APPRECALC EXAM FULL STUDYHGU

Calculator: Essential Mode

Ensure calculator is in RADIAN mode for all Trig and Polar questions.

Calculator: Storing a Function

Press [Y=] to enter the function. On the home screen, use [ALPHA] + [TRACE] to call Y1 for evaluations (e.g., Y1(5)).

Calculator: Finding Zeros

[2nd] + [TRACE] > 2: Zero. Set Left Bound, Right Bound, and Guess to find x-intercepts.

Calculator: Finding Max/Min

[2nd] + [TRACE] > 3: Minimum or 4: Maximum. Useful for finding peaks/valleys of polynomial or trig models.

Calculator: Regression Steps

1. [STAT] > [EDIT] to enter data in L1/L2. 2. [STAT] > [CALC] to choose model (LinReg, ExpReg, SinReg).

Calculator: Intersection of Curves

[2nd] + [TRACE] > 5: Intersect. Select both curves and guess to find where two functions are equal.

Calculator: Residual Plots

Go to [STAT PLOT]. Set Xlist to L1 and Ylist to RESID (found in the [LIST] menu). Check for patterns.

AP Exam: Rounding Rule

Round all final decimal answers to at least THREE decimal places unless the problem specifies otherwise.

AP Exam: Show the Setup

Always write the equation or expression you are solving (e.g., write the AROC formula) before giving the numerical answer.

AP Exam: Justification Logic

Avoid 'calculator speak.' Instead of saying 'I used 2nd Trace,' say 'The function has a maximum of Y because it changes from increasing to decreasing at X.'

AP Exam: Interval Awareness

Check if the prompt limits your answer to an interval (e.g., 0 ≤ x < 2π). Don't give answers outside the requested range.

AP Exam: MCQ Guessing

There is no penalty for guessing. Never leave a bubble blank on the Multiple Choice section.

Identifying Models from Tables

Constant first differences = Linear. Constant ratios = Exponential. Constant second differences = Quadratic.

Difference of Cubes Formula

a³ + b³ = (a + b)(a² - ab + b²) AND a³ - b³ = (a - b)(a² + ab + b²).

Synthetic Division Steps

Use the root 'c' from (x-c). Bring down first coeff, multiply by c, add to next coeff, repeat until the remainder.

Polar: Converting z = x + yi

Convert to r(cosθ + i sinθ). r = √(x² + y²) and θ = tan⁻¹(y/x).

Polar: Rate of Change Calculation

Treated like slope: use the change in r-values over the change in θ-values. r is like 'y', θ is like 'x'.

Rate of Change: UP vs DOWN Tracing

UP tracing on a polar graph = distance from pole is increasing. DOWN tracing = distance is decreasing.

Limaçon Ratios

r = a + b cosθ. If a/b < 1, there is an inner loop. If a/b = 1, it is a Cardiod (heart-shaped).

Rose Curve Petal Rule

r = a cos(nθ). If n is odd, there are 'n' petals. If n is even, there are '2n' petals.

Rational Function: Slant Asymptote

Occurs when degree of numerator is exactly one more than the denominator. Found using long division.

Inverse Trig: Range of Arccos

The output angle for cos⁻¹(x) must be between 0 and π (inclusive).

Inverse Trig: Range of Arcsin/Arctan

The output angle must be between -π/2 and π/2.

Semi-Log Plot Interpretation

A linear trend on a semi-log plot (y-axis is log-scale) confirms the original data follows an exponential model.

Concavity Definition

Concave Up: Rate of change is increasing. Concave Down: Rate of change is decreasing.

Factoring: Difference of Cubes

a³ + b³ = (a + b)(a² - ab + b²); a³ - b³ = (a - b)(a² + ab + b²).

Factoring by Grouping

Used for 4 terms. Split in half, factor out the GCF of each side, then factor out the common binomial.

Synthetic Division: Setup

Use the root (c) from the divisor (x - c). Bring down the first coefficient, multiply by c, add to the next coefficient, repeat.

Remainder Theorem

If a polynomial f(x) is divided by (x - c), the remainder is f(c).

Rational Root Theorem

Possible rational roots = ±(factors of constant term / factors of leading coefficient).

End Behavior: Even Degree (+ Leading Coeff)

As x → ∞, f(x) → ∞; As x → -∞, f(x) → ∞ (Both ends up).

End Behavior: Odd Degree (+ Leading Coeff)

As x → ∞, f(x) → ∞; As x → -∞, f(x) → -∞ (Down on left, up on right).

Multiplicity: Bounce vs Cross

Even multiplicity (e.g., (x-2)²): Bounces. Odd multiplicity (e.g., (x-2)³): Crosses.

Rational Function: Domain

All real numbers except where the denominator equals zero.

Vertical Asymptote (VA)

Occurs at x-values that make the denominator zero but do NOT cancel out.

Hole (Removable Discontinuity)

Occurs at x-values that make both numerator and denominator zero (factors that cancel).

Horizontal Asymptote (HA): Case 1

Degree of numerator < Degree of denominator: HA is y = 0.

Horizontal Asymptote (HA): Case 2

Degree of numerator = Degree of denominator: HA is y = (lead coeff / lead coeff).

Slant (Oblique) Asymptote

Occurs when numerator degree is exactly 1 higher than denominator degree. Found via long division.

Rational Function: x-intercepts

Set the numerator equal to zero (after canceling holes) and solve for x.

Exponential Form to Log Form

b^x = y is equivalent to log_b(y) = x.

Natural Log (ln)

A logarithm with base 'e' (approx 2.718). ln(e) = 1.

Log Property: Product Rule

log_b(mn) = log_b(m) + log_b(n).

Log Property: Quotient Rule

log_b(m/n) = log_b(m) - log_b(n).

Log Property: Power Rule

log_b(m^p) = p * log_b(m).

Change of Base Formula

log_b(a) = log(a) / log(b) or ln(a) / ln(b).

Exponential Growth Model

A(t) = P(1 + r)^t where r is the growth rate.

Semi-Log Graph: Purpose

If a set of (x, y) data points forms a straight line on a semi-log plot, the relationship is exponential.

Unit Circle: Quadrant 2 Signs

Sine (+), Cosine (-), Tangent (-).

Unit Circle: Quadrant 3 Signs

Sine (-), Cosine (-), Tangent (+).

Unit Circle: Quadrant 4 Signs

Sine (-), Cosine (+), Tangent (-).

Reference Angle

The acute angle formed by the terminal side and the x-axis.

Pythagorean Identity 1

sin²θ + cos²θ = 1

Pythagorean Identity 2

1 + tan²θ = sec²θ

Pythagorean Identity 3

1 + cot²θ = csc²θ

Double Angle: sin(2θ)

sin(2θ) = 2sinθcosθ

Double Angle: cos(2θ)

cos(2θ) = cos²θ - sin²θ OR 2cos²θ - 1 OR 1 - 2sin²θ

Sum Identity: sin(A + B)

sin(A)cos(B) + cos(A)sin(B)

Difference Identity: cos(A - B)

cos(A)cos(B) + sin(A)sin(B)

Trig Transformation: Amplitude

|a| in y = a sin(bx). The half-distance from max to min.

Trig Transformation: Period (Sin/Cos)

Period = 2π / |b|

Trig Transformation: Period (Tan/Cot)

Period = π / |b|

Inverse Sine Domain Restriction

[-π/2, π/2] or [-90°, 90°]

Inverse Cosine Domain Restriction

[0, π] or [0°, 180°]

Polar Coordinate: r

r = √(x² + y²). The directed distance from the pole.

Polar Coordinate: θ

θ = tan⁻¹(y/x). Be careful with the quadrant!

Polar to Rectangular: x

x = r cosθ

Polar to Rectangular: y

y = r sinθ

Complex Number: Polar Form

z = r(cosθ + i sinθ)

Rose Curve: Petals

r = a cos(nθ). If n is odd: n petals. If n is even: 2n petals.

Cardioid

A limaçon where a = b (e.g., r = 2 + 2sinθ). Heart-shaped.

Limaçon with Inner Loop

A limaçon where a < b (e.g., r = 1 + 2cosθ).

Lemniscate

r² = a² cos(2θ). Looks like a figure-eight or infinity symbol.

Polar Rate of Change: Increasing Distance

Distance from pole is increasing if: (r > 0 and r' > 0) OR (r < 0 and r' < 0).

Polar Rate of Change: Tracing

UP tracing on a graph means distance from the pole is increasing.

Average Rate of Change (AROC)

Slope formula: [f(b) - f(a)] / [b - a].

Concavity Change

Occurs at a Point of Inflection. Rate of change switches from increasing to decreasing (or vice versa).

Residual

Actual y-value minus Predicted y-value (y - y_hat).

Linear Regression Appropriateness

A model is appropriate if the residual plot shows a random scatter of points with no pattern.

Term

Definition

Polynomial Functions: Graphical Features

Point of Inflection: Concavity changes. Global Max/Min: Absolute highest/lowest. Local Max/Min: Relative peaks/valleys. Zero: x-intercept.

Zeros of Functions: Graphical Behavior

Crosses x-axis: Factor has an ODD exponent. Bounces off x-axis: Factor has an EVEN exponent.

Rational Functions: Holes

Occur if a factor cancels out from both the numerator and denominator after factoring.

Rational Functions: Asymptotes

Vertical: Solve Denominator=0 (after canceling). Horizontal: Compare degrees. numdenom (Slant asymptote/Long Division).

Concavity: Graphical & Numerical

Concave Up: 'Like a Cup' (rate of change is increasing). Concave Down: 'Like a Frown' (rate of change is decreasing).

Rate of Change (Average)

Formula: (f(b) - f(a)) / (b - a). Verbal: 'For every increase in x, there is an average change in y'.

Identifying Functions (Differences)

1st Differences equal: Linear. 2nd Differences equal: Quadratic. 3rd Differences equal: Cubic.

Exponential Graphs

r > 1: Growth (Increasing/Concave Up). 0 < r < 1: Decay (Decreasing/Concave Up). Horizontal Asymptote usually at y=0.

Logarithm Basics

log(x) is base 10. ln(x) is base e. log_b(a) = x means b^x = a.

Logarithm Rules

Product: log(mn)=log(m)+log(n). Quotient: log(m/n)=log(m)-log(n). Power: log(m^p)=p*log(m).

Solving Exp/Log Equations

Get common bases or use logs. ALWAYS check for extraneous solutions (cannot take the log of a negative number).

Semi-Log Plots

The y-axis is scaled by powers of 10. If an exponential function is plotted on a semi-log scale, it appears LINEAR.

Binomial Theorem

Coefficients come from Pascal's Triangle. For (Ax+By)^n, the first term's power decreases from n to 0 while the second term's power increases.

Trig: Unit Circle Basics

Sin = y-coord. Cos = x-coord. Tan = y/x (Slope). Radians: 180° = π.

Trig Signs (ASTC)

Q1: All (+). Q2: Sine (+). Q3: Tangent (+). Q4: Cosine (+).

Sine vs. Cosine Graphs

Sine starts at midline (0,0). Cosine starts at max (0,1). Both have period 2π.

Trig Transformations

f(x) = a*sin(b(x+c)) + d. |a|=Amplitude. Period = 2π/|b|. c=Phase Shift. d=Midline/Vertical Shift.

Tangent Graph

Period = π. Vertical asymptotes at odd multiples of π/2. Zeros at multiples of π.

Inverse Trig Restricted Domains

arcsin/arctan: [-π/2, π/2]. arccos: [0, π]. Used to find angles from ratios.

Polar Coordinates (r, θ)

r = distance from pole. θ = angle from polar axis. To Rectangular: x=r cosθ, y=r sinθ.