This document contains a math worksheet with 7 problems involving graphing and analyzing functions. The problems include sketching graphs of functions with given properties, finding examples of functions with vertical tangents or local maxima, graphing polynomial functions, determining intervals where functions are positive or negative, and using templates to analyze additional functions by finding domains, intercepts, asymptotes, critical points, and concavity.

1. Math 220 Worksheet 12
7/7/2016
Name: Group:
1. Sketch the graph of a function f that is continuous on [1, 5] and has the given properties:
f has an absolute minimum at 1, an absolute maximum at 5, a local maximum at 2, and a
local minimum at 4.
2. Give an example of a function which has both a vertical tangent line and a local maximum
at the origin. You do not need to justify your answer.
3. Find the graph of the following:
f(x) = 2x3
+ 3x2
− 12x − 13
Use the given template.
4. Find a, b, c, and d so that the function f(x) = ax3
+ bx2
+ cx + d has a local minimum at
the origin and a local maximum at (2, 2).
5. Graph the following function:
f(x) =
3x − 1
5x + 2
6. For each of the following, sketch a graph of a function that satisfies the properties:
(a) f (x) > 0 and f (x) < 0 for all x;
(b) f (x) < 0 and f (x) > 0 for all x;
7. Here are some extra functions you can draw:
(a) f(x) =
x + 1
x2 + 1
(b) f(x) =
x2
− 2x + 2
x − 1
(c) f(x) = 5x2/3
− x5/3
(d) f(x) = etan−1(x)
(Note that e
π
2 = 4.8 and e−π
2 = 0.2)

2. Use the following template:
1. Find the domain of the function
2. Find the x-intercept of the function
3. Find the horizontal and vertical asymptotes of the function.
4. Find f (x) and simplify as much as possible and then find the critical points of f(x).
5. Fill out the following table to determine the intervals where the function increases and
decreases.
x
f (x)
Inc/Dec
6. Use the table to determine local and absolute maximum and minimum
7. Find f (x) and simplify as much as possible and then solve f (x) = 0
8. Fill out the following table to determine the intervals where the function increases and
decreases.
x
f (x)
Concave Up/Down
9. Use all the information about asymptotes, increasing/decreasing and concave up/down to
give a rough sketch of the function: