Square Root Graph: Understanding x- & y- Intercepts

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Discussion Overview

The discussion revolves around the graph of the function y = √x - 4, focusing on its x- and y-intercepts, the behavior of the graph in different quadrants, and the implications of defining functions. Participants explore the nature of square roots, the definition of functions, and the graphical representation of various mathematical concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about whether the graph extends into the negative x-axis, noting that square roots of negative numbers yield complex results.
  • Another participant points out that the graph of y = √x - 4 should resemble a sideways parabola, suggesting it is limited to the first quadrant due to the definition of a function.
  • Some participants clarify that the choice of the positive square root is a convention to ensure the function is well-defined.
  • There is a discussion about the implications of defining functions and the concept of one-to-one mappings, with some participants questioning the necessity of such definitions.
  • Several participants debate whether a sideways parabola can be considered a function, referencing the vertical line test and the nature of outputs for given inputs.
  • One participant mentions that graphs can be plotted even if they are not functions, using the example of a unit circle.
  • There is confusion regarding the representation of functions versus non-functions, with questions about how to graph different types of equations.
  • Some participants emphasize that when mathematicians write y = √x, they refer specifically to the positive square root, which influences how the graph is drawn.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement. While there is some consensus on the convention of using the positive square root for functions, there is ongoing debate about the implications of this choice, the nature of functions, and how to graph various mathematical entities. The discussion remains unresolved on several points, particularly regarding the graphical representation of non-functions.

Contextual Notes

Limitations include varying interpretations of the definition of a function, the implications of choosing positive versus negative square roots, and the conditions under which certain graphs can be plotted. The discussion does not resolve these complexities.

Who May Find This Useful

This discussion may be useful for students and educators in mathematics, particularly those interested in graphing functions, understanding the properties of square roots, and exploring the definitions and implications of mathematical functions.

  • #31
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 [itex]y= \sqrt{x}[/itex] is defined as "the positive number whose square is x" and is a function. Then [itex]y= \sqrt{x}- 4[/itex] is the part of a parabola that starts at (0,-4) and rises upward and two the right. The functions domain is [itex][0,\infty)[/itex] and its range is [itex][-4,\infty)[/itex].
 
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  • #32
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.
 
  • #33
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.
 
  • #34
Yersinia Pestis said:
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.
 
  • #35
Think of it as the branch of y = x^2 that is in the 1st quadrant "flipped" over y = x.
 
  • #36
lLovePhysics said:
What's so important about functions anyways? All they have are two y outputs for every x input right? Would it make a big difference (and get marked wrong) if I drew a horizontal parabola rather than just the top half??
No, they have one y output for every x input! If you were specifically asked to graph the function [itex]y= \sqrt{x}[/itex] 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.

There is nothing terribly important about "functions" (except that they are somewhat simpler than "relations") in mathematics but they tend to be very important in applications of mathematics to science because of the requirement of "repeatability": if you do an experiment twice, with everything set up exactly the same way, you should get exactly the same result- one input, one output. "This causes that" gives functions.

Think of it this way: If you were to go to a store and find different products that had the same price, you would not be surprised, right? The "product" is not a function of the price. On the other hand, if you found exactly the same product, same size, brand, and everything, in the same store for two different prices, you would know that something was wrong: the price is a function of the product.
 

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