Curve sketching

Curve sketchingEmily Cooper and Ashli Haas

Emily Cooper and Ashli Haas 2011IntroductionThe purpose of curve sketching is to graph the curve of a function by using differentiation.

How to sketch the curve Step 1 Find critical points (x-intercepts)Step 2 Fine horizontal, vertical or oblique asymptotesStep 3 Take the derivative of the original equationStep 4 Sign lineStep 5 Determine where the graph is increasing and decreasingStep 6 - GraphFinding x-interceptsFinding x-interceptsAsymptotesAn asymptote to a curve is a straight line to which the curve approaches, but never touches, as the distance from the origin increases.There are three types of possible asymptotes: Vertical, Horizontal and Oblique.

Vertical AsymptotesVertical asymptotesVertical asymptotes are x-values and are not able to be crossed.

Horizontal asymptotesTo calculate the horizontal asymptotes, take the ratio of the coefficients of the variable raised to the highest power in the given problem.

Horizontal asymptotesIf the powers of the variables are equal, the h.a. is the ratio of the coefficients.If the power of the variable in the denominator is larger, then y=0If the power of the variable in the numerator is larger, there is an oblique asymptote

Horizontal AsymptotesHorizontal AsymptotesHorizontal asymptotes are y-values and can be crossed no more than once.

Oblique AsymptotesOblique asymptotesOblique asymptotes are neither horizontal nor vertical but diagonal instead.

Oblique asymptotesOblique asymptotes cannot be crossed.

DerivativeTake the derivative of the original equation.To take a derivative, you multiply each coefficient by the corresponding variables exponent then lower the exponent by 1.For example:f(x) = 20x3 3x5f(x) = 60x2 15x4Simplify the derivativef(x) = 15x2 (4-x2)f(x) = (3x)(5x)(2-x)(2+x)

DerivativeTo take the derivative of a fraction, you must take the denominator and multiply it by the derivative of the numerator. Then subtract from that the numerator multiplied by the derivative of the denominator. Then divide by the denominator squared.

DerivativeDerivative & Sign lineUsing the derivative, you must find critical points for your sign line.Factor your derivative to find the portions on the side of the sign line. Then solve for x in each of those to find points on your sign line.Derivative & Sign lineSign lineSign lines are meant to tell where the graph of the equation is increasing or decreasing.Use the portions of the derivative to do the sign line.To find critical points on the sign line, solve for x for each of the portions. Those are the points listed on the top of your line.

Sign LineWhen the graph of a portion is positive, it is shown as a solid line. When the graph of portion becomes negative, it is shown as a dashed lineDetermine whether the graph is negative or positive by using your critical points

Sign lineThe points that are decreasing will have a negative slope between the given intervals and will follow the asymptotes downwards.

The points that are increasing will have a positive slope between the given intervals and will follow the asymptote upwards.Graph the curveGraph the curveExample 1Example 2Try me!Solution #1Solution #2Solution #3