Rational Functions



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Rational Functions

A rational number gets its name because it is the ratio of two integers. Rational functions are named for similar reasons. A rational function is the ratio of two polynomial functions.

If f(x) is a rational function, then f(x) = p(x), where p(x) and q(x) are polynomial

q(x)

functions. The polynomials can be of any degree, but in this investigation they will have degree one or zero.

THE RECIPROCAL FUNCTION

One simple example of a rational function is the reciprocal function 1/x. This will be the first function in your investigation.

1. In a new sketch, define the coordinate axes. Adjust the scale so that the x- axis fits your screen between about -10 and 10.

2. Choose Graph/Plot New Function. Define the new function f(x): f(x) = 1/x

Q1: Does f(x) satisfy the definition of a rational function? Explain.

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Q2: For what values of x is f(x) zero? For what values is it undefined?

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The graph y = f(x) is a hyperbola. Like all hyperbolas, it has two asymptotes, lines that the curve approaches at the extremes.

Q3: What are the equations of the asymptotes of this curve? Asymptotes are lines, so your answer should be the equations of two lines.

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TRANSFORMATIONS OF THE RECIPROCAL FUNCTION

3. On the same grid, plot another rational function: g(x) = 4x - 10

x - 3

Q4: What are the equations of the asymptotes of the graph of g(x)?

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Here’s how it works. The function g(x) is a fraction. Divide the denominator into the numerator, and leave a remainder.

g(x) = 4x - 10 = 4(x – 3) + 2 = 2 +4

x – 3 x – 3 x - 3

When you view the function this way, you can see g(x) in terms of the reciprocal function. Since f(x) = 1/x, it follows that g(x) = 2f(x - 3) + 4. Expressed as a transformation, this stretches the parent function, f(x), vertically by a ratio of 2, and translates it right 3 units and up 4 units.

[pic]

Q5: How does this transformation help explain the positions of the asymptotes of g(x)? You can verify this by plotting a third function as a transformation of f(x). Its graph should fall right on top of the graph of g(x).

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4. Plot h(x) with the following definition: h(x) = 2f(x - 3) + 4

Q6: Below are three new definitions to try for g(x). In each case, express the function in terms of f(x), as in the example above. Find the asymptotes of each graph. Check your work by plotting g(x) and h(x) in the sketch.

a) g(x) = -3x - 12

x + 5

b) g(x) = 4x – 17

4x – 16

c) g(x) = 15x + 103

5x + 35

Q7: Given the two asymptotes, it is possible to find any number of different rational functions that fit them. Define two different functions having the asymptotes x = 6 and y = -4. Start by showing the functions as transformations of f(x), and then express them as ratios of polynomials.

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