{(y+1)^2}{9}$$. I want to make the horizontal ellipse's range $y \leq 0.838$. So, when I also have to write the domain for the function, would I write x is all real numbers (even though some x values would cause it to go above the range set) or would I say $x$cannot be between $-5.56 $ and $5.56$ (even though this would also take out the points on the bottom part of the ellipse- which I don't want)?
All in all, what would I write for the domain if my range for the function stated is $y \leq 0.838$, and I still want to have my bottom part of the ellipse? Thanks!
Your ellipse is centered at $(0,1)$. You want $y\leq 0.838$, which is in the bottom half of the ellipse, so you will want to take the negative square root when you solve for $y$. However, simply writing $$y= -1-\sqrt{9-\frac{9}{49}x^2}$$ would give you all points on the ellipse with $y \leq 1$. To restrict to only $y \leq 0.838$, you need to solve for where your ellipse intersects the line $y=0.838$. This give the equation: $$1=\frac{x^2}{49}+\frac{1.838^2}{9}.$$ You will find two values of $x$ that solve this equation, and those will be the endpoints of your desired domain.