# Square root graph

Gotcha, I keep forgetting about that little plussy-minusy thing.

But if, as you are saying, they are both functions, how does one represent non-functional graphs? (is my wording right? I'm trying to pick stuff up here.)

Gib Z
Homework Helper
But wait up a second there cristo, if we had the function $f(x)=\sqrt{x}$ then yes its only in the first quadrant. But since this function is $f(x)=\sqrt{x} -4[/tex], its as if the graph of the function [itex]\sqrt{x}$ was moved down by 4, so it is not only in the first quadrant, but in the fourth as well. ie domain: x equal or greater than 0, range y more or equal to -4.

Gotcha, I keep forgetting about that little plussy-minusy thing.

But if, as you are saying, they are both functions, how does one represent non-functional graphs? (is my wording right? I'm trying to pick stuff up here.)

Okay, okay. I think people are getting confused here. Here's my take on it:

A graph is simply a plot of all of the x-y pairs that satisfy a given equation. Any 2-variable equation you can think up can be graphed.

A function is a subset of graphs. It's a type of graph. A function is a graph for which every x input gives a single y output. The 'functionality,' if you will, of a graph can be tested by, as someone said, the vertical line test.

You can graph any equation, but that does not make it a function. You can graph a circle, and in order to do so you'd just draw a circle on the graph. A circle, though, is not a function.

function:
$$y=\sqrt{x}$$
Because it always gives the positive root,

not function:
$$y=\pm\sqrt{x}$$

Because it gives two answers for each x.

Despite the curse of a useless GCSE curriculum, I think I've finally got it. Also, LaTeX is really freaking cool.

cristo
Staff Emeritus
But wait up a second there cristo, if we had the function $f(x)=\sqrt{x}$ then yes its only in the first quadrant. But since this function is $f(x)=\sqrt{x} -4[/tex], its as if the graph of the function [itex]\sqrt{x}$ was moved down by 4, so it is not only in the first quadrant, but in the fourth as well. ie domain: x equal or greater than 0, range y more or equal to -4.

Of course. In the above I may have mixed up talking about y=sqrtx and y=sqrtx-4. However, the important point still holds-- the graph of y=sqrtx-4 is not a "tipped over" parabola.

HallsofIvy
Homework Helper
Once again, no one said you can[t graph a relation that is not a function. A circle is the graph of a relation that is not a function. It fails the "vertical line test": any vertical line that passes through the graph does so only once.
The graph of (y+4)2= x is a "parabola lying on its side" but y is NOT a function of x. If x= 4, then y can be either -2 or -6: the vertical line x= 4 passes through the graph at both (4,-2) and (4,-6).

However, the function $y= \sqrt{x}$ is defined as "the positive number whose square is x" and is a function. Then $y= \sqrt{x}- 4$ is the part of a parabola that starts at (0,-4) and rises upward and two the right. The functions domain is $[0,\infty)$ and its range is $[-4,\infty)$.

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I recently got marked wrong on a calc test for this... my teacher gave us y= √ x (not ±) and x=3-2ysquared. We were supposed to find the area between the two curves and I got the wrong answer since y= √ x (not ±) was only the positive half of the parabola. She argued that it was the full parabola since we should've turned it into x= ysquared. Is that right? can you just suare both sides and magically get both halfs of the parabola? This doesn't make sense to me.

The square root of x on a graph starts from zero and is infinite. X > zero for the square root of x where x is all real numbers. Square root of x - 4 is x > -4 where x is all real numbers and is infinite.

Remember, every real number has a positive and a negative square root. Don't feel too bad about that mistake, most graphing application programmers seem to forget as well.

So, basically, it should look a bit like y=x^2 tipped on its side.

What do you mean bit like y=x^2.
you mean half parabola in 4th quadrant.

Think of it as the branch of y = x^2 that is in the 1st quadrant "flipped" over y = x.

HallsofIvy
No, they have one y output for every x input! If you were specifically asked to graph the function $y= \sqrt{x}$ and graphed the entire parabola, yes you would be marked wrong. If you were asked to graph x= y2 (so y is a "relation", not a function of x, though now x is a function of y) then you should graph the entire parabola.