AP Precalculus Flashcards

Unit 1 Polynomial and Rational Functions

• Average Rate of Change (AROC) on an interval $[a, b]$:
  • Given two points $(a, f(a))$ and $(b, f(b))$, the AROC is the slope of the secant line.
    AROC = (b) - f(a) \over b-a
  • Secant Line: A line formed between two points $a$ and $b$, using the Average Rate of Change formula.
• Relating Function Behavior, Concavity, and Rate of Change
  • Increasing: positive rate of change
  • Decreasing: negative rate of change
  • Concave Up: rate of change is increasing
  • Concave Down: rate of change is decreasing
  • Point of Inflection: rate of change changes from increasing to decreasing, or vice versa.
• Function Behavior Summary
  • $f(x)$ decreases, rate of change increases: concave up
  • $f(x)$ increases, rate of change increases: concave up
  • $f(x)$ decreases, rate of change decreases: concave down
  • $f(x)$ increases, rate of change decreases: concave down
• Odd/Even Functions
  • Odd: $f(-x) = -f(x)$; passes through the origin.
  • Even: $f(-x) = f(x)$; reflects over the y-axis.
• Important Justification Phrase: "Over equal-length input-value intervals…"

Unit 1 Polynomial and Rational Functions

• Zeros, Multiplicity, and Factors
  • $(x - a)^1$: Crosses x-axis (linearly)
  • $(x - a)^2$: Bounces (quadratically)
  • $(x - a)^3$: Bends (cubically)
• Intercepts
  • x-intercept(s): Let $y = 0$
  • y-intercept: Let $x = 0$
• Complex Conjugate: If $(a + bi)$ is a factor, then $(a - bi)$ is also a factor.
• End Behavior - Limit Notation
  • $\lim_{x \to c} f(x) = L$
• Degree and Leading Coefficients
  • Odd Degree: End behaviors are opposites
  • Even Degree: End behaviors are the same
  • Positive Leading Coefficient: Up on the right
  • Negative Leading Coefficient: Down on the right
• Transformations of Polynomial Functions: $F(x) = a \cdot f[b(x - h)] + k$
  • $a$-value: Vertical dilation by a factor of $a$. Vertical reflection across x-axis for a < 0.
  • $b$-value: Horizontal dilation by a factor of $\frac{1}{b}$
  • $h$-value: Horizontal translation in the direction of $-h$
  • $k$-value: Vertical translation. Up $+k$, Down $-k$
• Analysis of Table Problems
  • Linear Function: Input and output values change consistently.
  • Quadratic Function: Input changes consistently and the 2nd Differences of output values are equal.
  • Cubic Function: Input changes consistently and the 3rd Differences of output values are equal.
  • Exponential Function: Input changes consistently and output values change proportionally.
  • Logarithmic Function: Input values change proportionally and output values change consistently.

Unit 1 Polynomial and Rational Functions

• Asymptotes, Holes, and Discontinuities
  • General Form of Rational Functions: $f(x) = \frac{ax^n + …}{bx^m + …}$
  • Vertical Asymptotes: Occur when a factor in the denominator does not cancel out with factors in the numerator. $x = c$.
    $\lim_{x \to c} R(x) = \pm \infty$
  • Horizontal Asymptotes
    • Case I: If n < m, then $y = 0$
    • Case II: If $n = m$, then $y = \frac{a}{b}$
    • Case III: If n > m, then no Horizontal Asymptote
  • Slant/Oblique Asymptotes: If n > m and the degree of the numerator is one degree larger than the degree of the denominator, use long division (ignoring the remainder) to find the equation of the slant or oblique asymptote.
• Holes: Occur when a common factor cancels out after simplifying the rational expressions. Find the location of the hole by evaluating the simplified expression at the x-value of the hole.
• Pascal’s Triangle - Binomial Expansion
  • Row 0: $(x + y)^0 = 1$
  • Row 1: $(x + y)^1 = x + y$
  • Row 2: $(x + y)^2 = x^2 + 2xy + y^2$
  • Row 3: $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
• Know your Parent Functions
  • $y = x$
  • $y = x^2$
  • $y = x^3$
  • $y = \frac{1}{x}$

Unit 2 Exponential and Logarithmic Functions

• Arithmetic and Geometric Sequences
  • Arithmetic sequences (Linear): have the same differences between terms (add): $d = a<em>2 - a</em>1, a<em>3 - a</em>2, …$
    • $d$ = common difference between consecutive terms
    • $a<em>n = a</em>k + d(n - k)$
    • $a<em>n = a</em>0 + dn$
  • Geometric sequences (Exponential): have common ratios between consecutive terms.
    • Common ratio: $r = \frac{a<em>2}{a</em>1} = \frac{a<em>3}{a</em>2} = \frac{a<em>n}{a</em>{n-1}}$
    • Each term in a geometric sequence can be obtained recursively from its previous term by multiplying by $r$.
    • $g<em>n = g</em>0 (r)^n$
    • $g<em>n = g</em>k (r)^{n-k}$ for $g_k$ is the kth term
• General Form Equations
  • Linear: $f(x) = mx + b$
  • Exponential: $f(x) = a \cdot b^{x-h} + k$
• Properties of Exponents: If a, b, m and n are real numbers with a > 0 and b > 0, then
  • $(ab)^m = a^m \cdot b^m$
  • $(b^m)^n = b^{mn}$
  • $b^m \cdot b^n = b^{m+n}$
  • $b^{\frac{1}{k}} = \sqrt[k]{b}$
  • $b^{-n} = \frac{1}{b^n}$
  • $(\frac{a}{b})^n = \frac{a^n}{b^n}$
• Characteristics of Exponential Functions: $y = a \cdot b^x$
  • For consistent input values, output values change proportionally
  • $a$ and $b$ are non-zero constants; $b \neq 1$
  • Concave up or concave down everywhere
  • Horizontal asymptote at $y = k$
  • Growth for a > 0, b > 1
  • Domain $(-\infty, \infty)$
  • Decay for a > 0, 0 < b < 1
  • Increasing or decreasing everywhere
  • $\lim_{x \to \pm \infty} f(x) = \pm \infty$ or 0

Unit 2 Exponential and Logarithmic Functions

• Converting between Exponential and Logarithmic Form
  • $b^p = n \iff \log_b n = p$
• Characteristics of Logarithmic Functions: $y = \log_b (x - h) + k$
  • Input values change proportionally for consistent output values
  • $a$ and $b$ are non-zero constants; b > 0, b \neq 1
  • Concave up or concave down everywhere
  • Vertical asymptote at $x = h$
  • Growth for a > 0, b > 1
  • Range $(-\infty, \infty)$
  • Decay for a > 0, 0 < b < 1
  • Increasing or decreasing everywhere
  • $\lim_{x \to 0^+} f(x) = \pm \infty$
• Basic Properties of Logarithms (Also valid for base 10 and base e). For b > 0, b \neq 1, x > 0, and any real number $y$
  • $\log_b 1 = 0$
  • $\log_b b = 1$
  • $\log_b b^y = y$
  • $b^{\log_b x} = x$
  • $\ln 1 = 0$
  • $\ln e = 1$
  • $\ln e^y = y$
  • $e^{\ln x} = x$
  • $\lim_{x \to \infty} f(x) = \pm \infty$
• Properties of Logarithms: For any positive numbers $M, N$, and $b$ where $b \neq 1$, and $p$ any real number:
  • Product Property
    • $\log<em>b (MN) = \log</em>b M + \log_b N$
    • $\ln (MN) = \ln M + \ln N$
  • Quotient Property
    • $\log<em>b (\frac{M}{N}) = \log</em>b M - \log_b N$
    • $\ln (\frac{M}{N}) = \ln M - \ln N$
  • Power Property
    • $\log<em>b (N^p) = p \cdot \log</em>b N$
    • $\ln(N)^p = p \cdot \ln N$

Unit 2 Exponential and Logarithmic Functions

• Composition of Functions: $f \circ g (x) = f(g(x))$
• Inverse Functions: A one-to-one function has an inverse. A function in which input values and output values are “switched,” such that an ordered pair for a function $f$ is $(a, b)$ becomes $(b, a)$ on the inverse function $f^{-1}(x)$.
  • Composition of inverse functions “undo” each other: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
  • Note: The domain and range of logarithmic and exponential functions are opposites of each other.
• Semi-Log Plots
  • The y-axis is a logarithmic scale, usually powers of 10.
  • The beginning y-value is $10^0 = 1$
  • A data set that behaves in an exponential model will appear linear when the y-axis is logarithmically scaled.
  • For an exponential function $y = ab^x$, the corresponding linear model for the semi-log plot is $y = \log(b)x + \log(a)$
• Know Your Parent Functions
  • $y = 2^x$
  • $y = e^x$
  • $y = \log_2 x$
  • $y = \ln x$

Unit 3 Trigonometric and Polar Functions

• Know Your Parent Functions
  • $y = x$
  • $y = x^2$
  • $y = x^3$
  • $y = \sqrt{x}$
  • $y = \frac{1}{x}$
  • $y = 2^x$
  • $y = \log_2 x$
  • $y = e^x$
  • $y = \ln x$
• Know Your Trigonometry Functions
  • $y = \sin x$
  • $y = \cos x$
  • $y = \tan x$
  • $y = \csc x$
  • $y = \sec x$
  • $y = \cot x$
• Trigonometric Values
  • $\theta$ (radians): $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi$
  • $\sin \theta: \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1, 0$
  • $\cos \theta: \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0, -1$
  • $\tan \theta: \frac{\sqrt{3}}{3}, 1, \sqrt{3}, DNE, 0$
• Even Functions: If $f(x)$ is even, then $f(-x) = f(x)$
  • $\cos(-\theta) = \cos(\theta)$
  • $\sec(-\theta) = \sec(\theta)$
• Odd Functions: If $f(x)$ is odd, then $f(-x) = -f(x)$
  • $\sin(-\theta) = -\sin(\theta)$
  • $\csc(-\theta) = -\csc(\theta)$
  • $\tan(-\theta) = -\tan(\theta)$
  • $\cot(-\theta) = -\cot(\theta)$
• Where Trig Functions are Positive
  • QI: All functions positive
  • QII: Sine, Cosecant
  • QIII: Tangent, Cotangent
  • QIV: Cosine, Secant

Unit 3 Trigonometric and Polar Functions

• Trig Identities
  • Reciprocals
    • $\csc(x) = \frac{1}{\sin(x)}$
    • $\sec(x) = \frac{1}{\cos(x)}$
    • $\sin(x) \csc(x) = 1$
    • $\cos(x) \sec(x) = 1$
  • Pythagorean
    • $\sin^2 x + \cos^2 x = 1$
    • $1 + \tan^2 x = \sec^2(x)$
    • $\cot^2 x + 1 = \csc^2(x)$
  • Double Angle
    • $\sin 2x = 2 \sin x \cos(x)$
    • $\cos 2x = \cos^2 x - \sin^2(x) = 2 \cos^2 x - 1 = 1 - 2 \sin^2(x)$
• Domain Restrictions on Inverse Trig Functions
  • $y = \sin^{-1}(x)$
  • $y = \cos^{-1}(x)$
  • $y = \tan^{-1}(x)$
• Sum and Difference Formulas
  • $\sin(A + B) = \sin A \cos B + \cos A \sin B$
  • $\sin(A - B) = \sin A \cos B - \cos A \sin B$
  • $\cos(A + B) = \cos A \cos B - \sin A \sin B$
  • $\cos(A - B) = \cos A \cos B + \sin A \sin B$
• $(x, y) = (\cos \theta, \sin \theta)$

Unit 3 Trigonometric and Polar Functions

• Transformations of Sinusoidal Functions: $y = A \cdot \sin(B(x + C)) + D$ or $y = A \cdot \cos(B(x + C)) + D$
  • $\sin x$ is odd; origin symmetry
  • $\cos x$ is even; y-axis symmetry
  • A-value: Amplitude $A = \frac{max - min}{2}$. a < 0 reflects graph over x-axis
  • B-value: Number of cycles in $2\pi$. Period $= \frac{2\pi}{B}$
  • C-value: Phase Shift. Opposite of value in equation
  • D-value: Vertical Shift. $y = D$ is midline. $D = \frac{max + min}{2}$
• Tangent Transformations: $y = A \cdot \tan(B(x + C)) + D$
  • $\tan x$ odd; origin symmetry
  • No amplitude
  • Period $= \frac{\pi}{B}$
  • Phase Shift
  • D-value is location for points of inflection on tangent curve
  • Vertical Asymptotes: $\theta \neq \frac{\pi}{2} + k\pi$ where $k$ is any integer
• Relationships between Polar and Rectangular Coordinates
  • To change from polar to rectangular coordinates, use the formulas:
    • $x = r \cos \theta$
    • $y = r \sin \theta$
  • To change from rectangular to polar coordinates, use the formulas:
    • $x^2 + y^2 = r^2$
    • $\tan \theta = \frac{y}{x} (x \neq 0)$

Unit 3 Trigonometric and Polar Functions

• SPIRAL CURVES: $r(\theta) = \theta$, spiral with increasing radius
• Let $a$ be a positive real number
• CIRCLES with radius $a$. Domain: $[0, 2\pi]$
  • $r = a \cos \theta$
  • $r = a$
  • $r = a \sin \theta$
• CARDIOIDS have the following form where the curve passes through the pole and a > 0, Domain: $[0, 2\pi]$
  • $r = a \pm b \cos \theta$, $\frac{a}{b} = 1$
  • $r = a \pm b \sin \theta$, $\frac{a}{b} = 1$
• CONVEX LIMACON (without an inner loop) have the following form where the curve does not pass through the pole, a > 0, b > 0, and a > b. Domain: $[0, 2\pi]$
  • $r = a \pm b \cos \theta$, \frac{a}{b} > 2
  • $r = a \pm b \sin \theta$, \frac{a}{b} > 2
• LIMACON (with an inner loop) have the following form where the curve passes through the pole twice, a > 0, b > 0, and a < b. Domain: $[0, 2\pi]$
  • $r = a \pm b \cos \theta$, \frac{a}{b} < 1
  • $r = a \pm b \sin \theta$, \frac{a}{b} < 1
• ROSE CURVES have the following form and have graphs that are rose shaped. In $n \neq 0$ is even, the rose has $2n$ petals; if $n \neq \pm 1$ is odd, the rose has $n$ petals, for $a \neq 0$. Domain, $n = even$: $[0, 2\pi]$ Domain, $n = odd$: $[0, \pi]$
  • $r = a \cos n\theta$, $n = 3$, odd
  • $r = a \sin(n\theta)$, $n = 4$, even