Final answer:
To find y-intercepts in a graph, set x to zero and solve for y; for example, in the equation y = mx + b, the term b is the y-intercept. Horizontal asymptotes are found by comparing degrees of polynomial terms in rational functions, while vertical asymptotes are found by setting the denominator to zero and solving for x. X-intercepts are determined by setting y to zero and solving for x.
Step-by-step explanation:
To find asymptotes and intercepts in the graph of a mathematical function, you can use different approaches depending on the type of function you are dealing with. To find the y-intercept, set the x variable to zero and solve for y. This gives you the point where the graph crosses the y-axis. The provided equation y = mx + b shows that the y-intercept is represented by b, the constant term. For example, if a linear equation has a y-intercept of 9, we know that the graph crosses the y-axis at the point (0, 9).
Finding horizontal asymptotes for rational functions involves looking at the degrees of the numerator and denominator polynomials and comparing them. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. For vertical asymptotes, set the denominator equal to zero and solve for x, but only if these points do not cancel out with factors in the numerator.
To determine intercepts such as the x-intercept, set the y variable to zero and solve for x. This will give you the point(s) where the graph crosses the x-axis. For our straight-line example, we would solve 0 = 3x + 9 and find that x-intercept occurs when x is -3.