Unit 1

1.1 - Change in Tandem

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…

1. Verbally: as the input values increase, the output values always increase

2. Analytically: for all a and b in the interval ___, if a < b, then f(a) < f(b)

Decreasing function if…

1. Verbally: as the input values increase, the output value always decreases

2. 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…

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)

1.2 - Rates of Change

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

1. Get as close as you can to the point (at least 3 decimal places)

   • (1, 2) & (1.001, 2.0019995)

2. 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)

1.3 - Rates of Change in Linear and Quadratic Functions

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 input-value 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

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 equal-length input-value 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)

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)

1. here the bottom is varying (18 + 2 = 20; 20 + 0 = 20; 20 - 2 = 18; 18 - 4 = 14)

2. Then you would do that again to find a commonality (+2, 0, -2, -4) → ( 0 - 2 = -2; -2 - 0 = -2; -4 - (-2) = -2)

3. 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

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)

1.4 - Polynomial Functions and Rates of Change

Polynomial

p(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + … + a₁x² + a₁x + a₀

a_nxⁿ 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

1.5A - Polynomial Functions and Complex Zeros & 1.5B - Even and Odd Polynomials

If the real zero, a, of (x - a), has an even multiplicity, then the graph will bounce off the x-axis

Example:

• Given the function: (x + 1)²(x - 3)³(x + 4)¹

Complex Zeros - non-real zeros that will always come in pairs (Ex. 3 ± 2i)

• a ± bi

Even functions = symmetrical over the y-axis

• 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) = x⁶ - 4x² → f(-x) = (-x)⁶ - 4(-x)² → f(-x) = x⁶ - 4x²

Odd functions = symmetrical over the origin

• has the property f(-x) = -f(x)

1.6 - Polynomial Functions and End Behavior

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)

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

1.7 - Rational Functions & End Behavior

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)

1.8 - Rational Functions & Zeros

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: (x-int)(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 x-intercept 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

1.9 - Rational Functions & Vertical Asymptotes

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

1. 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)

2. 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)

1.10 - Rational Functions & Holes

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

As you can see, the closer we get to 3, the closer the values get to 2

1.11A - Equivalent Representations & Binomial Theorem & 1.11B - Polynomial Long Division & Slant Asymptotes

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 = C₁aⁿb⁰ + C₂a^{n - 1}b¹ + C₃a^{n - 2}b² + … + C_{n - 1}a¹b^{n - 1} + C_na⁰bⁿ

  • 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

  • 1x³5⁰ + 3x²5¹ + 3x¹5² + 1x⁰5³ → 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)

1.12 - Transformations of Functions

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 y-values change, the x-values do not

  • Multiply the y-values 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 x-values change, the y-values do not

  • Multiply the x-values by the inverse of b

  • 1/b > 1: horizontal shrink

  • 1/b < 1: horizontal stretch

    

Reflections

• -f(x) → reflect over x-axis → vertical change

• f(-x) → reflect over y-axis → horizontal change

Transformations Numerically

Ex. let g(x) = 3f(x - 2) + 1, find g(3)

1. g(x) = 3f(x - 2) + 1 → set up equation

2. g(3) = 3f(3 - 2) + 1 → plug in the 3 for x in the modified equation

3. g(3) = 3f(1) + 1 → simplify

4. g(3) = 3(-12) + 1 → find what f(1) is and substitute that in

5. 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

1.13 - Function Model Selection

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

  

1.14 Function Model Construction

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.

Unit 2

2.1 - Change in Arithmetic and Geometric Sequences

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 = a₀ + dn

  • a₀ 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 = g₀rⁿ

  • g₀ is the initial value (zero term) and r is the common ratio

    

• Equation for any term → g_n = g_kr^{(n - k)}

  • g_k is the k^th term of the sequence

    

2.2 - Change in Linear and Exponential Functions

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 (x₁, y₁) of a linear function is similar to (k, a_k) of an arithmetic sequence

Exponential Functions vs Geometric Sequences

• Again, they are basically the same except with discrete points for the sequence

• The point (x₁, y₁) of an exponential function is similar to (k, g_k) of a geometric sequence

How each output value changes analytically

• Arithmetic → Over equal-length 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 equal-length input value intervals, if the output values of a function change proportionally, then the function is exponential (multiplying the ratio)

• “Over equal-length input intervals” means that the x-values 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

1. Make sure the function is increasing over equal-length input intervals

2. Test to see whether the function is linear or exponential

   1. Linear functions change by adding or subtracting a constant amount

   2. Exponential functions change by multiplying or dividing a constant amount

3. This example is increasing x2 from 8-16 and 16-32

4. So this function is exponential because over equal-length intervals of 1, f(x) proportionally changes by a ratio of 2

   1. If this was linear you could say, the function is linear because over equal-length 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. Find how many times r would have to be multiplied to get from 4 to 16 (in this case it is 2)

2. This means we would have to multiply 4 ⋅ r ⋅ r to get 16 → 4r² = 16

3. Then, you solve for r → r² = 16/4 = 4 → r = √4 = 2

4. So r = 2

• OR… given the points (a, b) & (c, d), you can find the common ratio using this equation →

2.3 - Exponential Functions

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 Exponential growth Increasing Ex. f(x) = 1(2)x	a > 0; 0 < b < 1 Exponential decay Decreasing Ex. f(x) = 1(0.5)x
a < 0; b > 1 a < 0, so the function remains below 0 Decreasing Ex. f(x) = -2x	a < 0; 0 < b < 1 a < 0, so the function remains below 0 Increasing Ex. f(x) = -(0.5)x

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 end-behavior 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 equal-length input-value 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

    

2.4 - Exponential Function Manipulation

Negative Exponent Property → b^-n = 1/bⁿ

Product Property → bⁿ * b^m = b^{n + m}

Product Property → (b^m)ⁿ = b^mn

Transformations of Exponential functions

Horizontal Translations & Vertical Dilations

ac(b)^x = b^kb^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 × 2³(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 → (2³)^x → 8^x

The kth Root of b

2.5 - Exponential Function Context & Data Modeling

Finding the exponential function in the form f(x) = ab^x + k through a table

1. Find the ratio (b) (in this case b=2)

   1. Find the differences

      • 9-6 = +3; 15-9= +6; 27-15= +12; 51-27= +24

   2. Then, you can find the ratio

      1. 3 × 2 = 6 × 2 = 12 × 2 = 24

2. Then find a and k by setting up a system of equations

   • 6 = a(2)⁰ + k → 6 = a + k

   • 9 = a(2)¹ + k → 9 = 2a + k

     • 9 = 2a + k

     • -6 = -a - k

     • a = 3

   • 6 = 3 + k → k = 3

The number e

• e ≈ 2.718…

• The natural base e is often used as the base in exponential functions that model contextual scenarios

Percent Increase:

• Start with the initial amount a and grow with a % increase

• y = a(1 + %increase)^x

Percent Decrease:

• Start with the initial amount a and decay with a % decrease

• y = a(1 - %decrease)^x

Half-Life:

• An initial amount a that shrinks by half every h

• f(t) = a(1/2)^t/h

Doubling time:

• An initial amount a that doubles every d

• f(t) = a(2)^t/d

Equivalent Forms

• If t represents the number of weeks, then the base of f(t) = (1.01)^t indicates that the quantity increases by a factor of 1.01 every week

  • If you want the equation for every year, then that equation would look like this → (1.01⁵²)^t/52

    • This is because every 52 weeks, t = 52/52, means that it would account for 1 year

    • since we’re counting years, 1.01⁵² means that in 1 year, we’d have (1.01 × 1.01 × 1.01 … 1.01) 52 times per year

    • So when t = 52 weeks / 52 (one year), 1.01 would’ve increased 52 times

  • If you wanted an equation for every day, then it would look like this → (1.01^1/7)^t

    • This is because once t is > 7, the function will equal 1.01 which is the same as if it where (1.01)¹ weeks

2.6 - Competing Function Model Validation

The only way to know what model fits an equation best is to use context clues

Residual Value - a measure of how much a regression curve vertically misses a data point

Residuals → Actual - Predicted (AP)

The difference between the predicted and actual values is the error

Error → Predicted - Actual (PA)

• A positive error means that the model overestimated (you can see the y=0 line is above the point)

• A negative error means that the model underestimated (you can see the y=0 line is below the point

Residual Plot Patterns

If there is a clear pattern in the residual plot, it is a bad regression model

2.7 - Composition of Functions

The composition of functions means we take one function and substitute it into the other function

• f(g(x) or f ∘ g(x)

• Ex. f(x) = 2x + 3; g(x) = 5x²; find f(g(5))

  • 5(5)² → 5(25) → 125

  • 2(125) + 3 = 253

Identity Function

• When f(x) = x

• then f(g(x)) = g(x)

Finding a Composed Function

• f(x) = x² - 1 & g(x) = √x ; find f(g(x))

  • h(x) = (√x)² -1

  • h(x) = x - 1

Domain of a Composed Function

• When finding the domain of a function h(x) = f(g(x)), there are 2 things to consider

  1. because g(x) is the input, we can only use x-values in the domain of g

  2. any restrictions on the domain of f(g(x)) must be included

• Ex. problem above in Finding a Composed Function

  • g(x) is the input, so it can only be from [0, ∞)

  • and there are no restrictions in x - 1, so the domain is [0, ∞)

Decomposition of Functions

• if there is a radical, one function can simply be √x (this applies with 1/x as well)

• Ex. h(x) = √x³ + 1 ; find f(g(x))

  • f(x) = √x

  • g(x) = x³ + 1

• Ex2. h(x) = 1/(x² + 1) ; find g(f(x))

  • f(x) = x² + 1

  • g(x) = 1/x

2.8 - Inverse Functions

The inverse of a function is a reverse mapping of a function where (x, y) → (y, x)

They should reflect across the line y=x

If the graph turns (has a max/min) then it is no longer invertible because there will be output values that are the same for different input values

Find the Inverse of a function

1. instead of (for example) y = mx + b, switch the x and y so it is → x = my +b

2. solve for y

Finding the domain and range

• The domain and range swap for inverse functions

  

Composition of f and f^-1

• f(f^-1(x)) = f^-1(f(x)) = x

• If both functions, when composed with each other, cancel out, then they are inverses of each other