Graphs of rational functions: vertical asymptotes  High School Math  Khan Academy  Summary and Q&A
TL;DR
Determine the possible graph of a polynomial function based on the given information about its denominator and potential vertical asymptotes or removable discontinuities.
Key Insights
 π The given function f(x) is in the form of a rational function, with a polynomial numerator and a polynomial denominator.
 βΊοΈ The denominator, x^2  x  6, can be factored into (x  3)(x + 2), revealing the potential values of x that make the denominator equal to zero.
 βΊοΈ The function f(x) will have a vertical asymptote or a removable discontinuity at x = 3 or x = 2, depending on whether g(x) can be factored as (x  3) times another polynomial.
Transcript
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Questions & Answers
Q: What is the given function?
The given function is f(x) = g(x) / (x^2  x  6), where g(x) is a polynomial.
Q: How can we determine the potential points where the denominator equals zero?
By factoring the denominator, x^2  x  6, we find (x  3)(x + 2). So, the denominator equals zero when x = 3 or x = 2.
Q: What does it mean if the denominator equals zero at a particular point?
If the denominator equals zero at x = a, then f(x) will have either a vertical asymptote or a removable discontinuity at x = a, depending on whether g(x) can be factored into (x  a) times another polynomial.
Q: How can we determine the graph of f(x) from the given choices?
We compare the given graph choices to the information we have. If a graph choice has a vertical asymptote or removable discontinuity at x = 3 or x = 2, it could be a possible graph of f(x). Choice C is the only one that aligns with these conditions.
Summary & Key Takeaways

The function is defined as f(x) = g(x) / (x^2  x  6), where g(x) is a polynomial.

The denominator of f(x), x^2  x  6, can be factored into (x  3)(x + 2).

The function f(x) will have a vertical asymptote or a removable discontinuity at x = 3 or x = 2, depending on whether g(x) can be factored into (x  3) times another polynomial.

Four graph choices are given, and only choice C aligns with the information provided.