What is the equation of the function represented below?
Let’s call this function 𝑔 of 𝑥. And based on its shape, it seems to be a square root graph.
So what does the square root graph look like? Well, let’s pass some points. So if we were to plug
in zero, the square root of zero is zero, so we have the point zero, zero. The square root of one is
one, so we have the point one, one. The square root of two is a decimal. So let’s say we didn’t
have a calculator, what’s the next number that we actually know the square root of? Not three,
but we do know the square root of four, it’s two. So we have the point four, two. And then our
next point that we would know, we don’t know the square root of five, six, seven, or eight, but
the square root of nine would be three. Now that would be off of our graph, but that’s okay.
So here is the graph of 𝑦 equals the square root of 𝑥. So our graph — our function in red, what
we’re trying to figure out — has a very similar shape. It’s just upside down to the left, and
maybe some other things; we’re not quite sure. So that’s called a transformation of a function;
it’s been transformed; it’s been moved. So if you take the square root function 𝑓 of 𝑥, which is
the same thing as 𝑦, which is the same thing of 𝑔 of 𝑥, it’s equal to 𝑎 square root 𝑥 minus ℎ plus 𝑘.
If 𝑎 would be negative, it would flip over the 𝑥-axis. If the absolute value of 𝑎, meaning we’re not
looking at the sign anymore, if it would be bigger than one, it would be a vertical stretch, kind of
makes it taller looking. If the absolute value of 𝑎 is less than one, it would be a vertical compression,
kind of making it shorter.
Now ℎ, if ℎ is positive, it will shift the graph right. If ℎ is negative, it shifts the graph left. Now one
thing to pay close attention to, it’s 𝑥 minus ℎ. So if you would plug in a positive number, like three,
you would plug in three and it would look like 𝑥 minus three, where the three isn’t negative, we’ve
just plugged in a positive. So if it’s a positive, if ℎ is positive, it will look like it’s a negative when you
plug it in. And if you plug in a negative, say negative three, it would make it turn positive. So just
And then lastly 𝑘, if it’s positive, it will move the graph up. And if it’s negative, it will move the
So let’s first begin with looking at 𝑘. So let’s look at the original point zero, zero and decide if the
original point on our new graph has moved up or down at all. It hasn’t. It’s only moved left. So our
𝑘 would be zero. So if we’re gonna call this 𝑔 of 𝑥, we need to plug in zero for 𝑘. And as we just
noticed, it was shifted left three. So we need to plug in negative three. And then next, it’s been
flipped upside down, so 𝑎 needs to be negative. So now we need to decide, has it been stretched?
So let’s begin by taking our original and flipping it upside down. So from our pink original point,
we had to go right one and down one and then right four and down two.
So let’s see if we do that on the red graph. We did go right one and down one, and then we did
go right four and down two. And as we originally said, if we would go over nine, the square root
of nine is three. And that would be here. So it hasn’t been stretched at all. All of our numbers
have been the same. It hasn’t been multiplied by anything to make those points move. So we
can let 𝑎 be one. So let’s simplify what we have. Negative one, we can just leave it as a negative
sign. And then minus negative three would really be a plus three. And then we don’t really have
to write the plus zero.
So our final answer would be 𝑔 of 𝑥 equals negative square root 𝑥 plus three.