Common mistakes include not labeling axes correctly, inaccurate plotting of key points (intercepts, turning points, asymptotes), and incorrect behavior of the graph near asymptotes or at extreme values of x.
To accurately identify asymptotes, analyze the function for values of x that make the denominator zero (vertical asymptotes) and examine the functions behavior as x approaches positive or negative infinity (horizontal or oblique asymptotes).
Find the first derivative of the function, set it equal to zero, and solve for x. These x-values are potential turning points. Use the second derivative test or analyze the sign of the first derivative around these points to determine if they are maxima or minima.
Labeling key points like intercepts, turning points, and asymptotes provides a clear and complete representation of the functions behavior, making it easier to understand and verify the graphs accuracy.
Input the function into your GC and compare the GC-generated graph with your hand-drawn graph. Pay close attention to intercepts, turning points, asymptotes, and the overall shape to identify and correct any discrepancies.
For functions with modulus signs, consider different cases based on the sign of the expression inside the modulus. Sketch the graph without the modulus first, then reflect the part of the graph below the x-axis about the x-axis.
Graph each piece of the function separately over its specified interval. Pay close attention to the endpoints of each interval, noting whether they are included (closed circle) or excluded (open circle) in the graph.
Calculate additional points between the key points to get a better sense of the graphs shape. This is especially helpful for curves or functions with complex behavior.
By recognizing transformations like translations, reflections, stretches, and compressions, you can start with a basic function (e.g., y = x^2, y = sin x) and apply the transformations step-by-step to obtain the graph of the more complex function.