First, I appreciate your help, and this is very kind of all of you to help us.
Now, this problem we are both sick of: we are to use transformations of f(x)= 1/x or f(x)=1/x^2 to graph the rational function: f(x)= 1/x -4. I tried using the information you gave me, but my graphs look nothing like the examples.
I get confused too, with the various smooth lines and directions, and the dotted lines. Can you please explain?
This web site does not have a way of showing graphs or images, so I am not able to draw the graph for you. If the function really is (1/x) -4 and not 1/(x-4), then it will have a value of y = -4 at very large and very small values of x. As you approach x=0 from the left, the graph turns downwand sharply, going off the botton of the graph. The same thing happens in reverse when you approach x=0 from the right, namely, the curve goes quickly off the top of the graph. The set of two separate curves is called a hyperbola.
For a plotted example of what f(x)= y = 1/x looks like, see the third graph at this web site:
http://www.jimloy.com/geometry/analytiz.htm
Hi,
You need to draw f(x) =1/x first and then transform it to f(x) = (1/x) - 4
To draw f(x) = 1/x, all you need to do is find several points, example:
let x=1, then f(1)=1, you get one point (1,1)
let x = 0.5, f(0.5)=1/0.5=2, another point (0.5,2).
Try to get 5-10 points and then connect then together using a smooth curve.
That is your f(x)=1/x
f(x)=(1/x) -4 is obtained by shifting the entire graph of f(x)=1/x down (negative y direction) by 4 units.(By the way this procedure is called transformation).
Contact me if you need further help, I can email you an attachment.
drwls, thank you for the information. Yes the question is the first. I moved 4 points down the y-axis to the -4(?). Now, I need to graph it. I don't understand which way the curve goes (rt. or left).Thank you.
Of course, I'd be happy to help you understand graphing rational functions using transformations! When transforming a rational function, such as f(x) = 1/x or f(x) = 1/x^2, and applying a vertical translation, there are a few steps you can follow:
Step 1: Identify the parent function
The parent function is the original function before any transformations are applied. In this case, the parent function is f(x) = 1/x.
Step 2: Apply the vertical translation
The given function f(x) = 1/x - 4 has a vertical translation of 4 units downwards. To apply this transformation, subtract 4 from the parent function's equation. The transformed function becomes f(x) = 1/x - 4.
Step 3: Analyze the key features
Now, let's discuss the key features of the transformed function. The most important aspects to consider are:
- Vertical asymptotes: These are the values of x where the function approaches positive or negative infinity. Vertical asymptotes occur when the denominator of the function becomes zero. For the parent function f(x) = 1/x, the vertical asymptote is x = 0 (since division by zero is undefined).
- Horizontal asymptotes: These are the values of y that the function approaches as x approaches positive or negative infinity. For the parent function f(x) = 1/x, there is a horizontal asymptote at y = 0. This means that as x becomes extremely large or small, the function approaches zero.
- x-intercepts and y-intercept: The x-intercepts occur when the numerator of the function becomes zero. However, in this case with f(x) = 1/x - 4, there are no x-intercepts because the numerator is always 1. The y-intercept occurs when x is zero, so the y-intercept of the transformed function f(x) = 1/x - 4 is (0, -4).
Step 4: Plot the graph
Now, you can plot the graph of the transformed function f(x) = 1/x - 4 using the information gathered from the previous steps.
- Start by plotting the vertical asymptote at x = 0. This will be a vertical dotted line that extends from negative to positive infinity.
- Then, plot the horizontal asymptote at y = 0. This will be a horizontal line crossing the y-axis.
- Since there are no x-intercepts, the graph will not intersect the x-axis.
- Finally, plot the y-intercept at (0, -4).
Remember, the function f(x) = 1/x - 4 will have a similar shape to the parent function f(x) = 1/x, but it will be shifted downwards by 4 units.
I hope this explanation helps you understand how to graph a rational function using transformations. If you have any further questions, please feel free to ask!