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AP Precalculus Guide

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…

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)


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

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 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)

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)

  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

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)


1.4 - Polynomial Functions and Rates of Change

Polynomial

p(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x2 + a1x + a0

anxn is the leading term | n is the degree | an 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)

Left Side

Right Side

Up

limx → -∞ f(x) = ∞

limx → ∞ f(x) = ∞

Down

limx → -∞ f(x) = -∞

limx → ∞ f(x) = -∞

Left side

x → -∞

Right Side

x → ∞

Even degree and Positive leading coefficient

limx → -∞ f(x) = ∞

limx → ∞ f(x) = ∞

Even degree and Negative leading coefficient

limx → -∞ f(x) = -∞

limx → ∞ 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

  • limx → -∞ f(x) = 0 | limx → ∞ 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

Rules of horizontal asymptotes

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

    • limx → 2- f(x) = -∞

  • 2+ means approaching the vertical asymptote on the right side from right to left

    • limx → 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

(x+3)(x-3)/x(x-3)

Writing limits for holes:

  • From the left:

    • As x approaches 3 from the left, f(x) approaches 2

    • limx → 3- f(x) = 2

  • From the right:

    • As x approaches 3 from the right, f(x) approaches 2

    • limx → 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


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

Each number is made by adding the 2 numbers above it

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 + C2an - 1b1 + C3an - 2b2 + … + Cn - 1a1bn - 1 + Cna0bn

    • 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

asymptote at <br />y = x + 1

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

  • limx → f(x) =

  • limx → - 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

x

0

1

2

3

4

f(x)

-20

-12

0

8

14

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 → an = a0 + dn

    • a0 is the initial value (zero term) and d is the common difference

  • Equation for any term → an = ak + d(n - k)

    • ak is the kth 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 → gn = g0rn

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

  • Equation for any term → gn = gkr(n - k)

    • gk is the kth 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, ak) of an arithmetic sequence

Slope-intercept form

f(x) = b + mx

Arithmetic Sequence

an = a0 + dn

Point-slope form

f(x) = y1 + m(x - x1)

A.S. for the kth term

an = ak + d(n - k)

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, gk) of a geometric sequence

Exponential Function

f(x) = abx

Geometric Sequence

gn = g0rn

Shifted Exponential Function

f(x) = y1r(x - x1)

G.S. for the kth term

gn = gkr(n - k)

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

x

5

6

7

f(x)

8

16

32

  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

2

3

4

16

  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) = abx

  • 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

    • limx → ±∞ abx = 0

    • limx → ±∞ abx = ±∞

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

      6/4 ≠ 10/6<br />4/2 = 8/4

2.4 - Exponential Function Manipulation

Negative Exponent Property b-n = 1/bn

Product Property bn * bm = bn + m

Product Property(bm)n = bmn

Transformations of Exponential functions

Horizontal Translations & Vertical Dilations

ac(b)x = bkbx = 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(2x) → 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

    • (bc)x → bc and c ≠ 0

    • Ex. 23x → (23)x → 8x

The kth Root of b

V

AP Precalculus Guide

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…

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)


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

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 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)

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)

  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

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)


1.4 - Polynomial Functions and Rates of Change

Polynomial

p(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x2 + a1x + a0

anxn is the leading term | n is the degree | an 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)

Left Side

Right Side

Up

limx → -∞ f(x) = ∞

limx → ∞ f(x) = ∞

Down

limx → -∞ f(x) = -∞

limx → ∞ f(x) = -∞

Left side

x → -∞

Right Side

x → ∞

Even degree and Positive leading coefficient

limx → -∞ f(x) = ∞

limx → ∞ f(x) = ∞

Even degree and Negative leading coefficient

limx → -∞ f(x) = -∞

limx → ∞ 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

  • limx → -∞ f(x) = 0 | limx → ∞ 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

Rules of horizontal asymptotes

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

    • limx → 2- f(x) = -∞

  • 2+ means approaching the vertical asymptote on the right side from right to left

    • limx → 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

(x+3)(x-3)/x(x-3)

Writing limits for holes:

  • From the left:

    • As x approaches 3 from the left, f(x) approaches 2

    • limx → 3- f(x) = 2

  • From the right:

    • As x approaches 3 from the right, f(x) approaches 2

    • limx → 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


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

Each number is made by adding the 2 numbers above it

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 + C2an - 1b1 + C3an - 2b2 + … + Cn - 1a1bn - 1 + Cna0bn

    • 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

asymptote at <br />y = x + 1

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

  • limx → f(x) =

  • limx → - 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

x

0

1

2

3

4

f(x)

-20

-12

0

8

14

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 → an = a0 + dn

    • a0 is the initial value (zero term) and d is the common difference

  • Equation for any term → an = ak + d(n - k)

    • ak is the kth 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 → gn = g0rn

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

  • Equation for any term → gn = gkr(n - k)

    • gk is the kth 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, ak) of an arithmetic sequence

Slope-intercept form

f(x) = b + mx

Arithmetic Sequence

an = a0 + dn

Point-slope form

f(x) = y1 + m(x - x1)

A.S. for the kth term

an = ak + d(n - k)

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, gk) of a geometric sequence

Exponential Function

f(x) = abx

Geometric Sequence

gn = g0rn

Shifted Exponential Function

f(x) = y1r(x - x1)

G.S. for the kth term

gn = gkr(n - k)

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

x

5

6

7

f(x)

8

16

32

  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

2

3

4

16

  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) = abx

  • 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

    • limx → ±∞ abx = 0

    • limx → ±∞ abx = ±∞

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

      6/4 ≠ 10/6<br />4/2 = 8/4

2.4 - Exponential Function Manipulation

Negative Exponent Property b-n = 1/bn

Product Property bn * bm = bn + m

Product Property(bm)n = bmn

Transformations of Exponential functions

Horizontal Translations & Vertical Dilations

ac(b)x = bkbx = 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(2x) → 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

    • (bc)x → bc and c ≠ 0

    • Ex. 23x → (23)x → 8x

The kth Root of b

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