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

Are you Patricia?

Answer 2

no

Answer 3

The domain is going to be the possible x values. The range is going to be the possible y values.

It's known that for polynomials (horizontal parabolas are polynomials) that the domain is all numbers.

However for parabolas, it at some point bends around... The place where it bends will either be the highest point or the lowest point. You need to find that high/low point and basically then you know the range.

Answer 4

The high point is the vertex, so it helps to put the parabola in vertex form.

Answer 5

ok

Answer 6

Vertex form is: \[
y = a(x-h)^2+k
\]Where \(
(h,k)\) is the vertex. What we are really interested in here is \(k\), because it is the value of the high/low point. However \(a\) is also important because if it is negative, then \(k\) is the highest point, and it is positive then \(k\) is the lowest point.

Answer 7

You get vertex form by completing the square.