1 Basics

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Basics
Arithmetic
• The distributive rule
a(b + c) = ab + ac.
• (a + b)2 = a2 + 2ab + b2 .
a
c
+ =
.
b
d
bd
• Multiplying fractions
a
c
ac
&times; = .
b
d
bd
• Dividing fractions
a −1
b
More generally,
a
b
c
d
=
=
b
.
a
.
bc
Exponentiation
• If n is a positive whole number, then an is equal to the product of n copies
of a.
• ab ac = ab+c .
• (ab )c = abc .
• a−b =
• a1/n =
1
.
ab
√
n
a
Points.
• Points in the plane are represented by pairs of numbers (x, y). Here, x is
the horizontal coordinate of the point and y is the vertical coordinate of
the point.
• The distance between two points (x1 , y1 ) and (x2 , y2 ) is
(x2 − x1 )2 + (y2 − y1 )2
by Pythagoras’s Theorem.
Straight lines.
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1/2
• Slope-intersect form:
y = mx + b.
m is the slope, b is the y-intercept.
• Point-slope form:
y − y0 = m(x − x0 ).
m is the slope, (x0 , y0 ) is a point that the line goes through.
• The slope of the line between the points (x1 , y1 ) and (x2 , y2 ) is
y2 − y1
.
x2 − x1
• Horizontal lines: y = b. Vertical lines: x = c.
• Parallel lines have the same slope.
• The slopes of perpendicular lines multiply to −1: If the lines
y = m1 x + b1 and y = m2 x + b2
are orthogonal, then m1 m2 = −1.
• A quadratic equation is an equation of the form
ax2 + bx + c = 0
where x is a variable and a, b and c are constants.
• We can solve either by
– Factoring: if we can write
ax2 + bx + c = (x − d)(x − e)
for some numbers e and d, then the solutions of the equation are
x = d and x = e.
1/2
−b &plusmn; b2 − 4ac
x=
.
2a
The equation has two solutions if b2 − 4ac is positive, one solution if
b2 − 4ac is zero, and no solutions if b2 − 4ac is negative.
Parabolas and circles.
• y = ax2 + bx + c is the equation of a parabola if a 6= 0 (if a = 0 it is the
equation of a line).
2
• The equation of a circle with centre (a, b) and radius r is
(x − a)2 + (y − b)2 = r2 .
Which is the same as saying
distance between (x, y) and (a, b) = r.
2
Functions
The domain of a function is the collection of possible inputs to the function,
the range of a function is the collection of possible outputs of the function.
Some functions you should be familiar with.
• The absolute value function is defined by
x
if x ≥ 0
|x| =
−x if x &lt; 0.
Its graph looks like this.
•
3
Limits
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