Math 134 Biocalculus Week 1 Summary
Algebra
Make sure that you know how to perform (correctly!) algebraic manipulations such as
simplification, factorization, rationalization and completion of the square. In particular,
recall that
√
x + y 6=
√
√
x + y,
1 1
1
6= + ,
x+y
x y
(x + y)2 6= x2 + y 2 .
Never ever make these mistakes! :-)
Functions
A function f is a rule that assigns to each element x in a set D exactly one element, called
f (x), in a set E. The set D is called the domain of f , and the range of f is the set of all
possible values of f (x) as x varies through the domain.
The graph of a function consists of all points in the xy-plane such that y = f (x) with x in
the domain of f .
A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects
the curve more than once (this is known as the vertical line test; it ensures that the function
f (x) assigns only one element to the points x in the domain of f ).
Functions can be defined using different formulae for different parts of their domains; those
are known as piecewise defined functions. A very useful example is the absolute value function
f (x) = |x| which is defined by
(
x
for x ≥ 0,
|x| =
.
−x for x < 0.
A function is increasing on some interval I if f (x1 ) < f (x2 ) for all x1 , x2 ∈ I such that
x1 < x2 . It is called decreasing on I if f (x1 ) < f (x2 ) for all x1 , x2 ∈ I such that x1 > x2 .
A linear function f (x) is a function of the form
f (x) = mx + b,
for some constants m and b. Its graph is a line with slope m and y-intercept b.
A polynomial function f (x) is a function of the form
f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0 ,
for some constants an , an−1 , . . . , a1 , a0 and some positive integer n. The degree of f (x) is n.
When n = 2, f (x) is called quadratic, while it is called cubic if n = 3.
A power function is a function of the form
f (x) = xa
for some constant a. If a is a positive integer, then f (x) is a particular example of a polynomial function. If a = 1/n with n a positive integer, then f (x) is a root
√ function (for example,
for a = 1/2 it is the familiar square root function f (x) = x1/2 = x). For a = −1, it is the
reciprocal function f (x) = 1/x.
A rational function is a function of the form
f (x) =
P (x)
Q(x)
where P (x) and Q(x) are polynomial functions.
An algebraic function is a function that can be constructed using algebraic operations (such
as addition, subtraction, multiplication, division, and taking roots) starting with a polynomial function. All rational functions are clearly algebraic.
Completing the Square
A function f (x) = ax2 + bx + c can be rewritten as
f (x) = a(x + p)2 + q
2
b
b
and q = c − 4a
. The function f (x) is a parabola with its vertex at (−p, q). It
where p = 2a
opens upward of a > 0 and opens downward if a < 0.
Transformations of Functions
• y = f (x) + k shifts the graph of f (x) up by k units if k > 0 and down by |k| units if
k < 0.
• y = f (x − k) shifts the graph of f (x) by k units to the right if k > 0 and down by |k|
units to the left if k < 0.
• y = cf (x), c > 1 stretches the graph of f (x) vertically by a factor of c.
• y = cf (x), 0 < c < 1 compresses the graph of f (x) vertically by a factor of 1/c.
• y = f (cx), c > 1 compresses the graph of f (x) horizontally by a factor of c.
• y = f (cx), 0 < c < 1 stretches the graph of f (x) horizontally by a factor of 1/c.
Geometry
Lines
Given two points P1 (x1 , y1 ) and P2 (x2 , y2 ) in the xy-plane,
∆x = x2 − x1
∆y = y2 − y1
is called the change in x, or the “run”
is called the change in y, or the “rise”.
If x1 6= x2 , the line passing through the points P1 and P2 is non-vertical, and its slope is
m=
rise
∆y
y2 − y1
=
=
x2 − x1
∆x
run
If P1 6= P2 but ∆y = 0 then m = 0 and the line is horizontal. If P1 6= P2 but ∆x = 0 then
the line is vertical, and its slope is undefined. A vertical line is not the graph of a function,
since it does not pass the vertical line test.
Typically we give the equation of a line in the form
y = mx + b,
where m is the slope and b the y-intercept. Two lines y = m1 x + b1 and y = m2 x + b2 are
parallel if m1 = m2 . They are perpendicular if m1 = −1/m2 .
To find the equation of the line with a given slope m = a passing through a point P (x1 , y1 ),
we can use the point-slope formula:
y − y1 = m(x − x1 ).
We expand and gather terms to get an equation of the form y = mx + b.
To find the equation of the line passing through two points P1 (x1 , y1 ) and P2 (x2 , y2 ), we first
determine that the slope of the line is
y2 − y1
m=
,
x2 − x1
and then substitute into the point-slope formula to get
y2 − y1
y − y1 =
(x − x1 ).
x2 − x1
We expand and gather terms to get an equation of the form y = mx + b.
Circles, parabolas, ellipses and hyperbolas
The equation of a circle with centre (h, k) and radius r is
(x − h)2 + (y − k)2 = r2 .
A parabola is the graph of a function of the form
y = ax2 + bx = c.
The equation of an ellipse with centre (h, k) has the form
(x − h)2 (y − k)2
+
= 1,
a2
b2
for some constants a and b.
The equation of a hyperbola with “centre” (h, k) has the form
(x − h)2 (y − k)2
−
= 1,
a2
b2
for some constants a and b.
Trigonometry
Angles can be given either in degrees or radians. A complete revolution corresponds to 360o
or 2π radians.
The two basic trigonometric functions are sin θ and cos θ. If you pick a point (x, y) on the
unit circle, with angle θ with respect to the positive side of the x-axis (positive angle meaning
counterclockwise rotation), then the trigonometric functions are defined as
x = cos θ,
y = sin θ.
These two trigonometric functions can also be obtained from the length of the sides of a
right triangle, as:
opposite
adjacent
sin θ =
,
cos θ =
.
hypotenuse
hypotenuse
The four other trigonometric functions can be obtained from sin θ and cos θ as:
tan θ =
sin θ
,
cos θ
cot θ =
cos θ
,
sin θ
sec θ =
1
,
cos θ
csc θ =
1
.
sin θ
Trigonometric functions satisfy many identities, which you can easily find online or in Appendix D of the textbook. The most important one is
sin2 θ + cos2 θ = 1,
and the two other similar identities that can be obtained from this one through the relations
above:
tan2 θ + 1 = sec2 θ,
1 + cot2 θ = csc2 θ.
Functional Composition
Given two functions f and g, the composition of f with g is a function defined by
f ◦ g(x) = f (g(x)).
The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the domain
of f .