AP Precalculus Guide
A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly 1 output value
Positive Function  the output values are above 0
Negative Function  the output values are below 0
Increasing function if…
Verbally: as the input values increase, the output values always increase
Analytically: for all a and b in the interval ___, if a < b, then f(a) < f(b)
Decreasing function if…
Verbally: as the input values increase, the output value always decreases
Analytically: for all a and b in the interval ___, if a < b then f(a) > f(b)
Zero: when the output value is zero
Concave up: bowl facing UP → slope increasing
Concave down: bowl facing DOWN → slope decreasing
Point of Inflection: point where the concavity changes
Justification on stating whether or not the function is increasing or decreasing…
x  4  6  7  10 
f(x)  1  1.01  1.04  1.06 
The function f is increasing on the interval 4 < x < 10 or (4, 10) because for all a and b values, if a < b, then f(a) < f(b)
rate of change = slope
Rate of Change of a point
Ex. Estimate the rate of change at x = 1 for the function f(x) = ½x² + 3x  ½
Get as close as you can to the point (at least 3 decimal places)
(1, 2) & (1.001, 2.0019995)
Now calculate the slope
(2.0019995  2) ÷ (1.001  1) = 1.9995 ≈ 2
Positive ROC  indicates that as one quantity increases or decreases the other quantity does the same (same as if it were to say a function is increasing)
Negative ROC  indicates that as one quantity increases, the other decreases (same as if it were to say a function is decreasing)
Average Rate of Change = Slope of a SECANT Line (calculates overall change in a quantity across a given interval)
average rate of change = f(a)  f(b) / a  b
The average rate of change for a linear function is constant. Regardless of the inputvalue interval length, the average rate of change always stays the same
Does not matter where you pick your points, the slope is always the same
x  15  16  17  18  19 
f(x)  18  20  22  24  26 
NOTE: linear functions only have an increase/decrease in the y values, meanwhile quadratics have an increase/decrease in the average rate of change (the amount of differences is the degree)
The amount of differences you get from this indicates the degree. 1 is linear, 2 is quadratic, 3 is cubic, 4 is quartic, 5 is quintic, etc.
The average rate of change for a quadratic doesn’t stay the same. For consecutive equallength inputvalue intervals, the average rate of change of the rate of change of a quadratic function is constant.
here the x is going up by one (15 +1 = 16; 16 +1 = 17)
x  15  16  17  18  19 
f(x)  18  20  20  18  14 
The rate of change of the rate of change: what is the change in the slope/ROC (This will always be 0 for linear functions)
here the bottom is varying (18 + 2 = 20; 20 + 0 = 20; 20  2 = 18; 18  4 = 14)
Then you would do that again to find a commonality (+2, 0, 2, 4) → ( 0  2 = 2; 2  0 = 2; 4  (2) = 2)
Which means the rate of change of the rate of change is 2
The function is concave down because the rate of change is decreasing over equal length input intervals
Example problem on quiz:
The function p is given by p(x) = g(x + 1)  g(x). If p(x) = 2, which of the following statements must be true?
g(x + 1)  g(x) is really the slope (y2 [which is f(x + 1)]  y1 [which is f(x)]) / 1
p(x) = 2 means that the slope is 2
This means the graph always has a positive slope and that because p is positive and constant, g is increasing
Confusing Terms
Positive Rate of Change  Negative Rate of Change 
The independent variable increases, the dependent variable also increases  As the independent variable increases the dependent variable decreases 
Increasing Rate of Change  Decreasing Rate of Change 
The rate of change (slope) itself is increasing Ex. (car speed goes from 20 to 30 in one hour and 30 to 70 in the next hour)  The rate of change (slope) itself is decreasing Ex. (car speed increases from 30 to 40 in 1 hour and then to 45 in the next hour, the acceleration is decreasing) 
Positive Average Rate of Change  Negative Average Rate of Change 
Whether the rate of change between an interval or 2 points (secant line) is positive  Whether the rate of change between an interval or 2 points (secant line) is negative 
Increasing Average Rate of Change  Decreasing Average Rate of Change 
The rate of change over an interval itself is increasing Ex. (In a 3 hour period, a car goes 10 mph at 1 hour, 15 mph a 2 hours and 20 mph at 3 hours, the aroc of the speed is increasing)  The rate of change over an interval itself is decreasing Ex. (In a 3 hour period, a car goes 20 mph at 1 hour, 15 mph a 2 hours and 10 mph at 3 hours, the aroc of the speed is decreasing) 
Increasing + Positive ROC  Increasing + Negative ROC 
→ The function is increasing → The graph is concave up  → The function is decreasing → The graph is concave up 
Decreasing + Positive ROC  Decreasing + Negative ROC 
→ The function is increasing → The graph is concave down  → The function is decreasing → The graph is concave down 
Increasing / Decreasing ROC → The graph is concave Up / Down
Positive / Negative ROC → The function is Increasing / Decreasing (± slope; imagine a positive or negative linear line)
Polynomial
p(x) = a_{n}x^{n}^{ }+ a_{n1}x^{n1} + a_{n2}x^{n2 }+ … + a_{1}x^{2} + a_{1}x + a_{0}
a_{n}x^{n} is the leading term  n is the degree  a_{n} is the leading coefficient
Local / Relative Maximum  if the polynomial switches from increasing to decreasing
Local / Relative Minimum  if the polynomial switches from decreasing to increasing
an included endpoint of a polynomial with a restricted domain may also be the local minimum or maximum
Global Maximum  the greatest of the local maximums (If it goes to infinity, there is none)
Global Minimum  the least of the local minimums
Two Zeros = if you have 2 zeros of a polynomial, there must be at least 1 local extrema between the two
Even degree = absolute extreme
Positive leading coefficient = absolute minimum
Negative leading coefficient = absolute maximum
If the real zero, a, of (x  a), has an even multiplicity, then the graph will bounce off the xaxis
Example:
Given the function: (x + 1)^{2}(x  3)^{3}(x + 4)^{1}
Complex Zeros  nonreal zeros that will always come in pairs (Ex. 3 ± 2i)
a ± bi
Even functions = symmetrical over the yaxis
has the property f(x) = f(x)
You can prove that the function is even by substituting in x and see if you get the same original function
Ex. f(x) = x^{6}  4x^{2} → f(x) = (x)^{6}  4(x)^{2} → f(x) = x^{6}  4x^{2}
Odd functions = symmetrical over the origin
has the property f(x) = f(x)
The left or right side of a polynomial will either go up or down.
The limit expresses the end behavior of a function.
The Left Side
The x values are approaching negative ∞
lim x → ∞ f(x)
The Right Side
The x values are approaching positive ∞
lim _{x → ∞ }f(x)
Left Side  Right Side  
Up  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Down  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Left side x → ∞  Right Side x → ∞  
Even degree and Positive leading coefficient  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Even degree and Negative leading coefficient  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Even degree means that the left and right side will behave the same
Odd degree means that the left and right side will behave opposite
Rational Function  the ratio of two polynomials where the polynomial in the denominator cannot equal 0
Usually can be described as:
Labeling some of the properties of the function:
Domain: (∞, 3) U (3, 3) U (3, ∞) There is a hole and vertical asymptote
lim_{x → ∞ }f(x) = 0  lim_{x → ∞ }f(x) = 0
0 is the horizontal asymptote because both limits approach this number
For inputs with a larger magnitude, the polynomial in the denominator dominates the polynomial in the numerator
Ways to remember:
BOBO BOTNA EATS DC
BOBO  Bigger on bottom = 0 (BOB0)
BOTNA  Bigger on top = No Asymptote (BotNA)
EATS DC  Exponents Are The Same = Divide Coefficients (EATSDC)
When there is an unfactored polynomial in a rational function, try to factor both numerator and denominator to make it easier to see holes and asymptotes
This is a helpful equation when working with a polynomial that is both factored on the numerator and denominator: (xint)(hole) / (hole)(vertical asymptote)
Example function: (x + 3)(x  2) / (x + 4)(x  2)
X intercepts (zeros) will be any factors on the numerator that aren’t holes
(x + 3); so that means that there is an xintercept at x = 3
Holes will appear on both the numerator and denominator (This is the value you cannot plug into the equation)
(x  2) / (x  2); meaning that x = 2 does not exist
Vertical asymptotes are any factors that aren’t holes left in the denominator
(x + 4); so that means that there is a vertical asymptote at x = 4
Finding the limit as x approaches a number/asymptote
2^{} means approaching the vertical asymptote on the left side from left to right
lim_{x → 2 }f(x) = ∞
2^{+} means approaching the vertical asymptote on the right side from right to left
lim_{x → 2+ }f(x) = ∞
Order of dominance
Hole > 0
If there is a zero with a hole, then there will be no zero, but a hole instead
Ex. (x + 1)(x + 1) / (x + 1)(x + 2) means that there will be a hole at x = 1 instead of a zero because there is a hole of (x + 1)
Vertical Asymptote > Hole
If there is a vertical asymptote the same as a hole, then that hole will be a vertical asymptote
Ex. (x + 1) / (x + 1)(x + 1) means that there is a vertical asymptote at x = 1 instead of a hole because there is a vertical asymptote of (x + 1)
Writing limits for holes:
From the left:
As x approaches 3 from the left, f(x) approaches 2
lim_{x → 3 }f(x) = 2
From the right:
As x approaches 3 from the right, f(x) approaches 2
lim_{x → 3+ }f(x) = 2
Using a Table to Determine Limits
x  2.9  2.99  3  3.01  3.1 
f(x)  2.0345  2.0033  undefined  1.9967  1.9677 
As you can see, the closer we get to 3, the closer the values get to 2
Using pascal’s triangle, you can expand binomials. The rows of pascal’s triangle start from 0 (the top) and so on.
For example, if you had (x + 3)^{4} then the row you would use would be → 1 4 6 4 1
Binomial Theorem:
The formula is (x + y)^{n} where n would be the row #
(a+b)^{n} = C_{1}a^{n}b^{0} + C_{2}a^{n  1}b^{1} + C_{3}a^{n  2}b^{2} + … + C_{n  1}a^{1}b^{n  1} + C_{n}a^{0}b^{n}
C is the coefficient corresponding to the row in pascal’s triangle
n is the power
Using the numbers in pascal’s triangle, expand the polynomial
As the power of x decreases, the power of y increases
Ex. (x + 5)^{3}
1x^{3}5^{0} + 3x^{2}5^{1} + 3x^{1}5^{2} + 1x^{0}5^{3} → x³ + 15x² + 75x + 125
Slant Asymptote  When the degree of the numerator is one higher than the degree of the denominator
End behavior of slant asymptote:
Finding the asymptote of the slant asymptote with long division or synthetic division will give you the linear equation. The end behavior of that function (in this case, y = x + 1) is the end behavior of the rational function
lim_{x → }_{∞}_{ }f(x) = ∞
lim_{x → }_{∞}_{ }f(x) = ∞
Long Division
Long division can be used to divide any polynomial by another one
Synthetic Division:
Synthetic division can only be used when dividing a polynomial by a factor like (ax + b)
Vertical Translations (anything done outside of the function)
f(x) + d → up by d units
f(x)  d → down by d units
Vertical Dilations (anything done outside of the function)
a × f(x)
When doing a vertical dilation, only the yvalues change, the xvalues do not
Multiply the yvalues by a
a > 1: vertical stretch
a < 1: vertical shrink
Horizontal Translations (anything done inside the function)
f(x  d) → right by d units
f(x + d) → left by d units
Horizontal Dilations (anything done inside the function)
f(bx)
b is inverted (if it was f(3x) that would be a horizontal dilation of 1/3)
When doing a horizontal dilation, only the xvalues change, the yvalues do not
Multiply the xvalues by the inverse of b
1/b > 1: horizontal shrink
1/b < 1: horizontal stretch
Reflections
f(x) → reflect over xaxis → vertical change
f(x) → reflect over yaxis → horizontal change
Transformations Numerically
x  0  1  2  3  4 
f(x)  20  12  0  8  14 
Ex. let g(x) = 3f(x  2) + 1, find g(3)
g(x) = 3f(x  2) + 1 → set up equation
g(3) = 3f(3  2) + 1 → plug in the 3 for x in the modified equation
g(3) = 3f(1) + 1 → simplify
g(3) = 3(12) + 1 → find what f(1) is and substitute that in
g(3) = 35 → solve
Transformations through domain & range
Ex. Given the graph for f has a domain of [4, 3] and a range of (3, 9]. Let g(x) = f(x + 5) + 2.
f(x)
Domain: [4, 3]
Range: (3, 9]
g(x)
Domain: [4  5, 3  5] → [9, 2]
f(x + 5) → move left 5 → subtract 5 from the domain
Range: (3 * 1, 9 * 1] → (3 + 2, 9 + 2] → (1, 7]
Do the inversions and dilations first then the translations
Transformations Algebraically
Ex. Given f(x) = x²  3x + 2, let g(x) = f(x  3) + 2, find g(x)
g(x) = f(x  3) + 2 → set up equation
[(x  3)²  3(x  3) + 2] + 2 → substitute the input values in g(x) for x in f(x)
(x²  6x + 9  3x + 9 + 2) + 2 → distribute
(x²  9x + 20) + 2 → combining like terms
g(x) = x²  9x + 22 → solved
Perimeter will be linear
Area will be quadratic
Volume will be cubic
Restricted Domain & Range
Restricted domain and range is give between brackets and is usually given in a word problem or between a piece of a function
Piecewise Functions
Range of a piecewise function
Types of Regression
Linear
Quadratic
Cubic
Quartic
Exponential
Logarithmic
Logistic
Sine
Inversely Proportional
When something is inversely proportional, the equation y = k / x is used
k is some constant
Ex. The number of workers at a job is inversely proportional to the time it takes to complete a task. If there are 15 workers and it takes 20 minutes to do a task, then how many workers would it take to complete a task in 3 minutes?
n = k / t → number of workers = k / time
15 = k / 20 → set up equation to solve for k
k = 15 × 20 = 300 → solve for k
n = 300 / 3 → input k into new equation to solve for workers
n = 100 workers → it would take 100 workers to complete a task in 3 minutes
Piecewise Functions Algebraically
Piecewise functions can be shown like the image above. The function on the left is in the bounds of the inequality on the right.
Sequence → an ordered list of numbers; each listed number is a term
Ex. {1, 3, 5, 7, 9, …} (… means it keeps going continuously)
1 would be the first term not the zero term
Arithmetic Sequences
Linear
If each successive term in a sequence has a common difference (constant ROC), the sequence is arithmetic
Equation for the nth term → a_{n} = a_{0} + dn
a_{0} is the initial value (zero term) and d is the common difference
Equation for any term → a_{n} = a_{k} + d(n  k)
a_{k} is the k^{th} term of the sequence
Geometric Sequence
Exponential
If each successive term in a sequence has a common ratio (constant proportional change) the sequence is geometric
If dividing is involved (for example, {4, 2, 1, …} is ÷2 each time), then the ratio will be multiplied by a fraction (÷2 → ×1/2)
Equation for the nth term → g_{n} = g_{0}r^{n}
g_{0} is the initial value (zero term) and r is the common ratio
Equation for any term → g_{n} = g_{k}r^{(n  k)}
g_{k} is the k^{th} term of the sequence
Linear Functions vs Arithmetic Sequences
They are basically the same, except when using a sequence vs a function, the domain and points are discrete, meaning the domain only includes specific points and not all numbers in between
The point (x_{1}, y_{1}) of a linear function is similar to (k, a_{k}) of an arithmetic sequence
Slopeintercept form f(x) = b + mx  Arithmetic Sequence a_{n} = a_{0} + dn 
Pointslope form f(x) = y_{1} + m(x  x_{1})  A.S. for the kth term a_{n} = a_{k} + d(n  k) 
Exponential Functions vs Geometric Sequences
Again, they are basically the same except with discrete points for the sequence
The point (x_{1}, y_{1}) of an exponential function is similar to (k, g_{k}) of a geometric sequence
Exponential Function f(x) = ab^{x}  Geometric Sequence g_{n} = g_{0}r^{n} 
Shifted Exponential Function f(x) = y_{1}r^{(x  x}_{1}^{)}  G.S. for the kth term g_{n} = g_{k}r^{(n  k)} 
How each output value changes analytically
Arithmetic → Over equallength input value intervals, if the output values of a function change at a constant rate, then the function is linear (adding the slope)
Geometric → Over equallength input value intervals, if the output values of a function change proportionally, then the function is exponential (multiplying the ratio)
“Over equallength input intervals” means that the xvalues of the function or a table are increasing by a constant amount (ex. 1, 2, 3, 4 or 2, 4, 6, 8)
Finding whether a set of points is linear or exponential
x  5  6  7 
f(x)  8  16  32 
Make sure the function is increasing over equallength input intervals
Test to see whether the function is linear or exponential
Linear functions change by adding or subtracting a constant amount
Exponential functions change by multiplying or dividing a constant amount
This example is increasing x2 from 816 and 1632
So this function is exponential because over equallength intervals of 1, f(x) proportionally changes by a ratio of 2
If this was linear you could say, the function is linear because over equallength input intervals of ___, f(x) has a constant rate of change of ___
Finding a common difference/ratio from a set of points
Common Difference
Given the points (a, b) & (c, d), you can find the common difference using the slope formula →
Common Ratio
1  2  3 
4  16 
Find how many times r would have to be multiplied to get from 4 to 16 (in this case it is 2)
This means we would have to multiply 4 ⋅ r ⋅ r to get 16 → 4r² = 16
Then, you solve for r → r² = 16/4 = 4 → r = √4 = 2
So r = 2
OR… given the points (a, b) & (c, d), you can find the common ratio using this equation →
General form of an exponential function → f(x) = ab^{x}
a = initial value
a ≠ 0 & b > 0
Exponential growth → when a > 0 & b > 1
Exponential decay → when a > 0 & 0 < b < 1
Determining exponential function based on output values
Ex. 3 × 3 × 3 × 3 × 6
a, the input value, will be the odd one out: 6
the rest are the common ratio multiplied by however many times
f(x) = 6(3)^{x} where x = 4
Exponential Functions depending on parameters
a > 0; b > 1
 a > 0; 0 < b < 1

a < 0; b > 1
 a < 0; 0 < b < 1

Characteristics of Exponential Functions
They are always increasing or decreasing
There is no extreme (except on closed intervals)
Their graphs are always concave up or concave down
They have no inflection points
If the input values increase or decrease without bound, the endbehavior can be expressed as
lim_{x → ±∞} ab^{x} = 0
lim_{x → ±∞} ab^{x} = ±∞
Additive Transformation of an Exponential Function
g(x) = f(x) + k
If the output values of g are proportional over equallength inputvalue intervals, then f is exponential
Meaning that if g(x) has proportional output values, then f(x) will always be exponential
But if f(x) is proportional, that doesn’t mean g(x) will be proportional
Negative Exponent Property → b^{n} = 1/b^{n}
Product Property → b^{n} * b^{m} = b^{n + m}
Product Property → (b^{m})^{n} = b^{mn}
Transformations of Exponential functions
Horizontal Translations & Vertical Dilations
ac(b)^{x} = b^{k}b^{x} = b^{(x + k)}
Through the product property, when doing a horizontal translation, it is the same as doing a vertical dilation
Every horizontal translation of an exponential function, f(x) = b^{(x + k)}, is equivalent to a vertical dilation
Ex. 1(2)^{(x + 3)} → 1 × 2^{3}(2^{x}) → 8(2)^{x}
Horizontal translation left by 3 units / Vertical dilation of 8 units
Horizontal Dilations
Every horizontal dilation of an exponential function, f(x) = b^{(cx)}, is equivalent to a change of base
(b^{c})^{x} → b^{c} and c ≠ 0
Ex. 2^{3x} → (2^{3})^{x} → 8^{x}
The kth Root of b
A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly 1 output value
Positive Function  the output values are above 0
Negative Function  the output values are below 0
Increasing function if…
Verbally: as the input values increase, the output values always increase
Analytically: for all a and b in the interval ___, if a < b, then f(a) < f(b)
Decreasing function if…
Verbally: as the input values increase, the output value always decreases
Analytically: for all a and b in the interval ___, if a < b then f(a) > f(b)
Zero: when the output value is zero
Concave up: bowl facing UP → slope increasing
Concave down: bowl facing DOWN → slope decreasing
Point of Inflection: point where the concavity changes
Justification on stating whether or not the function is increasing or decreasing…
x  4  6  7  10 
f(x)  1  1.01  1.04  1.06 
The function f is increasing on the interval 4 < x < 10 or (4, 10) because for all a and b values, if a < b, then f(a) < f(b)
rate of change = slope
Rate of Change of a point
Ex. Estimate the rate of change at x = 1 for the function f(x) = ½x² + 3x  ½
Get as close as you can to the point (at least 3 decimal places)
(1, 2) & (1.001, 2.0019995)
Now calculate the slope
(2.0019995  2) ÷ (1.001  1) = 1.9995 ≈ 2
Positive ROC  indicates that as one quantity increases or decreases the other quantity does the same (same as if it were to say a function is increasing)
Negative ROC  indicates that as one quantity increases, the other decreases (same as if it were to say a function is decreasing)
Average Rate of Change = Slope of a SECANT Line (calculates overall change in a quantity across a given interval)
average rate of change = f(a)  f(b) / a  b
The average rate of change for a linear function is constant. Regardless of the inputvalue interval length, the average rate of change always stays the same
Does not matter where you pick your points, the slope is always the same
x  15  16  17  18  19 
f(x)  18  20  22  24  26 
NOTE: linear functions only have an increase/decrease in the y values, meanwhile quadratics have an increase/decrease in the average rate of change (the amount of differences is the degree)
The amount of differences you get from this indicates the degree. 1 is linear, 2 is quadratic, 3 is cubic, 4 is quartic, 5 is quintic, etc.
The average rate of change for a quadratic doesn’t stay the same. For consecutive equallength inputvalue intervals, the average rate of change of the rate of change of a quadratic function is constant.
here the x is going up by one (15 +1 = 16; 16 +1 = 17)
x  15  16  17  18  19 
f(x)  18  20  20  18  14 
The rate of change of the rate of change: what is the change in the slope/ROC (This will always be 0 for linear functions)
here the bottom is varying (18 + 2 = 20; 20 + 0 = 20; 20  2 = 18; 18  4 = 14)
Then you would do that again to find a commonality (+2, 0, 2, 4) → ( 0  2 = 2; 2  0 = 2; 4  (2) = 2)
Which means the rate of change of the rate of change is 2
The function is concave down because the rate of change is decreasing over equal length input intervals
Example problem on quiz:
The function p is given by p(x) = g(x + 1)  g(x). If p(x) = 2, which of the following statements must be true?
g(x + 1)  g(x) is really the slope (y2 [which is f(x + 1)]  y1 [which is f(x)]) / 1
p(x) = 2 means that the slope is 2
This means the graph always has a positive slope and that because p is positive and constant, g is increasing
Confusing Terms
Positive Rate of Change  Negative Rate of Change 
The independent variable increases, the dependent variable also increases  As the independent variable increases the dependent variable decreases 
Increasing Rate of Change  Decreasing Rate of Change 
The rate of change (slope) itself is increasing Ex. (car speed goes from 20 to 30 in one hour and 30 to 70 in the next hour)  The rate of change (slope) itself is decreasing Ex. (car speed increases from 30 to 40 in 1 hour and then to 45 in the next hour, the acceleration is decreasing) 
Positive Average Rate of Change  Negative Average Rate of Change 
Whether the rate of change between an interval or 2 points (secant line) is positive  Whether the rate of change between an interval or 2 points (secant line) is negative 
Increasing Average Rate of Change  Decreasing Average Rate of Change 
The rate of change over an interval itself is increasing Ex. (In a 3 hour period, a car goes 10 mph at 1 hour, 15 mph a 2 hours and 20 mph at 3 hours, the aroc of the speed is increasing)  The rate of change over an interval itself is decreasing Ex. (In a 3 hour period, a car goes 20 mph at 1 hour, 15 mph a 2 hours and 10 mph at 3 hours, the aroc of the speed is decreasing) 
Increasing + Positive ROC  Increasing + Negative ROC 
→ The function is increasing → The graph is concave up  → The function is decreasing → The graph is concave up 
Decreasing + Positive ROC  Decreasing + Negative ROC 
→ The function is increasing → The graph is concave down  → The function is decreasing → The graph is concave down 
Increasing / Decreasing ROC → The graph is concave Up / Down
Positive / Negative ROC → The function is Increasing / Decreasing (± slope; imagine a positive or negative linear line)
Polynomial
p(x) = a_{n}x^{n}^{ }+ a_{n1}x^{n1} + a_{n2}x^{n2 }+ … + a_{1}x^{2} + a_{1}x + a_{0}
a_{n}x^{n} is the leading term  n is the degree  a_{n} is the leading coefficient
Local / Relative Maximum  if the polynomial switches from increasing to decreasing
Local / Relative Minimum  if the polynomial switches from decreasing to increasing
an included endpoint of a polynomial with a restricted domain may also be the local minimum or maximum
Global Maximum  the greatest of the local maximums (If it goes to infinity, there is none)
Global Minimum  the least of the local minimums
Two Zeros = if you have 2 zeros of a polynomial, there must be at least 1 local extrema between the two
Even degree = absolute extreme
Positive leading coefficient = absolute minimum
Negative leading coefficient = absolute maximum
If the real zero, a, of (x  a), has an even multiplicity, then the graph will bounce off the xaxis
Example:
Given the function: (x + 1)^{2}(x  3)^{3}(x + 4)^{1}
Complex Zeros  nonreal zeros that will always come in pairs (Ex. 3 ± 2i)
a ± bi
Even functions = symmetrical over the yaxis
has the property f(x) = f(x)
You can prove that the function is even by substituting in x and see if you get the same original function
Ex. f(x) = x^{6}  4x^{2} → f(x) = (x)^{6}  4(x)^{2} → f(x) = x^{6}  4x^{2}
Odd functions = symmetrical over the origin
has the property f(x) = f(x)
The left or right side of a polynomial will either go up or down.
The limit expresses the end behavior of a function.
The Left Side
The x values are approaching negative ∞
lim x → ∞ f(x)
The Right Side
The x values are approaching positive ∞
lim _{x → ∞ }f(x)
Left Side  Right Side  
Up  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Down  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Left side x → ∞  Right Side x → ∞  
Even degree and Positive leading coefficient  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Even degree and Negative leading coefficient  lim_{x → ∞} f(x) = ∞  lim_{x → ∞} f(x) = ∞ 
Even degree means that the left and right side will behave the same
Odd degree means that the left and right side will behave opposite
Rational Function  the ratio of two polynomials where the polynomial in the denominator cannot equal 0
Usually can be described as:
Labeling some of the properties of the function:
Domain: (∞, 3) U (3, 3) U (3, ∞) There is a hole and vertical asymptote
lim_{x → ∞ }f(x) = 0  lim_{x → ∞ }f(x) = 0
0 is the horizontal asymptote because both limits approach this number
For inputs with a larger magnitude, the polynomial in the denominator dominates the polynomial in the numerator
Ways to remember:
BOBO BOTNA EATS DC
BOBO  Bigger on bottom = 0 (BOB0)
BOTNA  Bigger on top = No Asymptote (BotNA)
EATS DC  Exponents Are The Same = Divide Coefficients (EATSDC)
When there is an unfactored polynomial in a rational function, try to factor both numerator and denominator to make it easier to see holes and asymptotes
This is a helpful equation when working with a polynomial that is both factored on the numerator and denominator: (xint)(hole) / (hole)(vertical asymptote)
Example function: (x + 3)(x  2) / (x + 4)(x  2)
X intercepts (zeros) will be any factors on the numerator that aren’t holes
(x + 3); so that means that there is an xintercept at x = 3
Holes will appear on both the numerator and denominator (This is the value you cannot plug into the equation)
(x  2) / (x  2); meaning that x = 2 does not exist
Vertical asymptotes are any factors that aren’t holes left in the denominator
(x + 4); so that means that there is a vertical asymptote at x = 4
Finding the limit as x approaches a number/asymptote
2^{} means approaching the vertical asymptote on the left side from left to right
lim_{x → 2 }f(x) = ∞
2^{+} means approaching the vertical asymptote on the right side from right to left
lim_{x → 2+ }f(x) = ∞
Order of dominance
Hole > 0
If there is a zero with a hole, then there will be no zero, but a hole instead
Ex. (x + 1)(x + 1) / (x + 1)(x + 2) means that there will be a hole at x = 1 instead of a zero because there is a hole of (x + 1)
Vertical Asymptote > Hole
If there is a vertical asymptote the same as a hole, then that hole will be a vertical asymptote
Ex. (x + 1) / (x + 1)(x + 1) means that there is a vertical asymptote at x = 1 instead of a hole because there is a vertical asymptote of (x + 1)
Writing limits for holes:
From the left:
As x approaches 3 from the left, f(x) approaches 2
lim_{x → 3 }f(x) = 2
From the right:
As x approaches 3 from the right, f(x) approaches 2
lim_{x → 3+ }f(x) = 2
Using a Table to Determine Limits
x  2.9  2.99  3  3.01  3.1 
f(x)  2.0345  2.0033  undefined  1.9967  1.9677 
As you can see, the closer we get to 3, the closer the values get to 2
Using pascal’s triangle, you can expand binomials. The rows of pascal’s triangle start from 0 (the top) and so on.
For example, if you had (x + 3)^{4} then the row you would use would be → 1 4 6 4 1
Binomial Theorem:
The formula is (x + y)^{n} where n would be the row #
(a+b)^{n} = C_{1}a^{n}b^{0} + C_{2}a^{n  1}b^{1} + C_{3}a^{n  2}b^{2} + … + C_{n  1}a^{1}b^{n  1} + C_{n}a^{0}b^{n}
C is the coefficient corresponding to the row in pascal’s triangle
n is the power
Using the numbers in pascal’s triangle, expand the polynomial
As the power of x decreases, the power of y increases
Ex. (x + 5)^{3}
1x^{3}5^{0} + 3x^{2}5^{1} + 3x^{1}5^{2} + 1x^{0}5^{3} → x³ + 15x² + 75x + 125
Slant Asymptote  When the degree of the numerator is one higher than the degree of the denominator
End behavior of slant asymptote:
Finding the asymptote of the slant asymptote with long division or synthetic division will give you the linear equation. The end behavior of that function (in this case, y = x + 1) is the end behavior of the rational function
lim_{x → }_{∞}_{ }f(x) = ∞
lim_{x → }_{∞}_{ }f(x) = ∞
Long Division
Long division can be used to divide any polynomial by another one
Synthetic Division:
Synthetic division can only be used when dividing a polynomial by a factor like (ax + b)
Vertical Translations (anything done outside of the function)
f(x) + d → up by d units
f(x)  d → down by d units
Vertical Dilations (anything done outside of the function)
a × f(x)
When doing a vertical dilation, only the yvalues change, the xvalues do not
Multiply the yvalues by a
a > 1: vertical stretch
a < 1: vertical shrink
Horizontal Translations (anything done inside the function)
f(x  d) → right by d units
f(x + d) → left by d units
Horizontal Dilations (anything done inside the function)
f(bx)
b is inverted (if it was f(3x) that would be a horizontal dilation of 1/3)
When doing a horizontal dilation, only the xvalues change, the yvalues do not
Multiply the xvalues by the inverse of b
1/b > 1: horizontal shrink
1/b < 1: horizontal stretch
Reflections
f(x) → reflect over xaxis → vertical change
f(x) → reflect over yaxis → horizontal change
Transformations Numerically
x  0  1  2  3  4 
f(x)  20  12  0  8  14 
Ex. let g(x) = 3f(x  2) + 1, find g(3)
g(x) = 3f(x  2) + 1 → set up equation
g(3) = 3f(3  2) + 1 → plug in the 3 for x in the modified equation
g(3) = 3f(1) + 1 → simplify
g(3) = 3(12) + 1 → find what f(1) is and substitute that in
g(3) = 35 → solve
Transformations through domain & range
Ex. Given the graph for f has a domain of [4, 3] and a range of (3, 9]. Let g(x) = f(x + 5) + 2.
f(x)
Domain: [4, 3]
Range: (3, 9]
g(x)
Domain: [4  5, 3  5] → [9, 2]
f(x + 5) → move left 5 → subtract 5 from the domain
Range: (3 * 1, 9 * 1] → (3 + 2, 9 + 2] → (1, 7]
Do the inversions and dilations first then the translations
Transformations Algebraically
Ex. Given f(x) = x²  3x + 2, let g(x) = f(x  3) + 2, find g(x)
g(x) = f(x  3) + 2 → set up equation
[(x  3)²  3(x  3) + 2] + 2 → substitute the input values in g(x) for x in f(x)
(x²  6x + 9  3x + 9 + 2) + 2 → distribute
(x²  9x + 20) + 2 → combining like terms
g(x) = x²  9x + 22 → solved
Perimeter will be linear
Area will be quadratic
Volume will be cubic
Restricted Domain & Range
Restricted domain and range is give between brackets and is usually given in a word problem or between a piece of a function
Piecewise Functions
Range of a piecewise function
Types of Regression
Linear
Quadratic
Cubic
Quartic
Exponential
Logarithmic
Logistic
Sine
Inversely Proportional
When something is inversely proportional, the equation y = k / x is used
k is some constant
Ex. The number of workers at a job is inversely proportional to the time it takes to complete a task. If there are 15 workers and it takes 20 minutes to do a task, then how many workers would it take to complete a task in 3 minutes?
n = k / t → number of workers = k / time
15 = k / 20 → set up equation to solve for k
k = 15 × 20 = 300 → solve for k
n = 300 / 3 → input k into new equation to solve for workers
n = 100 workers → it would take 100 workers to complete a task in 3 minutes
Piecewise Functions Algebraically
Piecewise functions can be shown like the image above. The function on the left is in the bounds of the inequality on the right.
Sequence → an ordered list of numbers; each listed number is a term
Ex. {1, 3, 5, 7, 9, …} (… means it keeps going continuously)
1 would be the first term not the zero term
Arithmetic Sequences
Linear
If each successive term in a sequence has a common difference (constant ROC), the sequence is arithmetic
Equation for the nth term → a_{n} = a_{0} + dn
a_{0} is the initial value (zero term) and d is the common difference
Equation for any term → a_{n} = a_{k} + d(n  k)
a_{k} is the k^{th} term of the sequence
Geometric Sequence
Exponential
If each successive term in a sequence has a common ratio (constant proportional change) the sequence is geometric
If dividing is involved (for example, {4, 2, 1, …} is ÷2 each time), then the ratio will be multiplied by a fraction (÷2 → ×1/2)
Equation for the nth term → g_{n} = g_{0}r^{n}
g_{0} is the initial value (zero term) and r is the common ratio
Equation for any term → g_{n} = g_{k}r^{(n  k)}
g_{k} is the k^{th} term of the sequence
Linear Functions vs Arithmetic Sequences
They are basically the same, except when using a sequence vs a function, the domain and points are discrete, meaning the domain only includes specific points and not all numbers in between
The point (x_{1}, y_{1}) of a linear function is similar to (k, a_{k}) of an arithmetic sequence
Slopeintercept form f(x) = b + mx  Arithmetic Sequence a_{n} = a_{0} + dn 
Pointslope form f(x) = y_{1} + m(x  x_{1})  A.S. for the kth term a_{n} = a_{k} + d(n  k) 
Exponential Functions vs Geometric Sequences
Again, they are basically the same except with discrete points for the sequence
The point (x_{1}, y_{1}) of an exponential function is similar to (k, g_{k}) of a geometric sequence
Exponential Function f(x) = ab^{x}  Geometric Sequence g_{n} = g_{0}r^{n} 
Shifted Exponential Function f(x) = y_{1}r^{(x  x}_{1}^{)}  G.S. for the kth term g_{n} = g_{k}r^{(n  k)} 
How each output value changes analytically
Arithmetic → Over equallength input value intervals, if the output values of a function change at a constant rate, then the function is linear (adding the slope)
Geometric → Over equallength input value intervals, if the output values of a function change proportionally, then the function is exponential (multiplying the ratio)
“Over equallength input intervals” means that the xvalues of the function or a table are increasing by a constant amount (ex. 1, 2, 3, 4 or 2, 4, 6, 8)
Finding whether a set of points is linear or exponential
x  5  6  7 
f(x)  8  16  32 
Make sure the function is increasing over equallength input intervals
Test to see whether the function is linear or exponential
Linear functions change by adding or subtracting a constant amount
Exponential functions change by multiplying or dividing a constant amount
This example is increasing x2 from 816 and 1632
So this function is exponential because over equallength intervals of 1, f(x) proportionally changes by a ratio of 2
If this was linear you could say, the function is linear because over equallength input intervals of ___, f(x) has a constant rate of change of ___
Finding a common difference/ratio from a set of points
Common Difference
Given the points (a, b) & (c, d), you can find the common difference using the slope formula →
Common Ratio
1  2  3 
4  16 
Find how many times r would have to be multiplied to get from 4 to 16 (in this case it is 2)
This means we would have to multiply 4 ⋅ r ⋅ r to get 16 → 4r² = 16
Then, you solve for r → r² = 16/4 = 4 → r = √4 = 2
So r = 2
OR… given the points (a, b) & (c, d), you can find the common ratio using this equation →
General form of an exponential function → f(x) = ab^{x}
a = initial value
a ≠ 0 & b > 0
Exponential growth → when a > 0 & b > 1
Exponential decay → when a > 0 & 0 < b < 1
Determining exponential function based on output values
Ex. 3 × 3 × 3 × 3 × 6
a, the input value, will be the odd one out: 6
the rest are the common ratio multiplied by however many times
f(x) = 6(3)^{x} where x = 4
Exponential Functions depending on parameters
a > 0; b > 1
 a > 0; 0 < b < 1

a < 0; b > 1
 a < 0; 0 < b < 1

Characteristics of Exponential Functions
They are always increasing or decreasing
There is no extreme (except on closed intervals)
Their graphs are always concave up or concave down
They have no inflection points
If the input values increase or decrease without bound, the endbehavior can be expressed as
lim_{x → ±∞} ab^{x} = 0
lim_{x → ±∞} ab^{x} = ±∞
Additive Transformation of an Exponential Function
g(x) = f(x) + k
If the output values of g are proportional over equallength inputvalue intervals, then f is exponential
Meaning that if g(x) has proportional output values, then f(x) will always be exponential
But if f(x) is proportional, that doesn’t mean g(x) will be proportional
Negative Exponent Property → b^{n} = 1/b^{n}
Product Property → b^{n} * b^{m} = b^{n + m}
Product Property → (b^{m})^{n} = b^{mn}
Transformations of Exponential functions
Horizontal Translations & Vertical Dilations
ac(b)^{x} = b^{k}b^{x} = b^{(x + k)}
Through the product property, when doing a horizontal translation, it is the same as doing a vertical dilation
Every horizontal translation of an exponential function, f(x) = b^{(x + k)}, is equivalent to a vertical dilation
Ex. 1(2)^{(x + 3)} → 1 × 2^{3}(2^{x}) → 8(2)^{x}
Horizontal translation left by 3 units / Vertical dilation of 8 units
Horizontal Dilations
Every horizontal dilation of an exponential function, f(x) = b^{(cx)}, is equivalent to a change of base
(b^{c})^{x} → b^{c} and c ≠ 0
Ex. 2^{3x} → (2^{3})^{x} → 8^{x}
The kth Root of b