Chapter 1: Preview and Review
Section 1.1: Preliminaries
Definition: The absolute value of a real number a, denoted by |a|, is defined as
a if a ≥ 0
|a| =
−a if a < 0.
Theorem: (Properties of the Absolute Value)
Suppose that a > 0. Then
1. |x| = a if and only if x = ±a.
2. |x| < a if and only if −a < x < a.
3. |x| > a if and only if x > a or x < −a.
Example: Solve |2x + 4| = 16.
Example: Solve |2x + 3| = |5 − 3x|.
Example: Solve |4x − 6| ≤ 10.
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Example: Solve |12 − 4x| > 8.
Equations of Lines
Definition: The slope of a nonvertical line that passes through (x1 , y1 ) and (x2 , y2 ) is
m=
y2 − y1
.
x2 − x1
The slope of a vertical line is undefined.
Definition: The point-slope form of the equation of a line passing through (x1 , y1 ) with
slope m is
y − y1 = m(x − x1 ).
The slope-intercept form of the equation of a line with slope m and y-intercept (0, b) is
y = mx + b.
The standard form of the equation of a line is
Ax + By + C = 0,
where A, B, and C are integers.
Example: Find an equation of the line passing through (2, 5) and (4, 9).
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Definition: Two nonvertical lines are parallel if and only if they have the same slope. Two
lines with slopes m1 and m2 are perpendicular if and only if m1 m2 = −1. That is, their
slopes are negative reciprocals:
1
m1 = − .
m2
Example: Find an equation of the line passing through (−1, 8) and parallel to the line
3x + y − 7 = 0.
Example: Find an equation of the line passing through (1, 4) and perpendicular to the line
2y − 5x + 7 = 0.
Note: The equation of the horizontal line with y-intercept a is y = a. Similarly, the equation
of the vertical line with x-intercept b is x = b.
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Circles
Definition: A circle is the set of all points (x, y) a fixed distance r, called the radius, from
a given point (x0 , y0 ) called the center.
Theorem: (Equation of a Circle)
The equation of a circle with center (x0 , y0 ) and radius r is
(x − x0 )2 + (y − y0 )2 = r2 .
Example: Find an equation of the circle with center (2, 3) passing through (5, 7).
Example: Show that the equation
x2 + y 2 − 4x + 10y + 13 = 0
represents a circle and find the center and radius.
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Trigonometry
Angles can be measured in degrees or radians. The angle given by a complete revolution
contains 360◦ , or 2π radians. Thus, 360◦ = 2π radians, which gives the conversion formulas
◦
180
= 1 radian,
π
π
radians = 1◦ .
180
Example: Find the radian measure of 36◦ .
Example: Find the degree measure of −
7π
radians.
2
Definition: Consider the right triangle shown below.
Hypotenuse
Opposite
𝜃
Adjacent
The six trigonometric functions are defined as
sin θ =
Opposite
Hypotenuse
csc θ =
Hypotenuse
Opposite
cos θ =
Adjacent
Hypotenuse
sec θ =
Hypotenuse
Adjacent
Opposite
Adjacent
cot θ =
Adjacent
.
Opposite
tan θ =
5
Note: The exact trigonometric ratios for certain angles can be read from the triangles below.
Example: Find the exact trigonometric ratios for θ =
5π
.
6
2
Example: If cos θ =
and θ is a Quadrant IV angle, find the other five trigonometric
5
functions of θ.
6
The graphs of the trigonometric functions are given below.
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Theorem: (Trigonometric Identities)
Recall the following trigonometric identities
sin2 θ + cos2 θ = 1
sin(2θ) = 2 sin θ cos θ
1 − cos(2θ)
sin2 θ =
2
tan2 θ + 1 = sec2 θ
cos(2θ) = 2 cos2 θ − 1
1 + cos(2θ)
cos2 θ =
.
2
Example: Find all values of x in the interval [0, 2π) that satisfy the following equations.
(a) 2 cos x + sin(2x) = 0
(b) sec2 x =
√
3 tan x + 1
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Exponentials
Definition: An exponential is an expression of the form ax , where a is a real number called
the base and x is a real number called the exponent.
Theorem: (Properties of Exponentials)
Suppose that a, b, x, y are real numbers. Then
1. ax ay = ax+y
2. (ab)x = ax bx
ax
= ax−y
ay
a x ax
= x
4.
b
b
3.
5. a−x =
1
ax
6. (ax )y = axy
Example: Evaluate the following expressions.
(a) (24 2−3/2 )2
(b)
65/2 62/3
61/3
3
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Logarithms
Definition: A logarithm is an expression of the form loga x, where a > 0 is a real number
called the base of the logarithm. Note that loga x is defined only for x > 0.
Note: There is a correspondence between logarithms and exponentials.
y = loga x ⇐⇒ ay = x.
Example: Solve the following expressions for x.
(a) log5 x = 3
(b) log1/2 x = −4
(c) log4 64 = x
(d) log10 10, 000 = x
10
Theorem: (Properties of Logarithms)
Suppose that a, x, y > 0 and n is a real number. Then
1. loga (xy) = loga x + logb y
x
2. loga
= loga x − loga y
y
3. loga (xn ) = n loga x
Example: Simplify the following expressions.
(a) log3 5 + 3 log3 x
(b) log5 (x2 + 3) − log5 2 − log5 x
(c) loga 3 + 2 loga x −
1
loga y
2
Note: The most important logarithms are the common logarithm, which has base 10 and
the natural logarithm which has the irrational number e as its base. There is special
notation for these logarithms.
log10 x = log x
loge x = ln x
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