How do you do asymptoes or what ever you call them? and how do you do the equations for the different graphs for transformation?
Whenever you have a rational function, vertical asymptotes are possible. A rational function is a fraction where one polynomial is divided by another. If the denominator is zero and the numerator is not zero, then you have a vertical asymptote.
Consider a simple function
y = 1/x
when x = 0, no value of y is defined. There is a vertical asymptote at x=0.
There are lots of free online graphing web sites. Find one, and play around with rational functions.
Things like
(x^2-5x + 2)/(x-4)
and so on.
Now, when you have a rational function, there is always the possibility of a horizontal asymptote. If you have a function like
(3x^2 - 9x - 2)/(x^3 + x + 1)
Then as x gets huge, x^3 grows much faster than x^2 or x.
For example,
x=10 x^2=100 x^3=1000
x=100 x^2 = 10000 x^3 = 1000000
So, for large values of x, the above function looks just like
3x^2/x^3 = 3/x
As x gets huge, the quotient gets small, so the horizontal asymptote is y=0.
If you play around with the graphing tools, you'll see both of these kinds of asymptotes appearing.
As for transformations, do some google searches for translation and scaling, and there will be all kinds of good articles and pictures.
To understand asymptotes and equations for graph transformations, let's break it down step by step:
1. Asymptotes:
- Vertical Asymptotes: Vertical asymptotes occur when the function approaches infinity or negative infinity as the input approaches a specific value.
- To find the vertical asymptotes:
i. Determine the values that make the denominator of the rational function zero (if it is a rational function).
ii. These values set the vertical asymptotes of the graph.
- Horizontal Asymptotes: Horizontal asymptotes occur when the function approaches a constant value as the input tends to positive or negative infinity.
- To find the horizontal asymptotes:
i. Evaluate the limits as the input approaches positive and negative infinity, respectively.
ii. If the limits exist and are equal, the graph has a horizontal asymptote at that value.
2. Equation for Graph Transformations:
- When transforming a function's graph, you can use different types of transformations, such as translations (shifts), reflections, stretches, and compressions.
- Generally, transformations are applied to the basic function (e.g., y = x, y = x^2, etc.) and involve changing the variables.
- Here are some key transformations and their effects on the graph:
- Translation:
- Horizontal Translation: Replace x with (x - h) in the function equation to shift the graph horizontally h units to the right.
- Vertical Translation: Replace y with (y - k) in the function equation to shift the graph vertically k units up.
- Reflection:
- Horizontal Reflection: Replace x with -x in the function equation to reflect the graph over the y-axis.
- Vertical Reflection: Replace y with -y in the function equation to reflect the graph over the x-axis.
- Stretch and Compression:
- Horizontal Stretch/Compression: Replace x with (x/a), where 'a' is the stretch or compression factor. A stretch (a > 1) elongates the graph horizontally, while a compression (0 < a < 1) makes it narrower.
- Vertical Stretch/Compression: Replace y with (y/a), where 'a' is the stretch or compression factor. A stretch (a > 1) elongates the graph vertically, while a compression (0 < a < 1) makes it shorter.
Remember, practice is key when dealing with transformations and asymptotes. Working through examples and exercises will help you solidify your understanding and improve your skills.