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A short class with Sarah and her students. The other ways are under More.
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Today we're discussing rational functions, which are functions represented as the ratio of two polynomials. Can anyone explain what that means?
Does it mean it’s like a fraction with polynomials on top and bottom?
Exactly! For example, is a rational function. The numerator is a polynomial and the denominator is another polynomial . Now, what does it mean if equals zero?
That means the function would be undefined.
Correct! So we must exclude those values in our domain. Remember the mnemonic 'Keep Denominator Non-Zero' or KDNZ to recall this: if , we exclude it!
Let's talk about finding the domain of a rational function. Who remembers how we do that?
We set the denominator equal to zero and solve for , right?
Exactly right! For instance, in , if we set , what do we find?
We find . So, the domain is all real numbers except 5.
Good job! Always remember: the domain includes all real numbers except where the denominator is zero. Let's emphasize this by revisiting KDNZ.
Now, let's explore asymptotes in rational functions. Who knows what a vertical asymptote is?
Isn't it where the function tends to positive or negative infinity?
Correct! A vertical asymptote occurs at the values of that make the denominator zero after simplification. What about horizontal asymptotes?
They show how the function behaves as goes to infinity, based on the degrees of the polynomials.
Exactly! Remember: if the degree of the numerator is less than that of the denominator, there’s a horizontal asymptote at . A good trick to remember this is 'Grows Less, Go Low!'
Let's find x and y-intercepts now. Who remembers how to find the x-intercept?
By setting , right?
Exactly! And for the y-intercept, we evaluate . Let’s illustrate with an example: . What’s the x-intercept?
Setting gives .
Awesome! And the y-intercept by finding . Quick reminder, we can use 'X in the Zero for X-intercept!'
Defined as a ratio of two polynomials.
All values of that allow the function to remain defined, excluding points making the denominator zero.
Occurs when the denominator equals zero, showing where the function is undefined.
Illustrates the function's behavior at infinity based on the polynomial's degrees.
Points where the graph intersects the axes, found via zeroing the function or evaluating at zero.
When polynomials do dance; their ratio is a chance, avoiding zero, that's the glance!
Imagine two polynomial friends, one up (numerator) and the other down (denominator). They can only play together when the bottom friend doesn't go to zero, or else the game is off!
For intercepts: 'Zero at the top, and check at the stop' meaning find the x-intercept by zeroing the numerator and y-intercepts by substituting zero into the function.
D.A.V.E. - Domain Excludes Values that make the denominator zero!
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What is a rational function?
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What is the domain of a rational function?
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How to find vertical asymptotes?
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What is a horizontal asymptote?
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A function that can be expressed as the ratio of two polynomial functions.
The set of all possible input values for the function, excluding those that make the denominator zero.
A vertical line at which the function tends to infinity, occurring where the denominator is zero after simplification.
A horizontal line that represents the behavior of the function as approaches infinity, determined by comparing the degrees of the numerator and denominator.
Points where the function crosses the axes; found by setting the function to zero (x-intercept) or evaluating at zero (y-intercept).
A rational function is defined as a ratio of two polynomial functions, with its denominator not equal to zero.
Rational functions are significant in both theoretical and applied mathematics, involving ratios of polynomials. Key aspects include determining the domain, identifying asymptotes, intercepts, and methods for graphing and solving rational equations.
A rational function is defined as a function of the form where both and are polynomial functions and . This foundational concept sets the stage for various applications, including motion analysis, economic modeling, and scientific research.
Mastering rational functions is vital for further studies in calculus and understanding more complex algebraic functions. This section empowers students with the tools necessary to simplify expressions, find domains, identify asymptotes, and graph rational functions efficiently.