Basics of Special Graphs

[chasej]

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.

[b^3]

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 , the maximal domain will be all real numbers, , and as will the range. For something like , our function is defined when is any non-negative number, so that is our domain will be or (we use square brackets when we are including the endpoint and the round brackets when the endpoint is not included).

intercepts are the coordinates when the curve crosses the axis, which will occur when .
intercepts are the coordinates when the curve crosses the axis, which will occur when .

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 . As we make really big, that is as goes off to infinity, then the denominator will become really large, which means the whole fraction itself will become really small, it will approach . That is as , (remember it doesn't actually ever become ). So we will have an asymptote, . We will also have a vertical asymptote, for which the curve is not defined. When , we will get the vertical asymptote, , as we cannot divide by .

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 , then which means . But for all other values of , 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 , then we have which means . So our domain would be (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 , so so .

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.

[chasej]

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 , the maximal domain will be all real numbers, , and as will the range. For something like , our function is defined when is any non-negative number, so that is our domain will be or (we use square brackets when we are including the endpoint and the round brackets when the endpoint is not included).

intercepts are the coordinates when the curve crosses the axis, which will occur when .
intercepts are the coordinates when the curve crosses the axis, which will occur when .

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 . As we make really big, that is as goes off to infinity, then the denominator will become really large, which means the whole fraction itself will become really small, it will approach . That is as , (remember it doesn't actually ever become ). So we will have an asymptote, . We will also have a vertical asymptote, for which the curve is not defined. When , we will get the vertical asymptote, , as we cannot divide by .

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 , then which means . But for all other values of , 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 , then we have which means . So our domain would be (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 , so so .

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.