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 be careful.
And then lastly 𝑘, if it’s positive, it will move the graph up. And if it’s negative, it will move the graph down.
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.