Question

Your friend, Patricia, is having a hard time understanding the concept behind the domain and range of a parabola. Using complete sentences, explain the meaning of the domain and range of the graph of
y = x2 + 4x − 21 and how to find both. Keep in mind, your goal is to help Patricia understand the "concepts", not just how to use the steps.

Answer 1

If I were thinking about the domain of something, I would be trying to understand what interval(s) on the x-axis would have some corresponding value on the y-axis. It might help a little bit to think of situations where we know there isn't a y-value because x isn't defined. For example, square root of x. We know that we can't put stuff like -1 or -50 in there, so that domain would be from zero to positive infinity.

Then I would look again back at that parabola, and try to imagine if there were ANY possible values for x that would make it undefined? I think we'll see that it is defined for all real numbers, so I would then have to say that it's domain is all real numbers.

For the range, I would be looking at my equation and trying to figure out what was the lowest possible y-value it could have, and also the greatest y-value. At this point, you could either graph it, or use the method of your choice to identify the vertex. Then you would know what the lowest (or highest) value is.