NOTE: I accidentally deleted the original; This is supposed to be an “ultimate guide” but I don’t think we can finish it before the AP exam; However, all formulas for all units are posted!! I’ll leave the notes for Unit 1 here in case you want to read them, and will add more notes when they are ready!

Credit to: Shreya Shah, Brooke Vang :)

Important Overall Exam Notes and Tips:

• Repeat what is said in the question

• Read very carefully, and answer only what is asked (especially with limits- it might only ask for one)

__________________________________________________________________________________________________________UNIT 1:

Terms:

• Function: a mathematical relation that maps a set of input values to a set of output values such that each input value is called the domain of the function, and the set of output values is called the range of the function

• Inflection Point: where a curve changes from concave up to concave down or concave down to concave up

• x-intercepts: the x-intercepts of the graph of f, if any, are the solutions of the equation f(x)=0 or (x,0)

• y-intercepts: if x=0 is in the domain of a function y=f(x), then the y-intercept of the graph of f is the value of f and 0, f(0), or (0,y)

• Average Rate of Change (AROC): the average rate of change between any two points on a graph of f(x) is the slope through these points

• Instantaneous Rate of Change: using the average rate of change over small intervals; the rate that the y-values would change if the x-values changed at that point.

• Polynomial Function: a monomial or a sum of monomials

• Turning Point: a point where the graph of a function changes direction from upwards to downwards, or from downwards to upwards

• End Behavior: the direction of the graph of a function as you move to the left and to the right, away from the origin

• Relative/Local Extrema: y-values on the graph of a function where the function changes from increasing to decreasing or decreasing to increasing

• Absolute/Global Extrema: the highest or lowest y-values on the graph of a function or on a specified domain of a function

• Odd Function: must travel through the origin and f(-x)=-f(x)

• Even Function: reflection over y-axis and f(-x)=f(x)

• Imaginary Number: when the value of the radicand is negative, the root is an imaginary number; i=-1

• Complex Number: a number of the form a+bi where a is the real part of the complex number and bi is the imaginary part of the complex number

• Complex Conjugate: if (a+bi) is a factor then (a-bi) is also a factor

Formulas:

Average Rate of Change: AROC=y2-y1x2-x1

1. Find the y-values of the function for the given interval

2. Find the slope between those two points

Binomial Expansion: (ab)n

1. Use Pascal’s Triangle (n+1 terms)

Transformations to a Quadratic: f(x)=a(b(x-h)2+k

• a: possible vertical reflection and/or dilation by a factor of a

• b: possible horizontal reflection and/or dilation by a factor of 1b (must be factored out)

• h: possible horizontal translation in the direction -h

• k: possible vertical translation in the direction k

Important Notes: 

• Rule of Four (G.N.A.W.): four ways to represent a function

  • Graphically: by points on a coordinate plane where input values are on the horizontal axis and output values are on the vertical axis

  • Numerically: by a table or a set of ordered pairs that match input values with output values

  • Algebraically: by an equation in two variables

  • Words: by verbally describing how the input variable is related to the output variable

• Attributes of Polynomials:

  • If the graph of f has a positive rate of change, then the graph of f is increasing

  • If the graph of f has a negative rate of change, then the graph of f is decreasing

  • If the rate of change of f is increasing (slope increases), then the graph of f is concave up

    • The average rate of change over equal-length input-value intervals is increasing, so the graph is concave up

  • If the rate of change of f is decreasing (slope decreases), then the graph of f is concave down

    • The average rate of change over equal-length input-value intervals is decreasing, so the graph is concave down

• For any linear function, the average rate of change over any length input-value interval is constant

  • The average rate of change is changing at a rate of zero

• For any quadratic function, the average rate of change over consecutive equal-length input-value intervals can be given by a linear function

  • The average rate of change is changing at a constant rate

1.1 🠆 Change in Tandem

• If a function f is defined by an equation and no domain is specified, then the domain will be the largest set of real numbers for which the value of f(x) is a real number. Exclude any real numbers from a function’s domain that cause division by zero (undefined) or that result in an even root of a negative number (imaginary number).

• A function increases over an open interval of its domain if its graph rises from left to right on the interval. A function decreases over an open interval of its domain if its graph falls from left to right on the interval. A function is constant over an open interval of its domain if its graph is horizontal on the interval.

1.2 🠆 Rates of Change

• The average rate of change is the slope between any two points on the graph of f(x)

• Instantaneous rate of change is finding the rate of change of a function at a given point

• You can estimate the rate of change at a point by finding the average rate of change over a very small interval

• You can estimate rates of change at two different points by estimating individual rates of change using a tangent line

• The slope of the line tangent to the graph of f at any given point is the true rate of change of f at that point

1.3 🠆 Average Rate of Change in Linear and Quadratic Functions

• For any linear function, the average rate of change over any length input-value interval is constant

  • The average rate of change is changing at a rate of zero

• For any quadratic function, the average rate of change over consecutive equal-length input-value intervals can be given by a linear function

  • The average rate of change is changing at a constant rate

1.4 🠆 Polynomial Functions and Rate of Change

• Degree: the highest exponent (only on polynomials in standard form)Polynomial functions of an even degree will have either a global (absolute) maximum or a global (absolute) minimum

• A function is concave up on an open interval if the graph looks like a “U” or part of a smile

• A function is concave down on an open interval if the graph looks like an upside down “U” or part of a frown

• A function f has a point of inflection at c on an open interval if a graph changes concavity

  • It can occur at 3 different places:

    1. At a zero with a multiplicity of 3

    2. Estimate a possible point of inflection between two consecutive extreme values (at the midpoint between the relative maximum/relative minimum values)

    3. Estimate a possible point of inflection between a zero, multiplicity of 3, and the closest relative extreme value

__________________________________________________________________________________________________________UNIT 2:

Formulas:

Arithmetic (Linear) Sequences: an=a0+dn or an+d(n-k)

1. The common difference is added forward and subtracted backward

Geometric (Exponential) Sequences: gn=g0(r)n or gn=gk(r)n-k

1. The common ratio is multiplied forward and divided backward

Exponential Functions: y=a(b)x

1. a and b are non-zero constants and b1

2. Always has one horizontal asymptote y=k

3. The domain is all real numbers

4. All increasing or all decreasing

5. All concave up or all concave down

6. For consistent input values, output values change proportionally

7. Growth for r>1

8. Decay for 0<r<1

9. End behavior: xg(x)= or xg(x)=0

Exponential Properties: bm+n=bmbn, bmn=(bm)n, b-n=1bn, b1k=kb

Logarithmic Functions: y=alogbx

1. a and b are non-zero constants, b>0, b1

2. Always has one vertical asymptote x=h

3. The range is all real numbers

4. All increasing or all decreasing

5. All concave up or all concave down

6. Input values change proportionately for consistent output values

End behavior: x0+f(x)= or xf(x)=

Lim (function)

x to infinity, -infinity, ha, va, 0 (depends on the function!)

__________________________________________________________________________________________________________UNIT 3:

Formulas:

f(x+k)=f(x)

1. The period of the function is the smallest possible k

2. k is the translation amount or the period

• sin=yr

• cos = xr

• tan = yx or tan = sin cos

• csc=1sin

• sec=1cos

• cot=cossin

• radians = arc lengthradius

• Arc length: r(theta)

• Coordinate Point: (rcos,rsin)

• Period: pd = 2b

• Pythagorean Identity: sin2+cos2 = 1

• Pythagorean Identity: 1+cot2=csc2

• Pythagorean Identity: tan2+1=sec2

• Sine Double Angle: sin(2) = 2sincos

• Cosine Double Angle: cos(2) = cos2-sin2

• Sine Sum: sin(+) = sincos+sincos

• Sine Difference: sin(-) = sincos-sincos

• Cosine Sum: cos(+) = coscos-sinsin

• Cosine Difference: cos(-) = coscos+sinsin

Vertical Asymptotes: Only 4 Trig Functions

• tan: pi/ 2+ kpi

• csc: kpi

• cot: kpi

• sec: pi/2+kpi and 3pi/2+kpi

Polar Formulas:

• Rectangular-Polar: r = sqrt(x² + y²) & tanx= y/x (ADD PI IF x<0!!)

• Polar-Rectangular: (rcosx, rsinx)

Increasing: above the x-axis

Decreasing: below the x-axis