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Curve Sketching Derivatives, Intercepts & Asymptotes

Instructor: Russell Frith
This lesson presents a calculus-based procedure for graphing various types of functions. The methods used will be based on using derivatives, intercepts, and asymptotes for graphing functions.

Curve Sketching - Introduction

In beginning calculus, the emphasis is placed on deriving properties of functions. Function properties may be grouped into the following categories:

(1) domain, range, and symmetry,

(2) limits, continuity, and asymptotes,

(3) derivatives and tangents, and

(4) extreme values, intervals of increase and decrease, concavity, and points of inflection.

In this lesson, the information listed in the four broad categories is leveraged to sketch the graphs of functions which reveal the important features of functions. The following checklist is intended as a guide to sketching a curve y = f(x) by hand. When applying the checklist for sketching a function, not every item is relevant to every function. For instance, a graph may not have an asymptote or possess x-intercepts.

Intercepts

The y-intercept of the function is f(0) and this indicates where the curve intersects the y-axis.

To find the x-intercepts, set y = 0 and solve for x. This may involve having to use synthetic division for high-order polynomials, trigonometric identities for trigonometric functions, or logarithmic rules for exponential functions. This step may be omitted if the equation is computationally difficult to solve.

Asymptotes

From calculus, if the following limit calculation holds:

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then the line y = L is a horizontal asymptote of the curve y = f(x). If

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Horizontal asymptotes of rational functions can be found by looking at the exponents of the highest degree terms in the numerator and denominator.

  • If the term with the highest exponent in the numerator has the same exponent as the highest exponent term in the denominator then there is a horizontal asymptote at y equals the ratio of their coefficients.
  • If the exponent of the leading term in the denominator is less than the exponent of the leading term of the numerator then there no horizontal asymptote.
  • If the exponent of the leading term of the denominator is greater than the exponent of the leading term of the numerator then there is an asymptote at y equals zero.

The line x = a is a vertical asymptote if at least one of the following statements is true:

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For rational functions, the vertical asymptotes may be located by equating the denominator to zero after canceling any common factors.

In addition, rational functions may have slant asymptotes if the degree of the numerator is one more than the degree of the denominator. The slant asymptote is the quotient resulting from long division.

Using Derivatives

The critical numbers of a function are the numbers c where f '(c) = 0 or f '(c) does not exist. If f ' changes from positive to negative at a critical number c, then f(c) is a local maximum. If f ' changes from negative to positive at c, then f(c) is a local minimum. This procedure is known as the First Derivative Test.

Alternatively, if f '(c) = 0 and f ''(c) is not zero then f ''(c) > 0 implies that f(c) is a local minimum and f ''(c) < 0 implies that f(c) is a local maximum. This procedure is known as the Second Derivative Test. Furthermore, the curve is concave upward where f ''(x) > 0 and concave downward where f ''(x) < 0. Inflection points occur where the direction of concavity changes.

Example

Use the methods from this lesson to sketch the curve:

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(1) The domain is:

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(2) The x- and y-intercepts are both 0.

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