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A Summary of Curve Sketching
In our next lesson, we will be sketching the graphs of various functions by hand. To get ready
for this, you need to review some things that you already know about functions.
1)
How do you find the x- and y-intercepts of a function?
2)
Find the x- and y- intercepts of y = x3 – 4x.
3)
How do you know if a function is symmetrical about: (Hint! Section P.1)
a)
the y-axis?
b)
4)
5)
6)
the origin?
Can a graph be symmetrical about the x-axis? ________
Can a function be symmetrical about the x-axis? ________
Find the axis of symmetry for the following graphs:
a)
y = 2x4 – x2 + 2
b)
y = x3 – 4x
c)
x – y2 = 1
7)
What is the general rule about odd and even powers for a function that is symmetrical
about the y- axis?
8)
What is the general rule about odd and even powers for a function that is symmetrical
about the origin?
9)
What is the domain and range of the following functions:
1
a)
y= 2
b)
y = x -1
c)
x -4
10)
When is a function not continuous?
y = tanx
11)
Give an example of a function that is not continuous at x = 3.
12)
Give two cases of when a continuous function is not differentiable.
13)
What are relative extrema? What are absolute (global) extrema?
14)
In terms of a function, what does concavity mean?
15)
What is a point of inflection?
16)
When does a function have a vertical asymptote?
17)
Give an example of a function that has two vertical asymptotes.
18)
When does a function have a horizontal asymptote?
19)
Give an example of a function that has a horizontal asymptote.
20)
Can a function have more than one horizontal asymptote? If yes, when?
21)
When does a function have a slant (oblique) asymptote?
22)
Give an example of a function that has a slant asymptote.
23)
24)
25)
Can a function have a horizontal and a vertical asymptote? ___________
Can a function have a horizontal and a slant asymptote? ___________
Can a function have a slant and a vertical asymptote? ___________
Answer Key:
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By letting y = 0 and solving for x, and by letting x = 0 and solving for y.
Intercepts: (0,0), (2,0), (-2,0)
Symmetry:
y-axis  If plugging in –x for x yields an equivalent equation.
x-axis  If plugging in –y for y yields an equivalent equation.
origin If plugging in –x & -y yields an equivalent equation.
YES! A graph can be symmetrical about the x-axis.
NO! A function cannot be symmetrical about the x-axis.
Symmetry: a) y-axis symmetry b) origin symmetry c) x-axis symmetry (however,
not a function!)
The graph of a polynomial has symmetry with respect to the y-axis if each term of
the function has an even exponent or is a constant. THIS DOES NOT WORK
FOR FUNCTIONS THAT ARE NOT POLYNOMIALS.
The graph of a polynomial has symmetry with respect to the origin if each term of
the function has an odd exponent. (No constants!) THIS DOES NOT WORK
FOR FUNCTIONS THAT ARE NOT POLYNOMIALS.
Domain and range: a) D: all x = -2, 2 R: all y = 0
b) D: all x > 1 R: all y > 0
c) D: all x =
When the function has a hole, jump or asymptote. A function is cont. when f(c)
is defined, the limit of f(x) at c exists, and is = to f(c).
F(x) = 1/(x – 3)
Sharp turns and vertical tangent lines  f(x) = |x| and f(x) = x1/3 are both not
differentiable at x = 0. See page 97.
Hills and valleys on an open interval.
A graph is concave up if f’ is increasing and concave down if f’ is decreasing. A
graph is concave up if f” is + and concave down if f” is -.
A point of inflection is a point at which the concavity changes.
A function has a vertical asymptote at all x-values that make the denominator
zero.
f(x) = 1/(x2 – 9)  v.a.’s at x = +3.
y = L is a horizontal asymptote if the limit as x approaches infinity of f(x) = L.
f(x) = (2x – 5) / (3x + 2) has a h.a. at x = 2/3.
Functions that are not rational MAY approach different horizontal asymptotes to
the right and to the left.
When the degree of the numerator is exactly one more than the degree of the
denominator.
F(x) = (x^3+4x+1)/(x^2-2x)
Yes
No
Yes