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

y = −x2 + 4x − 21 , is your given equation. Now we simplify it to a whole square as:

y= -{(x^2-4x+4)+16)}
= -(x-2)^2 +16

Now you can see that, you can put any value of 'x' in the equation... ranging from +ve infinity to -ve infinity. So the domain will be from -ve infinity to +ve infinity... In other words, Domain is the set of all real nos. R. {x = R}

Now, for the range Y, you keep in mind that there are two parts, of which y is made of.

First is -(x-2)^2...........
To see how this part varies, let us see first how (x-2)^2 varies, (x-2)^2 is always positive or zero for any value of x.
So, -(x-2)^2 is always negative or zero for every value of x. That is to say that value of -(x-2)^2 ranges from { -ve infinity to zero}

But this is not the range... range is what we get after adding 16....

So the real range becomes [ (-ve infinity +16 ) to (0 +16)
y = [ -ve infinity to 16]

Answer 2

If you have studied parabola too, then you have an easier way to answer this question....

You must know that the equation of parabola,

y= -(x-h)^2 +k is the equation of a parabola that is concave--- downwards type of parabola. Its vertex lies at (h,k)...

Then you can draw the given parabola y = - (x-2)^2 +16 as :

You can see that it expands to infinity horizontally on both +ve and -ve sides. So, domain is all the set of real nos.

However, it starts at 16 and goes to -ve infinity on the vertical direction. So, the range is from 16 to -ve infinity.