Graphs of Negative Fractional Powers x^(-p/q)

In summary, the conversation discusses how graphs are drawn and how the value of p/q affects the graph. The domain and range are also mentioned, with the example of a fraction with a polynomial in the denominator and a square root. The concept of asymptotes is also brought up.
  • #1
confusedatmath
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0
Can someone explain how these graphs are drawn. How does the value of p/q affect this graph? How does the domain and range change? How are the asymptotes found?

The below is an image about what I'm talking about:

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Here is a question the deals with this type of graph (no idea how to solve it because I'm unfamiliar with these kind of graphs)

View attachment 1832
 

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  • #2
Think about the domain as where the function is defined. For example, if you think of a fraction with a polynomial in the denominator then it it is defined for all numbers except for the zeros of this polynomial. If you have a square root then it has to be defined for only positive integers. If you have both then you have to take care of both conditions. As for the range it is a little bit tricker to work with but for the most time you can guess it.

Take the following example

\(\displaystyle f(x)=\frac{1}{x+a}\)

What is the domain, range and horizontal and vertical asymptotes ?
 

FAQ: Graphs of Negative Fractional Powers x^(-p/q)

What do negative fractional powers in a graph represent?

Negative fractional powers in a graph represent the inverse relationship between the x-value and the y-value. As the x-value decreases, the y-value increases.

Are there any restrictions to using negative fractional powers in a graph?

Yes, there are restrictions to using negative fractional powers in a graph. The denominator of the fractional power cannot be equal to 0, and the numerator must be an odd number to ensure the graph is continuous and passes through the origin.

How do you graph negative fractional powers?

To graph negative fractional powers, plot several points by choosing values for x and then solving for y using the given power. Connect these points with a smooth curve to create the graph.

What is the relationship between the value of the fractional power and the steepness of the graph?

The value of the fractional power determines the steepness of the graph. The larger the value, the steeper the graph will be. For example, a graph with a power of x^(-1/2) will be steeper than a graph with a power of x^(-1/4).

Can negative fractional powers be used in real-world applications?

Yes, negative fractional powers can be used in real-world applications. They can represent various relationships such as radioactive decay, population growth, and depreciation of assets. They are also commonly used in physics and chemistry to represent inverse proportional relationships.

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