Key features include intercepts (x and y), asymptotes (horizontal, vertical, and oblique), stationary points (maxima, minima, and points of inflection), and behavior as x approaches positive or negative infinity.
Vertical asymptotes occur where the denominator equals zero. Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is one greater than the denominator, an oblique asymptote exists, found through polynomial division.
Stationary points, where the derivative equals zero, indicate local maxima, minima, or points of inflection. Analyzing the second derivative helps determine the nature of these points, providing crucial information about the graphs shape.
Understanding transformations like translations, reflections, stretches, and compressions allows you to modify a basic graph (e.g., y = x^2, y = sin x) to obtain the graph of a more complex function efficiently.
The first derivative indicates where the function is increasing or decreasing, while the second derivative indicates concavity (whether the graph is concave up or concave down). This information is essential for accurate sketching.
For functions involving modulus, consider the different cases where the expression inside the modulus is positive or negative. Sketch the graph for each case separately and then combine them, reflecting the negative part about the x-axis.