Final answer:
To analyze a function, determine zeros, holes, vertical and horizontal asymptotes, slant asymptotes, and domain. Examples include examining the intercepts and slope for a straight line.
Step-by-step explanation:
Finding Zeros, Asymptotes, and Domain of a Function
To find the zeros of a function, you need to determine the values of x for which the function equals zero. The holes in a function occur at values of x where a factor cancels out in a rational function, making the function undefined at that point. The vertical asymptote occurs where the function approaches infinity as x approaches a certain value.
The horizontal asymptote reflects the behavior of the function as x goes to infinity or negative infinity, indicating the function's end behavior. A slant asymptote may occur when the degree of the numerator is one higher than the denominator in a rational function, which results in a line that the graph approaches but never touches. Lastly, the domain of a function is the set of all possible x-values for which the function is defined.
An example of this would be a straight line with equation y = mx + b, where m is the slope and b is the y-intercept. In this case, a straight line does not have vertical asymptotes, holes, or a slant asymptote, and its domain is all real numbers.