Math Help

Basics of Curve Sketching – Slope, Concavity, Inflection Points - Steps

In a Nut Shell: It is frequently helpful to visualize the function at hand by constructing

a graph of the function. Locating maxima, minima, where the function increases, where

the function decreases, and other information helps in understanding the function.

Steps for basic curve sketching include:

First determine the domain of f(x). i.e. a ≤ x ≤ b

Determine the intercepts ˗ where the function intersects the x and y axes

The y-intercepts are at x = 0. The x-intercepts are at y = 0.

Locate any vertical, horizontal, or slant asymptotes:

Vertical asymptotes occur for x = a is lim f(x) = ±∞ or if lim f(x) = ±∞

x→ a^- x→ a⁺

Horizontal asymptotes occur at f(x) = L provided lim f(x) = L or if lim f(x) = L

x→ +∞ x→ -∞

y = mx + b is a slant asymptote if lim [y ˗ mx ˗ b] = 0

x → ± ∞

Set the first derivative of f(x) to zero to identify the critical points (where

slope of the function is zero. On either side of each critical point the slope

may be positive or be negative.

If df/dx > 0 the function is increasing. If df/dx < 0 the function is decreasing.

Calculate the second derivative of f(x) to determine concavity and the

inflection points.

If d²f/dx² = 0 then the slope is either changing from positive to negative

or from negative to positive.

If d²f/dx² > 0 then the curve is concave up; If d²f/dx² < 0 then the curve is

concave down