To review these concepts, go to Graphs of Increasing and Decreasing Functions and Asymptotes Study Guide.
Graphs of Increasing and Decreasing Functions and Asymptotes Practice Questions
Use the asymptotes, concavity, and intervals of increase and decrease to graph the following functions.
 f(x) = x^{2} – 30x + 10
 g(x) = –4x – x^{2}
 h(x) = 2x^{3} – 3x^{2} – 36x + 5
 k(x) = 3x – x^{3}
 f(x) = x^{4} – 8x^{3} + 5
Solutions

f(x) has no asymptotes.
f'(x) = 2x – 30, thus there is a local minimum at (15,–210). Because f"(x) = 2, the graph is always concave up.

g(x) has no asymptotes.
g'(x) = –4 – 2x = –2(2 + x), so there is a local maximum at (–2,4). Because g"(x) = –2, the graph is always concave down.

h(x) has no asymptotes.
h'(x) = 6x^{2} – 6x – 36 = 6(x – 3)(x + 2), so there is a local maximum at (–2,49) and a local minimum at (3,–76). Because h"(x) = 12x – 6 = 12 (x – ) there is a point of inflection at (, – 13).

k(x) has no asymptotes.
k'(x) = 3 – 3x^{2} = 3(1 – x)(1 + x), so there is a local minimum at (–1,–2) and a local maximum at (1,2). k"(x) = –6x, so there is a point of inflection at (0,0).

f(x) has no asymptotes.
f'(x) = 4x^{3} = 24x^{2} = 4x^{2}(x – 6), so there is a local minimum at (6, –427).
f"(x) = 12x^{2} – 48x = 12x(x – 4), so there are points of inflection at (0, – 5) and at (4, –251).

g(x) has a vertical asymptote at x = –2 and a horizontal asymptote at y = 1. The first derivative is g'(x) = , and the second is g''(x) =

h(x) = has vertical asymptotes at x = – 3 and x = 3, and a horizontal asymptote at y = 0. Because h '(x) = , there is a local maximum at . The second derivative is h '' (x) = .

has vertical asymptotes at x = 1 and x = 1, and a horizontal asymptote at y = 0. The first derivative is , and the second derivative is . There is a point of inflection at (0,0).

j(x) has a vertical asymptote at x = 0 but no horizontal asymptotes. Because j'(x) = = , there is a local maximum at (–1,–2) and a local minimum at (1,2). The second derivative is j"( x) =

f(x) has a horizontal asymptote at y = 0 but no vertical asymptotes. Because f(x) = = , there is a local minimum at (–1,–) and a local maximum at (1,) Because f'' (x) = = , there are points of inflection at ( ), (0,0), and ().
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