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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: chasej on April 06, 2013, 09:31:04 pm

Title: Basics of Special Graphs
Post by: chasej on April 06, 2013, 09:31:04 pm: I have a holiday homework sheet to work through and it's just completely confusing me. It's asking me to find things like the domain and range, asymptotes, x intercepts and y intercepts of lines.

All this terminology is completely confusing me. Can someone please explain to me the basic terminology for special graphs?

I've tried working through the textbook chapter 5 essentials 1/2 (gallery of graphs), as well as my notes but no where does it explain the terminology I need to know.

I can post a picture of the worksheet once I find my camera.
Title: Re: Basics of Special Graphs
Post by: b^3 on April 06, 2013, 09:45:24 pm: The domain of a function is the values for which the independent variable (normally x) is defined, while the range is the values for which the dependent variable (normally y) is defined. For say $y=x$ , the maximal domain will be all real numbers, $R$ , and as will the range. For something like $y=\sqrt(x)$ , our function is defined when $x$ is any non-negative number, so that is our domain will be $x\geq0$ or $[0,\infty)$ (we use square brackets when we are including the endpoint and the round brackets when the endpoint is not included).

$x$ intercepts are the coordinates when the curve crosses the $x$ axis, which will occur when $y=0$ .
$y$ intercepts are the coordinates when the curve crosses the $y$ axis, which will occur when $x=0$ .

Asymptotes are lines that the function will approach, it will get closer and closer to the line, but never touch it. Lets look at an example with $y=\frac{1}{x}$ . As we make $x$ really big, that is as $x$ goes off to infinity, then the denominator will become really large, which means the whole fraction itself will become really small, it will approach $0$ . That is as $x\rightarrow \infty$ , $y\rightarrow 0$ (remember it doesn't actually ever become $0$ ). So we will have an asymptote, $y=0$ . We will also have a vertical asymptote, for which the curve is not defined. When $x=0$ , we will get the vertical asymptote, $x=0$ , as we cannot divide by $0$ .

There are a few tricks for working out the maximal domains.
If you have a fraction, as we cannot divide by zero, the denominator of the fraction cannot be zero, that is if we have say $y=\frac{1}{x+3}$ , then $x+3\neq0$ which means $x\neq -3$ . But for all other values of $x$ , the curve is defined, so we have the domain of R\{-3} (that is all real numbers excluding -3).

Now we cannot square root a negative number (in methods), so if we have a square root, then whatever is underneath the square root has to be equal to or greater than zero. So say we have $f(x)=\sqrt{x-1}$ , then we have $x-1\geq0$ which means $x\geq 1$ . So our domain would be $[1,\infty)$ (remember we can't include infinity).

Now if we have a log, then whatever is in the log has to be positive, so that is whatever is inside the log has to be greater than zero. For example $y=\log_{e}(x+2)$ , so $x+2>0$ so $x>-2$ .

One last thing to note, a trick that a few people fall for time after time. If you have a square root on the bottom of a square root, then you will have to have a positive or zero number to satisfy the square root, but a non zero number to satisfy the fraction, which means in the end you'd have to find when what is under the square root is greater than zero.

Anyways, I hope that helps.
Title: Re: Basics of Special Graphs
Post by: chasej on April 06, 2013, 11:45:23 pm: Quote from: b^3 on April 06, 2013, 09:45:24 pm
The domain of a function is the values for which the independent variable (normally x) is defined, while the range is the values for which the dependent variable (normally y) is defined. For say $y=x$ , the maximal domain will be all real numbers, $R$ , and as will the range. For something like $y=\sqrt(x)$ , our function is defined when $x$ is any non-negative number, so that is our domain will be $x\geq0$ or $[0,\infty)$ (we use square brackets when we are including the endpoint and the round brackets when the endpoint is not included).

$x$ intercepts are the coordinates when the curve crosses the $x$ axis, which will occur when $y=0$ .
$y$ intercepts are the coordinates when the curve crosses the $y$ axis, which will occur when $x=0$ .

Asymptotes are lines that the function will approach, it will get closer and closer to the line, but never touch it. Lets look at an example with $y=\frac{1}{x}$ . As we make $x$ really big, that is as $x$ goes off to infinity, then the denominator will become really large, which means the whole fraction itself will become really small, it will approach $0$ . That is as $x\rightarrow \infty$ , $y\rightarrow 0$ (remember it doesn't actually ever become $0$ ). So we will have an asymptote, $y=0$ . We will also have a vertical asymptote, for which the curve is not defined. When $x=0$ , we will get the vertical asymptote, $x=0$ , as we cannot divide by $0$ .

There are a few tricks for working out the maximal domains.
If you have a fraction, as we cannot divide by zero, the denominator of the fraction cannot be zero, that is if we have say $y=\frac{1}{x+3}$ , then $x+3\neq0$ which means $x\neq -3$ . But for all other values of $x$ , the curve is defined, so we have the domain of R\{-3} (that is all real numbers excluding -3).

Now we cannot square root a negative number (in methods), so if we have a square root, then whatever is underneath the square root has to be equal to or greater than zero. So say we have $f(x)=\sqrt{x-1}$ , then we have $x-1\geq0$ which means $x\geq 1$ . So our domain would be $[1,\infty)$ (remember we can't include infinity).

Now if we have a log, then whatever is in the log has to be positive, so that is whatever is inside the log has to be greater than zero. For example $y=\log_{e}(x+2)$ , so $x+2>0$ so $x>-2$ .

One last thing to note, a trick that a few people fall for time after time. If you have a square root on the bottom of a square root, then you will have to have a positive or zero number to satisfy the square root, but a non zero number to satisfy the fraction, which means in the end you'd have to find when what is under the square root is greater than zero.

Anyways, I hope that helps.

Thanks so much. That helped a lot. I understand it for the most part now. Should be able to get the work done. :)