Math 150 Study Guide
Chapter P – Precalculus Prerequisites
1. Set Notation: formal set builder notation; inequality notation; interval notation;
and/intersection; or/union; empty set; compound inequality; graphing sets on real
line; use set notation to express sets of functions (domain, range, continuity
interval, increase/decrease interval, concave up/down interval)
2. Rule of Four: using various ways to represent mathematical concepts. Different
representations are more effective in different circumstances. Aim to select the
most efficient, accurate, correct method for a given problem. Conceptual
understanding increases when more than one representation can be used to
explain a concept.
a) Verbal – words, definitions
b) Numerical – numbers, tables of values, tabular, ordered pairs/quadruples
c) Algebraic – variables, expressions, equations, inequalities, functions, relations
d) Graphical – graphs in 1-D (numberline), 2-D (Cartesian plane), function mode,
parametric mode
3. TI: graphing functions, setting reasonable window dimensions, identifying xintercepts, function values for given x – values, maxima, minima, intersection
points, inflection points, equations of tangent lines at a given x - coordinate
4. Analytical Geometry: midpoint and weighted averages, distance formula
5. Absolute Value: x − c = d
a) Verbal:
• The absolute value of a quantity refers to the magnitude (amount) of a
quantity without any reference to direction.
• All scalar quantities are the absolute value of their vector counterparts.
For example, speed is the absolute value of velocity, which is a vector.
There is no name for the absolute value of (the vector) acceleration; I
refer to the absolute value of acceleration as “the magnitude of
acceleration”.
•
x − c = d means “the distance between x and c is d”, or “x is a distance of
d spaces away from the center value of c”
b) Numerical:
• the output of the absolute value function is always positive or zero
•
−5 = 5 = 5 , because both -5 and 5 are a distance of 5 spaces away from
( ) () () ()
zero: −5 − 0 = 5 − 0 = 5
•
if the number/quantity inside the absolute value function is already
positive, then the absolute value does not change the original value,
maintaining the positive quality of the number
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if the number/quantity inside the absolute value function is negative, then
the absolute value changes the sign of the original value so that the result
is positive
⎧⎪− x − c if x − c < 0
c) Algebraic: x − c = ⎨
if x − c ≥ 0
⎩⎪x − c
•
(
)
d) Graphical: the graph of y = x − c is a partial vertical reflection of the graph
of y = x − c
•
•
( ) is a partial vertical reflection of f (x ) above the x- axis;
similarly, − f ( x ) is a partial vertical reflection of f ( x ) below the x- axis
in general, f x
6. Distance: Understand similarities and differences among absolute value in 1-D,
distance formula, Pythagorean Theorem and equation of a circle to each other.
7. Linear Functions: see Chapter 2
8. Secant and Tangent Lines: Rates of change and difference quotient; average rates
of change/slope of a secant line, instantaneous rates of change/slope of a tangent
line; limit definition of the derivative
9. Absolute Value Functions: two piecewise defined linear functions. See #5.
10. Piecewise Functions: sketch each function “piece” on its restricted domain
11. Solving Equations: Solve equations, inequalities: from definition/meaning/verbally,
algebraically, graphically and numerically; sign charts; Understand limitations of
each technique, and be able to choose the most effective technique for a given
equation
or
inequality.
Rearrange
solutions
and
solve
related
equations/inequalities.
Verify solutions and look for extraneous solutions,
particularly in the context of modeling and solving word problems, in which case a
real world restriction may cause the domain of the problem to be a subset of the
real numbers and extraneous solutions to appear.
12. Sign Chart: a numerical technique that requires factored expressions to produce
graphs or solve inequalities
13. Solving Inequalities: solve graphically; see Chapter 2
14. Modeling and Optimization - Modeling, word problems, real world domains,
extraneous solutions, optimization (max/min word problems), scatter plots and
regressions, interpolation and extrapolation. See Chapter 2
15. Variables vs. Parameters:
a) Variables are quantities that can be varied throughout a given equation, model,
function, etc., while parameters are fixed throughout that same equation.
b) A function with unspecified coefficients like
y = ax 2 + bx + c
involves
parameters a, b and c and variables x and y. For each curve, a, b and c do not
change although they change from one curve to the next curve. However, the
variables x and y vary within a specific curve from one point to the “next”
point.
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Chapter 1 – Properties of Functions
16. Limit versus Function Values
• Evaluating limits numerically, graphically and algebraically
• interpreting the graphical significance of a limit
• fractions involving zero: zero/undefined/indeterminate
17. Relations, Functions and One-to-One Functions
a) Any graph or curve or equation relating x and y is a relation. Ex: a 1-2 relation
means that 1 x value maps to 2 different y values, such as in a circle, or a
horizontally oriented parabola.
b) A function is a special kind of relation for which each x value maps to only one
y value. Ex: a 2-1 function means that 2 different x values map to the same 1 y
value, such as in a vertically oriented parabola.
c) A one-to-one function is a special kind of function for which each x value maps
to only one y value. Ex: a 1-1 relation means that 1 x value (and no other) maps
to a specific 1 y value (and no other), such as in any exponential function (one
exponent correlates uniquely to one power).
• Functions “pass” the vertical line “test”: a vertical line never intersects a
function more than once.
d) Graphical consequences of a function being one-to-one:
• Never turn around.
• Are monotone (always) increasing OR monotone decreasing.
• Pass the horizontal line test.
• Can’t (for example) have more than 1 x – intercept; in general, can’t have
more than one point at a given height.
18. Domain and Domain Restrictions
a) The domain of a relation is the set of x values that make the y values defined.
b) Restrictions on the domain are caused by the following conditions.
• Denominators must be nonzero.
• Radicands of even roots must be nonnegative.
• Powers must be positive.
• Domains can be determined both graphically and algebraically.
c) Domains can be restricted to create one-to-one functions. Conventions used
to generate a restricted domains include:
• Keeping as much of the original domain as possible so that the resulting
function is one-to-one.
• Keeping the origin, if possible.
• When choosing between the right and left sides of an axis of symmetry,
keep the right side.
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19. Range
a) The range is the set of y values produced by a relation.
b) Although the range can sometimes be determined algebraically, it is more
typical to determine the range algebraically.
20. Library of Basic Functions
a) Constant Functions
b) Almost Constant Function - basic hole/removable discontinuity, y =
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
u)
v)
w)
x)
Basic Jump Function
Greatest Integer Function
Identity Function
Absolute Value Function
Squaring Function
Square Root Function
Cubing Function
Cube Root Function
Reciprocal Function
Reciprocal (“Inverse”) Square Function
Exponential (Base e) Function
Natural Logarithm (Base e) Functions
Logistic Function
Sine Function
Cosine Function
Tangent Function
Cosecant Function
Secant Function
Cotangent Function
Inverse Sine Function
Inverse Cosine Function
Inverse Tangent Function
x
x
= 1, x ≠ 0
21. Piecewise Functions
a) Piecewise functions are functions that are defined in pieces, on subdomains
(subsets of the original domains) of the original function.
b) The domain of a piecewise function need not be the entire real line. For
⎧⎪3x 2 − 1, x < 0
example, the domain of f x = ⎨ x
is x ∈° and x ≠ 0 .
x>0
⎪⎩e ,
()
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c) Absolute value functions can be expressed as piecewise functions:
⎧⎪− x , x < 0
x =⎨
. Note that defining x as x and not -x for x = 0 is really just
⎪⎩x , x ≥ 0
a matter of convention (somebody made that choice initially and it stuck).
22. Function and Limit Values
a) Function and limit values refer to the y or output value for its corresponding x
value.
()
b) Function value, f a = b . On the graph of f, the value of y IS b when the value
of x IS a.
c) Note: function values are independent of what happens AROUND the specific
x coordinate.
d) Some special function values are x-intercepts (values of x that make y = 0 )
and y-intercepts (the value of y that make x = 0 ), local and absolute extrema
(maxima and minima), and points of inflection.
()
e) Limit value, lim f x = b . On the graph of f, y APPROACHES a value (or height)
x →a
of b as x APPROACHES the value of a (viewed as tracing along the graph to
the right and/or left of the vertical line x = a ).
f) Note: limit values are independent of what happens AT the specific x
coordinate.
g) Some special limit expressions and their values indicate that there may be a
hole in the graph, a vertical asymptote or a horizontal asymptote.
( ) exists if the following two conditions
h) Note: the limit value given by lim f x
x →a
are true:
()
()
lim f ( x ) ≠ ±∞
• (2) The (overall) limit is finite:
⎧⎪2x − 1, x < −1
Function and limit values for a piecewise function, f ( x ) = ⎨
⎩⎪x + 4, x ≥ −1
•
(1) The left and right limits are equal:
lim f x = lim f x
x →a −
x →a +
x →a
2
•
•
( ) ( )
lim f ( x ) = ∃ because the left and right limits are unequal:
⎡
⎤ ⎡
⎤
⎢⎣ lim f ( x ) = lim (2x − 1) = −3⎥⎦ ≠ ⎢⎣ lim f ( x ) = lim ( x + 4 ) = 5 ⎥⎦
2
f −1 = −1 + 4 = 5
x →−1
2
x →−1
−
x →−1
−
x →−1
+
x →−1
+
23. Continuity and Types of Discontinuities:
()
a) Continuity: The function f x
is continuous at x = a if and only if the
following three conditions are met:
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•
•
•
()
The limit value lim f ( x ) exists:
()
The function value f a is defined:
f a =∃
( ) (recall 2 conditions!)
lim f ( x ) = f (a )
lim f x = ∃
x →a
x →a
The function and limit values are equal:
x →a
b) Interval of Continuity: the set of x values for which the function is continuous.
The interval of continuity is the same as the domain for such functions as
polynomials and rational functions, but different for other functions, such as
()
the greatest integer (step) function, y = int x .
c) Identify removable, jump and infinite discontinuities verbally, graphically,
algebraically and numerically:
Discontinuity
Verbal
Removable
A hole with or without a
point defined above or
below the hole
() ()
lim f x ≠ f a
Algebraic
x →a
Jump
The left and right
branches of a graph on
either side a
hypothetical vertical
line do not meet in
height
Infinite
A vertical
asymptote; a finite x
value corresponds to
an infinite (or
undefined) y - value
lim f x ≠ lim f x
lim f x = ±∞
()
x →a −
()
x →a +
x →a
()
Graphical
(for
example)
⎧ x 3 − 2x 2 − x − 2
,x ≠ 1
⎪
y=⎨
x −1
⎪−1,
x =1
⎩
y-coord of hole:
Numerical
(for
example)
()
x →−1−
()
y-coord of graph: f 1 = −1
( ) ()
∴ lim f x ≠ f 1
x →1
()
lim f ( x ) = 1
∴ lim f ( x ) ≠ lim f ( x )
lim f x = −1
lim f x = −2
x →1
⎧⎪x 2 + 2x , x < −1
y=⎨
⎪⎩x + 2, x ≥ −1
x →−1+
x →−1
−
x →−1
y=
x −2
x −1
As x → 1− , y → +∞
As x → 1+ , y → −∞
()
∴ lim f x = ±∞
+
x →1
d) The limit of a function as x approaches “a”
i. exists in the case of a removable discontinuity
ii. does NOT exist in the cases of a jump
iii. does NOT exist in the case of an infinite discontinuity
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24. Extrema and Intervals Increasing/Decreasing
a) Increasing on an interval is a positive change in x yields a positive change in y.
b) Decreasing on an interval is a positive change in x yields a negative change in y.
c) Constant on an interval is a positive change in x yields no change in y.
d) monotone = always
e) Interval of increase (or decrease): the set of x values for which the function
is increasing (or decreasing).
f) Local maximum: a y value that is greater than or equal to all other y values on
either side of that point.
g) Absolute maximum (also a local maximum): the largest y value in the entire
range of the function
h) Local minimum: a y value that is less than or equal to all other y values on
either side of that point.
i) Absolute minimum (also a local minimum): the smallest y value in the entire
range of the function
25. Concavity and Points of Inflection
a) A measure of the curvature of a relation.
b) Upwards concavity at a point is visible when a tangent (line) to the curve at a
point lies below the curve (except at the point of tangency).
c) Downwards concavity at a point is visible when a tangent to the curve at a point
lies above the curve (except at the point of tangency).
d) A point where the concavity changes is a point of inflection.
26. Boundedness
a) Bounded below: there is some number (height of graph) that is less than or
equal to all other y – coordinates.
b) Bounded above: there is some number (height of graph) that is greater than or
equal to all other y – coordinates.
c) Bounded: bounded above and below
d) Unbounded: bounded neither above nor below
27. Symmetry
a) Even
• Reflective symmetry about the y – axis
• Horizontal reflection produces the original graph
•
( ) ()
f −x = f x
b) Odd
• Rotational symmetry about the origin
• Horizontal reflection is equivalent to the vertical reflection
•
( )
()
f − x = −f x
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c) Neither
• a horizontal reflection produces neither the original graph nor a vertical
reflection of the original graph
•
( ) ()
()
f − x ≠ f x nor −f x
28. Asymptotes
a) Vertical Asymptote (infinite discontinuity): a finite x value that makes y
()
undefined, as in lim f x = ±∞ implies that there is a vertical asymptote in
x →a
()
f x at x = a .
b) Horizontal Asymptote (a special case of end behaviour): a finite y value that
makes x undefined, or, the constant that y approaches as x approaches the
()
left and/or right ends of the graph x → ±∞ , as in lim f x = b , so that there
x → ±∞
()
is a HA in f x of y = b .
29. End Behaviour:
a) The end behaviour refers to what the graph looks like when zoomed out, as
x → ±∞ .
b) A horizontal asymptote is one example of a kind of end behaviour (for rational
functions with a denominator of higher or equal degree compared to the
numerator ... we’ll see this in detail in Chapter 2).
c) Other kinds of end behaviour include slant asymptotes (for cubic over
quadratic
rational
y = 3x − 5x + 1 .
2
functions)
or
3x 2
for
the
quadratic
polynomial
The end behaviour model is a simpler function (generally
monomials) for the graph when viewed from “far away” (as x → ±∞ ).
30. Inverses
a) Verbally: a function’s inverse is the “undoing” function, operations that undo
the original applied operations but performed in the reverse order.
b) Graphically: a function’s inverse is the original function’s reflection through
the line y = x
c) Algebraically: the equation that results when the variables x and y are
swapped, and usually y is isolated. Inverses become trickier to determine
algebraically when x occurs more than one time in an equation.
d) Numerically: the swapping of the domain and range (the domain of the original
becomes the range of the inverse, and vice versa)
e) On the TI, function mode allows us to graph only functions; to graph relations,
we need to use parametric mode. The parameterization of y = x 2 + 3x − 2 is
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⎧⎪x = t
and so the inverse, which is a 1-2 relation, can be sketched in
⎨
2
⎩⎪y = t + 3t − 2
⎧⎪x = t 2 + 3t − 2
parametric mode as ⎨
.
⎪⎩y = t
31. Composition
a) Domain of sums, differences and products of two or more functions is the
intersection of the individual functions domains.
b) Domains of quotients of two functions: the domain of these combinations of
functions is the intersection of the individual functions domains and the
intersection of the denominator not equal to zero.
c) Compositions
• of two functions are not commutative in general
•
()
()
of inverses: if f and g are 1-1 functions, and if f ⎡⎢ g x ⎤⎥ = g ⎡⎢f x ⎤⎥ = x , then
⎣
⎦
⎣
⎦
f and g are inverses of each other. Recall that the inverses f and g are the
undoing functions of each other, or the reflection through the identity (the
thing you started with) function, y = x .
()
d) Domain of composition f ⎡⎢ g x ⎤⎥ is the intersection of the simplified
⎣
⎦
composition function with the inner function.
e) Decomposition is the process of identifying and separating the two (or more)
individual functions involved in a composition.
32. Transformations
•
•
•
•
(
)
(
)
Transformational Form: a y − k = f ⎡⎢b x − h ⎤⎥ , where f refers to any basic
⎣
⎦
function (from the library of basic functions, for example)
Horizontal transformations are changes to x BEFORE the basic function f is
applied
Vertical transformations are changes to y, or changes to x AFTER the basic
function f is applied
If b < 0 , there is a horizontal reflection; if a < 0 , there is a vertical
reflection
1
The horizontal scale is
•
•
value used here?
The horizontal translation is h and the vertical translation is k.
Perform transformations numerically (to coordinates in an ordered pair) and
graphically (to the shape of the graph) according to the algebraic order of
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b
and the vertical scale is
1
•
a
. Why is the absolute
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Math 150 Study Guide
operations: multiply first (so do reflections and
then scales), and then add (do
−x
x
y
=
e
2y = e
y = x last).
translations
33. Five Kinds of Reflections
()
a) Vertical Reflection (VR): y = −f x
()
is a vertical reflection of the graph of
y = f x through the x – axis (or line y = 0 ).
()
b) Partial Vertical Reflection (PVR): y = f x
is a partial vertical reflection
()
through the x – axis, for the part(s) of the graph of y = f x
that has
()
negative y – coordinates. The y – coordinates of the graph of y = f x
are
non-negative.
y = x2 − 3
y = x2 − 3
y = − x2 − 3
c) Horizontal Reflection (HR):
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d) Partial Horizontal Reflection (PHR):
y = ex
y =e
x
y =e
−x
e) Inverse Reflection ( y ↔ x ):
Chapter 2 – Polynomial and Rational Functions
34. Linear Functions:
a) polynomials of degree 1
b) their graphs are called lines
c) have constant slope (or rate of change) between any two points on the line
d) are the only functions for which the secant slope over an interval is the same
as the tangent slope at a point for x ∈ Domain.
e) Properties of lines: domain, range, intercepts, slope, 1-1 function
(
)
f) Point-slope form: y − k = m x − h is nearly transformational form; graph it by
( )
plotting the given point h, k and generating a second point using the
()
g) Slope-intercept (function) form: y = f x = mx + b . If m = 0 , then the linear
equation becomes a constant function (a polynomial of degree zero). Note that
on the TI-89, this form is referred to as y = ax + b , which is consistent with
the way coefficients of polynomials are usually named.
x
y
( )
( )
= 1 has an x – int at a, 0 and a y - int at 0, b .
a b
i) About the different equation forms: Recognize various forms for equations of
a line and a parabola, be able to extract properties of lines from various
equation forms, be able to graph from such forms, be able to determine the
h) Intercept form:
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Math 150 Study Guide
most appropriate form in given situations, and be able to rearrange from one
form to another
35. Quadratic Functions:
a) Polynomials of degree 2
b) Their graphs are called parabolas
c) Have non-constant (or changing) slope; in other words, m tan is different at
different points on the graph
d) Properties of lines: domain, range, intercepts, vertex, axis of symmetry,
horizontal tangent at the vertex (so m tan = 0 at vertex), 2-1 function, tangent
slopes of symmetric points (points at same heights reflected through the axis
of symmetry) are negatives
(
e) Vertex/pseudo-transformational form: y − k = a x − h
(
)(
f) Factored/root form: y = a x − r1 x − r2
)
)
2
()
g) Function (standard – I don’t like this name) form: y = f x = ax 2 + bx + c
h) About the different equation forms: Recognize various forms for equations of
a parabola, be able to extract properties of parabolas from various equation
forms, be able to graph from such forms, be able to determine the most
appropriate form in given situations, and be able to rearrange from one form
to another
i) Slick way to graph y = ax 2 + bx + c : graph y = ax 2 + bx by getting the x –
intercepts from factored form, then vertically translate by c.
j) Completing the square
• Is a technique for rewriting any quadratic so that the independent variable
appears only once; useful for solving quadratic equations, deriving the
quadratic formula from ax 2 + bx + c = 0 and expressing y = ax 2 + bx + c in
vertex (transformational) form.
•
Complete the square on y = ax 2 + bx + c and get the transformational form
y+
b 2 − 4ac
4a
2
⎛
b⎞
= a ⎜ x + ⎟ , where the vertex is
2a ⎠
⎝
⎛ b 4ac − b 2 ⎞
h, k = ⎜ − ,
⎟.
4a ⎠
⎝ 2a
( )
Note that k depends on c but h does not.
k) The Quadratic Formula x =
−b ± b 2 − 4ac
2a
can be used to solve quadratic
EQUATIONS of the form ax 2 + bx + c = 0 .
l) The Quadratic Discriminant D = b 2 − 4ac can tell information about the nature
of the roots of any quadratic equation ax 2 + bx + c = 0 :
• D > 0 means that there are two real and distinct roots
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•
•
•
•
D = 0 means that there is one real root of multiplicity two; furthermore,
this x – intercept is ALSO h, the x – coordinate of the vertex
D < 0 means that there are two complex conjugate roots; furthermore, the
values of ax 2 + bx + c will be either al positive or all negative for x ∈ °
D > 0 and a perfect square means that the two real roots are rational
D > 0 but not a perfect square means that the two real roots are irrational
conjugates
36. Power Functions:
a) monomials of the form y = k ⋅ x a where a and k are non-zero, real constants
b) basic power functions: 1 = x 0 , x, x 2 ,
1
x = x2,
3
1
1
x = x3,
= x −1 ,
1
= x −2
x
x
c) since there are two parameters in the equation of a power function (k and a)
2
( )
two points (ordered pairs, x , y ) are required to solve for k and a, thereby
“uniquely determining” the equation of the power function
37. Polynomial Functions:
a) End behaviour – there are 4 end behaviour shapes
b) Multiplicity of roots/behaviour of x – intercepts (zoomed in shape at x-int’s)
c) Sign charts
d) x –intercepts correspond to REAL roots (or zeroes) of the function
e) regression curves and solving for parameters of the function given points
f) write the equations of polynomials in both factored and expanded forms
(forwards and backwards) given the real (rational and irrational) and complex
roots
38. Real Zeroes of Polynomials:
(
a) Common factoring: ax + ay + az = a x + y + z
b) Factoring Special Products:
•
•
•
•
)
( )( )
sum of squares: x + a = ( x − ia ) ( x + ia )
sum/difference of cubes x ± a = ( x ± a ) ( x m ax + a )
perfect trinomials x + 2ax + a = ( x + a )
difference of squares x 2 − a 2 = x − a x + a
2
2
3
2
3
2
2
2
2
c) Rational Root Theorem and listing possible rational roots (PRR)
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d) Remainder and Factor Theorems and testing possible rational roots to identify
actual rational roots
e) Using successive long or synthetic divisions to express polynomials as a product
of linear and quadratic polynomials to fully factor and determine all complex
roots
39. Complex Numbers:
a) The imaginary number i is defined by i = −1 and therefore i 2 = −1 .
b) Complex number have the form a + bi ; a, b ∈° (may be zero or non-zero)
c) All numbers are complex. If b = 0 , then the number is purely real and if a = 0 ,
then the number is purely imaginary. A number that is not real but not purely
imaginary can either be called “non-real” or “complex”.
d) The complex conjugate of a + bi is a − bi ; change the sign of the IMAGINARY
term.
(
)(
e) Factoring sum of squares: m2 + n2 = m + in m − in
(
)
)(
f) Factoring sum or difference of cubes: a 3 ± b 3 = a ± b a 2 m ab + b 2
)
40. Complex Zeroes:
a) a polynomial of degree n has n zeroes, which may be real or non-real, and which
may or may not be repeated (depending on the multiplicity)
b) if a polynomial in expanded form has real coefficients, then any non-real roots
occur in complex conjugate pairs: a + bi and a − bi .
c) if a polynomial in expanded form has rational coefficients, then any irrational
roots occur in irrational conjugate pairs
d) a polynomial of degree n can be expressed as a product of linear factors,
where the n real and non-real roots may or may not be repeated:
(x − r ) ⋅ (x − r ) ⋅ (x − r ) ⋅… ⋅ (x − r )
1
2
n
3
41. Graphs of Rational Functions:
a) Ratios of polynomials
( ) , where the minimum degree of d (x ) is 1 (so the
d (x )
n x
denominator is NOT a constant in rational functions)
b) Their graphs don’t have special names (like parabola, etc.)
c) Rational functions may or may not have domain restrictions; if there are
domain restrictions, they are holes (removable discontinuities), jumps (jump
discontinuities) or vertical asymptotes (infinite discontinuities)
d) Properties of rational functions: holes, x - intercepts, vertical asymptotes, y –
intercept, non-vertical asymptotes, intersection points of the function and the
non-VA, the remainder over denominator term is a variable vertical translation
starting from the non-VA to produce the graph of the rational function
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r (x )
(
)
=
q
x
+
( ) d (x ) ( ) d (x )
e) Division algorithm: f x =
n x
f) Improper Form:
• degree of numerator greater than or equal to the degree of the numerator
• use in factored and reduced function form to find coordinates of any holes,
x – intercepts and equations of VA
g) Proper Form:
• degree of numerator less than the degree of the numerator
()
•
y = q x is the non-vertical asymptote
•
real zeroes of r x are x – coordinates where f x I q x
•
•
()
() ()
real zeroes of d ( x ) are the equations of the VAs
r (x )
Consider that
acts as a "variable vertical translation” on the more
d (x )
r (x )
basic graph of y = q ( x ) to get the rational function graph, y = q ( x ) +
d (x )
h) Transformational Form: some rational function equations can be rearranged
from the proper fraction form to look like a transformation of one of the two
basic rational functions, y =
1
or y =
1
. You are not expected to be able to
x
x2
express all rational function equations in transformational form.
i) Holes
x ⎧⎪1 if x ≠ 0
• Basic Hole: y = = ⎨
, and by plotting a hole in the
x ⎩⎪not defined if x = 0
( )
graph of y = 1 at (or removing the point) 0,1 , we get the graph of y =
x
.
x
•
•
Removable discontinuity
Finite (or “defined”) x values that make y indeterminate
•
Occur whenever n x and d x have a common factor that has real roots
•
If f m =
()
( )
0
0
()
()
()
but lim f x = n (the limit exists and is equal to n), then f x
x →m
( )
has a hole at m, n .
j) Vertical Asymptotes
•
Basic vertical asymptote of multiplicity one: The Reciprocal Function y =
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x
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Math 150 Study Guide
•
Basic vertical asymptote of multiplicity two: The Reciprocal Square
Function y =
1
•
•
•
•
x2
Infinite discontinuity
May NOT be touched or crossed
Finite (or “defined”) x values that make y undefined
Occur whenever the denominator has real roots
•
If lim f x = ∞ or - ∞ or ± ∞ (check the sign chart for one-sided and twox →m
()
()
sided limits), then f x has a vertical asymptote at x = m .
•
VA multiplicity of 1 or 2; multiplicity affects the sign chart
k) Jump Discontinuity
•
Basic Jump: y =
x
x
=
⎧⎪−1 if x < 0
=⎨
x ⎩⎪ 1 if x > 0
x
Occurs when an x –value makes y indeterminate AND the function involves
an absolute value
l) End Behaviour/Non-Vertical Asymptotes
• Zoomed out behaviour, what the graph looks like when you zoom out far
enough
• May or may not be touched or crossed
•
•
()
Found as the quotient of a rational function ( y = q x
from the proper
fraction form); the degree of the quotient polynomial prescribes the “type”
of non-VA: horizontal, slant, parabolic, etc.. Furthermore, the type depends
of the difference in the degrees of the numerator and denominator
polynomials
m) Special Case of Non-VA: Horizontal (constant quotient) Asymptote
• Non-VA of degree 0
• Infinite x produces finite y (compare with vertical asymptotes, for which
finite x produces infinite y)
•
()
()
lim f x = k means that y = k is a horizontal asymptote of f x
x → ±∞
n) About the different equation forms: Recognize various forms for equations of
rational function graphs, be able to extract properties from various equation
forms, be able to graph from such forms, be able to determine the most
appropriate form in given situations, and be able to rearrange from one form
to another
42. Solving Equations in One Variable:
a) get LCD
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b)
c)
d)
e)
consider domain restrictions to eliminate possible extraneous solutions
clear fractions by multiplying both sides of the equation by the LCD
solve the resulting polynomial using factoring, rational root theorem, etc.
eliminate any extraneous solutions
43. Solving Inequalities in One Variable:
a) never solve as an equation! (Well, there are a few instances when you can, but
in general, and usually, you cannot.)
b) solve graphically by comparing heights of two different functions (the left
side and right side functions)
c) be clever in rearranging terms in the inequality to make solving by graphing
easier. For example, rearrange x 2 + 2x + 5 > 0 as x 2 + 2x > −5 , both of which
are easy to sketch; you should quickly note that y = x 2 + 2x is ALWAYS
higher than y = −5 and therefore the solution is x ∈ ° .
d) Solve by sign chart (which is a numerical and partial graphical method), useful
when the complete graph is not required. VERY IMPORTANT NOTES:
• All terms must be on one side (let’s say the left) and ZERO on the other
• The expression on the left must be FULLY FACTORED; all factors
(expressions in parentheses) then must be joined either by multiplication or
division, NOT by addition and subtraction. (Why? Do you know what (-)
times (-) is? YES .. positive. But do you know what (-) + (-) is? No ... it
depends on HOW BIG each negative quantity is.
44. Graph: Know all steps and required information
a) absolute value, polynomial, power/root, greatest integer, rational, exponential,
logarithmic, logistic and piecewise functions and some relations (circles and
horizontal parabolas) by hand
b) use sign charts to aid in graphing functions
45. Modeling and Optimization:
a) Regression curves; scatter plots; interpolation and extrapolation
b) Real world domain restrictions and extraneous solutions
c) Writing word problems algebraically
d) Geometry formulas
•
•
Circle: C = 2πr and A = πr 2
Surface Area: sum of all surfaces’ areas
1
A ⋅ h , where n = 1,2 or 3
n base
e) Optimization (max/min word problems)
• Define all quantities using meaningful variables different from x and y
• Define independent quantities using lower case variables and dependent
quantities using upper case variables
•
Volume: V =
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•
•
•
Write equations for all dependent variables in terms of the independent
variables; use function notation.
Identify the dependent quantity to be optimized (of which you want the
maximum or minimum).
Identify any information that relates the independent variables and write
equations from that information. You’ll recognise this because a value for
the dependent quantity (like volume of 1000 cm3 ) will be given. Use this
equation to make a substitution for one independent variable in terms of
the other. Be clever in your choice ... keep the independent variable that
you wish to solve for (if you only need to solve for one independent
variable), or eliminate the variable that is algebraically simpler if you need
to solve for both independent variable.
• Once you have expressed the dependent quantity to be optimized in terms
of the single independent variable, rename the variables x and y.
• State the domain of the real world problem. Extrema can occur at
endpoints, so you need to check the y values of any closed interval domains.
f) Physics
• 1-dimensional motion of a particle: motion is a synonym for velocity (which
may be constant or change) of an object in one dimension (either vertically,
up and down, or horizontally, right and left, but not the “projectile motion”
where the object moves horizontally and vertically, concurrently. Projectile
motion is 2-D motion). The relevant quantities to be analysed include: time,
position (i.e., height is vertical position), velocity, speed, acceleration.
• free-fall: constant acceleration due to the force of gravity, therefore
linear velocity and quadratic height functions, and the (same) time-domain
of these functions
• speed is the absolute value (PVR) of velocity
• writing the position, velocity, speed and acceleration functions of time
given initial position, initial velocity and constant acceleration
• dimensional analysis (verification by units)
• ordered quadruples
Chapter 3 – Exponential, Logistic and Logarithmic Functions
47.Properties of logarithms
48.Population Modeling
49.Equivalent functions, domain restrictions, (true or false and explain), using
algebraic identities and graphing
50.Transformations
51. Finances: simple/compound interest and continuously compounded formulas
52.Richter Scale
53.pH
54.Newton’s Law of Cooling
55.Exponential Growth and Decay
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56.Radioactive Decay (exponential decay, where 0 < b < 1 or r < 0 , and b = 1 + r is
the growth factor)
57.System of Equations: solving intersection points and solving inequalities
58.Window Dimensions
59.Domain, range
60.simplifying/evaluating logs
61. Solving exponential equations (no restriction on exponents) – the variable is in the
exponent!
62.Solving logarithmic equations (powers must be positive) – the variable is in the
power
63.approximating the value of a logarithm
64.basic exponential and log graphs’ shape
65.matching functions and graphs based on transformations
66.tricky transformations: for a log graph, a HS of one factor is equivalent to a VT
of a different factor
67.condensing logs using log identities
68.inverse functions (switch x & y), complete the square on a quadratic!
69.determine domain for a combination of functions
70.solving for parameters, watch out for quadratic discriminant! (unit 3 test #27)
71. using inverses to solve for parameters
72.determine intersection points of an exponential and/or logarithmic system by
hand
73.knowing when you can divide an equation by a common factor (variable expression)
– cannot divide by log x , but you can divide by e x .
74.compare/know the difference between
ln 8
ln2
and
⎛ 8⎞
ln ⎜ ⎟ , and know that
⎝ 2⎠
ln 8
= log2 8 = 3 ; going backwards using (all) identities!
ln2
75.use let statements to change expressions into equations, and then change form
from exponential to logarithmic and back
76.solve exponential equations by getting like bases
log
77.inverses of functions numerically: 4
4
32
= 32 and logk k a = a
78.find repeated expressions in an equation and use LET to hide the details and
back-substitute
79.check for extraneous solutions wherever the variable is restricted
80.Domain Restrictions:
a) denominator cannot be zero
b) power must be positive
c) radicands of even roots must be positive or zero
d) hypotenuse must be the greatest side in a right triangle
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Chapter 4 – Trigonometric Functions
81. Degree-radian conversion
82.Standard position of angles
83.Trig tools: two special triangles, the CAST “Rule”, the Unit Circle
84.Pythagorean Triples (which do NOT have anything to do with the 30, 45, 60
degree angles of the two special triangles)
85.Domain restriction: hypotenuse must be the largest side of a triangle
86.inverse trig function is NOT a reciprocal trig function
87.Angles: acute, principal, coterminal, negative
88.Reference triangles (reflected from QI into the other three quadrants)
89.Basic trig function graphs and their transformations
90.Connecting HS to period to number of cycles in a 2π interval for trig functions
91. Writing coterminal angle formulas
92.Evaluating trig and inverse trig functions for acceptable (domain) values
93.Domain and Range for trig and inverse trig functions
94.angles are inputs of trig functions; ratios are outputs; vice versa for inverse trig
functions
95.Solving trig functions
a) graphically
b) algebraically
c) by hand and with TI (use of domain restrictions, “such that”)
96.Solving word problems involving acute triangles
97.Sinusoidal modeling:
a) ferris wheel height versus time
b) rabbit population versus
c) tide height versus time
98.Angular and linear distance: s = θr (circumference is a special case)
99.Angular and linear speed: v = ωr
100. Area of a sector of a circle (includes area of a complete circle)
101. Sketching a point on a terminal arm in standard position, and stating all trig
ratios for the resulting angle
102.
103.
104.
105.
getting reference angles: example
() (
trig (θ ) = ±trig (θ
trig θ = trig θcoterminal
)
reference
53π
7
) , depending on the quadrant (if a nonquandrantal angle)
⎛
x ⎞
Write an algebraic expression for expressions like sin ⎜ cos −1
⎟ and stating
x
+
2
⎝
⎠
domain restrictions
106. OMIT: Simple Harmonic Motion – we did not cover this
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Chapter 5 – Analytical Trigonometry
107. Know all fundamental trig identities
a) Reciprocal
b) Quotient
c) Pythagorean
d) negative angle
e) cofunction
f) sum/difference
g) double angle
h) power reducing/half angle
i) Apply fundamental trig identities to evaluate trig ratios of non-special angles
( 22.5o , 75o , 165o )
108. Apply fundamental trig identities to evaluate trig ratios of non-special angles
⎛
1
4⎞
expressed in terms of inverse trig functions: tan ⎜ cos −1 − sin−1 ⎟
3
5⎠
⎝
109. Apply fundamental trig identities to simplify more complicated trig
expressions in order to:
a) prove an identity
b) sketch a basic transformed sinusoid in disguise/identify period & amplitude
c) solve trig equations
110. Solve trig equations:
a) using factoring (never divide by an expression that may equal zero!)
b) using trig identities to simplify first
c) determine reference angles and quadrants (where necessary)
d) solve for all principal angles
a. solve for all coterminal angles to the principal solutions
b. solve a trig equation on the HOME screen and interpret @n1 (etc)
c. solve graphically for x - intercepts or intersection points
111. Recognize whether two seemingly equivalent functions (trig or other type)
have the same domain or not: TRUE or FALSE and EXPLAIN!!!
112. Create a geometric proof using lines and circles involving angles
113. Understand the difference between an always true, a sometimes true, and a
never true statement
114. DO NOT distribute non-linear functions:
( )
(
)
a. know that sec 75o = sec 30o + 45o ≠ sec 30o + sec 45o
b. know that sec 75o =
=
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cos 75
o
=
(
1
cos 30o + 45o
)
1
cos 30o cos 45 0 − sin30o sin 45o
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Math 150 Study Guide
≠
1
cos 30 cos 45
o
0
−
1
sin30 sin 45 0
o
= sec 30o sec 45o − csc 30o csc 45o and explain why not!
Chapter 6
115. Vectors
a) rectangular components
b) polar components
c) P -> R and R -> P conversion by hand and on the TI
d) Unit vector notation (uses rectangular components)
e) Adding vectors (must be done using rectangular components)
116. Complex Numbers
a) standard form (like rectangular components of a vector)
b) trigonometric form (like pole components of a vector)
Euler’s Formula: e iθ = cos θ + i sin θ = cisθ
multiplying two complex numbers (shortcut)
dividing two complex numbers (shortcut)
Vector valued functions: determine the magnitude (r) and direction ( θ ) for a
r
vector function at a specific time: v t = x t î + y t ĵ .
Example: for
c)
d)
e)
117.
() ( ) (
)
() ()
r
v t = 14t î + −5t 2 + 10t + 2 ĵ , at time t = 1 ,

a) The rectangular components : v 1 = 14ˆ
i − 7 ĵ = ⎡14, −7 ⎤
⎣
⎦
b) The polar components are:
( )
r = 142 + −7
118.
2
()
()
= 7 5 and θ IV = 360o − tan−1
1
 333.435o
2
There is NO PROJECTILE MOTION in parametric mode on the exam.
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