You need to be proficient in graphing polynomial, modulus, exponential, logarithmic, and trigonometric functions, as well as their transformations and combinations.
For |f(x)|, reflect the part of the graph below the x-axis about the x-axis. For f(|x|), reflect the part of the graph for x > 0 about the y-axis (and discard the original x < 0 part).
Be comfortable with translations (shifting the graph), reflections (over x or y axis), stretches and compressions (horizontally or vertically).
Vertical asymptotes occur where the denominator is zero (and the numerator is non-zero). Horizontal asymptotes are found by considering the limit of the function as x approaches positive and negative infinity.
First, understand the inner function g(x). Then, use the range of g(x) as the domain for the outer function f(x) to determine the shape of the composite function.
Extremely important! Always label intercepts, turning points (maxima and minima), points of inflection, and any asymptotes with their equations.
Use your G.C. to verify key features such as intercepts, turning points, and asymptotes. Also, consider the functions behaviour as x approaches positive and negative infinity.
Practice consistently! The more graphs you sketch, the faster and more accurate youll become. Focus on understanding the transformations and key features of each function type.
Avoid incorrect scaling, misidentifying asymptotes, not labeling key points, and misunderstanding the effects of transformations.
Graph each piece of the function over its specified domain. Pay close attention to the endpoints of each interval and whether they are included or excluded.