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Praxis 5165 Flashcards

Rational number

A number that can be written as a ratio of two integers $\dfrac{a}{b}$; every terminating or repeating decimal is rational.

Irrational number

A real number that cannot be written as a ratio of integers; its decimal never ends or repeats — e.g. $\sqrt{2}$, $\pi$, $e$.

Integers

The whole numbers and their negatives: $\dots, -2, -1, 0, 1, 2, \dots$. Closed under addition, subtraction, and multiplication.

Real numbers

All rational and irrational numbers together — every point on the number line.

Imaginary unit $i$

$i = \sqrt{-1}$, so $i^2 = -1$. It extends the reals to the complex numbers.

Powers of $i$

Cycle with period 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then repeat. Use $n \bmod 4$.

Complex number

A number $a + bi$ with real part $a$ and imaginary part $b$.

Adding complex numbers

Add real and imaginary parts separately: $(a+bi)+(c+di) = (a+c)+(b+d)i$.

Multiplying complex numbers

Use the distributive property (FOIL) and $i^2 = -1$: $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$.

Complex conjugate

The conjugate of $a + bi$ is $a - bi$; their product $a^2 + b^2$ is real.

Modulus of a complex number

$|a + bi| = \sqrt{a^2 + b^2}$ — its distance from the origin in the complex plane.

Absolute value

The distance of a number from 0: $|x| = x$ if $x \ge 0$, else $-x$. Always nonnegative.

Prime number

A whole number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, 11, ...).

Prime factorization

Writing a number as a product of primes, e.g. $84 = 2^2 \cdot 3 \cdot 7$.

Greatest common divisor (GCD)

The largest integer dividing two numbers; the product of their shared prime factors.

Least common multiple (LCM)

The smallest positive integer that is a multiple of both numbers.

Order of operations (PEMDAS)

Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Scientific notation

A number written as $a \times 10^n$ with $1 \le |a| < 10$, e.g. $3.2 \times 10^4 = 32{,}000$.

Square root of a non-perfect-square

Always irrational, e.g. $\sqrt{50} = 5\sqrt{2}$.

Simplifying a radical

Factor out perfect squares: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

Rationalizing a denominator

Multiply by a form of 1 to clear a radical: $\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$.

Vector

A quantity with magnitude and direction, written $\langle a, b \rangle$. Added component-wise.

Vector magnitude

The length of $\langle a, b \rangle$ is $\sqrt{a^2 + b^2}$.

Scalar multiplication of a vector

Multiply each component by the scalar: $k\langle a, b \rangle = \langle ka, kb \rangle$.

Matrix

A rectangular array of numbers; added and scaled entry-by-entry when dimensions match.

Matrix multiplication

Row-by-column dot products; defined only when the first matrix's columns equal the second's rows.

Determinant of a 2×2 matrix

For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$.

Identity matrix

A square matrix with 1s on the diagonal and 0s elsewhere; $AI = A$.

Closure (of a set under an operation)

A set is closed if applying the operation to its members always yields a member of the set.

Slope

Steepness of a line: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ (rise over run).

Slope-intercept form

$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Point-slope form

$y - y_1 = m(x - x_1)$, used to write a line from a slope and one point.

Standard form of a line

$Ax + By = C$, with $A$, $B$, $C$ integers.

Parallel lines

Lines with equal slopes; they never intersect.

Perpendicular lines

Lines whose slopes are negative reciprocals; their product is $-1$.

Horizontal line slope

Slope 0; equation $y = c$.

Vertical line slope

Undefined slope; equation $x = c$.

System of equations

Two or more equations solved together; the solution satisfies all of them — where graphs intersect.

Solving a system by substitution

Solve one equation for a variable and substitute into the other.

Solving a system by elimination

Add or subtract the equations to cancel a variable.

System with no solution

The lines are parallel — same slope, different intercept.

System with infinite solutions

The equations are multiples of each other — the same line.

Quadratic equation

An equation of the form $ax^2 + bx + c = 0$ with $a \ne 0$.

Quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which solves any quadratic.

Discriminant

$b^2 - 4ac$. Positive → two real roots; zero → one; negative → two complex roots.

Factoring a quadratic

Write $ax^2 + bx + c$ as a product of two binomials; set each factor to 0 for the roots.

Completing the square

Rewrite $ax^2 + bx + c$ as $a(x - h)^2 + k$ to find the vertex or solve.

Sum and product of roots

For $ax^2 + bx + c = 0$: sum $= -\dfrac{b}{a}$, product $= \dfrac{c}{a}$.

Difference of squares

$a^2 - b^2 = (a+b)(a-b)$.

Perfect-square trinomial

$(a+b)^2 = a^2 + 2ab + b^2$ and $(a-b)^2 = a^2 - 2ab + b^2$.

Product of exponents (same base)

$x^m \cdot x^n = x^{m+n}$.

Quotient of exponents (same base)

$\dfrac{x^m}{x^n} = x^{m-n}$.

Power of a power

$(x^m)^n = x^{mn}$.

Negative exponent

$x^{-n} = \dfrac{1}{x^n}$.

Zero exponent

$x^0 = 1$ for any nonzero $x$.

Fractional exponent

$x^{1/n} = \sqrt[n]{x}$ and $x^{m/n} = \sqrt[n]{x^m}$.

Solving inequalities

Solve like equations, but flip the inequality sign when multiplying or dividing by a negative.

Factor theorem

$(x - r)$ is a factor of polynomial $p(x)$ if and only if $p(r) = 0$.

Remainder theorem

The remainder when $p(x)$ is divided by $(x - r)$ is $p(r)$.

Rational expression

A ratio of polynomials; simplify by factoring and canceling common factors.

Adding rational expressions

Find a common denominator, combine numerators, then simplify.

Solving a radical equation

Isolate the radical, square both sides, then check for extraneous solutions.

Direct variation

$y = kx$; $y$ increases proportionally with $x$, constant $k$.

Inverse variation

$y = \dfrac{k}{x}$; $y$ decreases as $x$ increases, constant product $k$.

Arithmetic sequence

Constant common difference $d$: $a_n = a_1 + (n-1)d$.

Geometric sequence

Constant common ratio $r$: $a_n = a_1 \cdot r^{n-1}$.

Sum of an arithmetic series

$S_n = \dfrac{n}{2}(a_1 + a_n)$.

Absolute-value equation

$|x| = a$ (with $a \ge 0$) splits into $x = a$ or $x = -a$.

Cross-multiplication

From $\dfrac{a}{b} = \dfrac{c}{d}$, get $ad = bc$ to solve a proportion.

Function

A rule assigning each input exactly one output; passes the vertical-line test.

Function notation

$f(x)$ is the output for input $x$; $f(3)$ means evaluate at $x = 3$.

Domain of a function

The set of all allowed inputs (x-values).

Range of a function

The set of all resulting outputs (y-values).

Vertical-line test

A graph represents a function if no vertical line crosses it more than once.

Composition of functions

$(f \circ g)(x) = f(g(x))$; apply $g$ first, then $f$.

Inverse function

$f^{-1}$ undoes $f$: $f^{-1}(f(x)) = x$; its graph reflects $f$ over $y = x$.

Finding an inverse

Swap $x$ and $y$, then solve for $y$.

Even function

Symmetric about the y-axis: $f(-x) = f(x)$, e.g. $x^2$.

Odd function

Symmetric about the origin: $f(-x) = -f(x)$, e.g. $x^3$.

Vertical shift

$f(x) + k$ shifts the graph up $k$ units (down if $k < 0$).

Horizontal shift

$f(x - h)$ shifts the graph right $h$ units (left if $h < 0$).

Reflection over the x-axis

$-f(x)$ flips the graph vertically.

Vertical stretch

$a \cdot f(x)$ with $a > 1$ stretches the graph away from the x-axis.

Quadratic function graph

A parabola; opens up if $a > 0$, down if $a < 0$.

Vertex of a parabola

At $x = -\dfrac{b}{2a}$; the maximum or minimum point.

Axis of symmetry

The vertical line $x = -\dfrac{b}{2a}$ through a parabola's vertex.

Exponential function

$y = a \cdot b^x$; growth when $b > 1$, decay when $0 < b < 1$.

Exponential growth vs. decay factor

A 5% growth gives $b = 1.05$; a 5% decay gives $b = 0.95$.

Logarithm

The inverse of exponentiation: $b^x = y \iff \log_b y = x$.

Product rule for logs

$\log_b(xy) = \log_b x + \log_b y$.

Power rule for logs

$\log_b(x^n) = n \log_b x$.

Natural log and $e$

$\ln x = \log_e x$, where $e \approx 2.718$.

Asymptote

A line a graph approaches but never reaches; rational and exponential functions have them.

Vertical asymptote of a rational function

Occurs where the denominator is 0 but the numerator is not.

Zero (root) of a function

An input $x$ where $f(x) = 0$; the x-intercept of the graph.

y-intercept

The output when $x = 0$; the point $(0, f(0))$.

Piecewise function

A function defined by different rules over different parts of its domain.

Period of a sine/cosine function

For $y = \sin(bx)$, the period is $\dfrac{2\pi}{b}$.

Amplitude of a sinusoid

For $y = a\sin(bx)$, the amplitude is $|a|$ — half the peak-to-trough distance.

Increasing vs. decreasing function

Increasing where outputs rise as $x$ rises; decreasing where they fall.

End behavior of a polynomial

Determined by the leading term's degree and sign of its coefficient.

Limit

The value $f$ approaches as $x \to a$: $\lim_{x \to a} f(x)$.

Continuity at a point

$f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$ and both exist.

Indeterminate form 0/0

Resolve by factoring, rationalizing, or L'Hôpital's rule before evaluating the limit.

Derivative

The instantaneous rate of change and slope of the tangent line: $f'(x)$.

Limit definition of the derivative

$f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$.

Power rule

$\dfrac{d}{dx} x^n = n x^{n-1}$.

Constant multiple rule

$\dfrac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$.

Sum rule (derivatives)

$(f + g)' = f' + g'$.

Product rule

$(fg)' = f'g + fg'$.

Quotient rule

$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$.

Chain rule

$\dfrac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$.

Derivative of $\sin x$

$\dfrac{d}{dx} \sin x = \cos x$.

Derivative of $\cos x$

$\dfrac{d}{dx} \cos x = -\sin x$.

Derivative of $e^x$

$\dfrac{d}{dx} e^x = e^x$ — it is its own derivative.

Derivative of $\ln x$

$\dfrac{d}{dx} \ln x = \dfrac{1}{x}$.

Critical point

Where $f'(x) = 0$ or is undefined — a candidate maximum or minimum.

First derivative test

If $f'$ changes $+$ to $-$, it's a local max; $-$ to $+$, a local min.

Second derivative and concavity

$f'' > 0$ is concave up; $f'' < 0$ is concave down; $f'' = 0$ may be an inflection point.

Increasing/decreasing from the derivative

$f' > 0$ means increasing; $f' < 0$ means decreasing.

Antiderivative (indefinite integral)

$\int f(x)\,dx = F(x) + C$, where $F' = f$; reverses differentiation.

Power rule for integration

$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \ne -1$.

Definite integral

$\int_a^b f(x)\,dx$ gives the signed accumulated area under $f$ from $a$ to $b$.

Fundamental theorem of calculus

$\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is an antiderivative of $f$.

Tangent line equation

At $x = a$: $y = f(a) + f'(a)(x - a)$.

Average rate of change

$\dfrac{f(b) - f(a)}{b - a}$ — the slope of the secant line over $[a, b]$.

Optimization with calculus

Maximize or minimize by setting the derivative to 0 and testing the critical points.

Pythagorean theorem

For a right triangle with legs $a$, $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$.

SOH-CAH-TOA

$\sin = \dfrac{\text{opp}}{\text{hyp}}$, $\cos = \dfrac{\text{adj}}{\text{hyp}}$, $\tan = \dfrac{\text{opp}}{\text{adj}}$.

45-45-90 triangle

An isosceles right triangle with side ratio $1 : 1 : \sqrt{2}$.

30-60-90 triangle

Side ratio $1 : \sqrt{3} : 2$ (short leg : long leg : hypotenuse).

Triangle angle sum

The interior angles of any triangle sum to $180^\circ$.

Congruent figures

Same shape AND size — corresponding sides and angles are equal.

Similar figures

Same shape, proportional sides, equal angles — related by one scale factor.

Triangle congruence criteria

SSS, SAS, ASA, and AAS prove triangles congruent.

Triangle similarity criterion (AA)

Two pairs of equal angles make triangles similar.

Scale factor effect on area and volume

If lengths scale by $k$, areas scale by $k^2$ and volumes by $k^3$.

Area of a triangle

$A = \tfrac{1}{2} b h$.

Area of a rectangle

$A = lw$ (length times width).

Area of a trapezoid

$A = \tfrac{1}{2}(b_1 + b_2) h$.

Area of a circle

$A = \pi r^2$.

Circumference of a circle

$C = 2\pi r = \pi d$.

Equation of a circle

Centered at $(h, k)$ with radius $r$: $(x - h)^2 + (y - k)^2 = r^2$.

Arc length

A fraction of the circumference set by the central angle: $\dfrac{\theta}{360^\circ} \cdot 2\pi r$.

Sector area

$\dfrac{\theta}{360^\circ} \cdot \pi r^2$.

Volume of a rectangular prism

$V = lwh$.

Volume of a cylinder

$V = \pi r^2 h$.

Volume of a sphere

$V = \tfrac{4}{3}\pi r^3$.

Volume of a cone

$V = \tfrac{1}{3}\pi r^2 h$.

Distance formula

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Midpoint formula

$\left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)$.

Sum of interior angles of a polygon

$(n - 2) \cdot 180^\circ$ for an $n$-sided polygon.

Complementary angles

Two angles that sum to $90^\circ$.

Supplementary angles

Two angles that sum to $180^\circ$.

Vertical angles

Opposite angles formed by two intersecting lines; they are equal.

Parallel lines cut by a transversal

Corresponding and alternate angles are equal; co-interior angles are supplementary.

Inscribed angle theorem

An inscribed angle is half the central angle subtending the same arc.

Surface area of a cylinder

$SA = 2\pi r^2 + 2\pi r h$ (two bases plus the side).

Reflection over the line y = x

Maps the point $(a, b)$ to $(b, a)$.

Translation (geometric)

Slides every point the same distance and direction; preserves size and shape.

Rotation (geometric)

Turns a figure about a fixed center by a given angle; preserves size and shape.

Radian measure

$180^\circ = \pi$ radians; convert by multiplying by $\dfrac{\pi}{180^\circ}$.

Mean

The average: sum of values divided by the count. Sensitive to outliers.

Median

The middle value of an ordered data set; resists outliers.

Mode

The most frequently occurring value in a data set.

Range (statistics)

The difference between the largest and smallest values.

Standard deviation

A measure of spread around the mean; larger means more spread.

Variance

The square of the standard deviation; the average squared deviation from the mean.

Interquartile range (IQR)

$Q_3 - Q_1$; the spread of the middle 50% of the data.

Outlier

A value far from the rest of the data, often beyond $1.5 \times \text{IQR}$ from a quartile.

Right-skewed distribution

A long right tail; the mean is greater than the median.

Left-skewed distribution

A long left tail; the mean is less than the median.

Normal distribution

A symmetric bell curve; mean = median = mode.

Empirical (68-95-99.7) rule

About 68% of data lies within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean.

z-score

$z = \dfrac{x - \mu}{\sigma}$; how many standard deviations a value is from the mean.

Probability

Favorable outcomes ÷ total outcomes, a value from 0 to 1.

Independent events

Outcomes don't affect each other: $P(A \text{ and } B) = P(A)P(B)$.

Mutually exclusive events

Cannot occur together: $P(A \text{ or } B) = P(A) + P(B)$.

General addition rule

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

Conditional probability

$P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}$.

Complement rule

$P(\text{not } A) = 1 - P(A)$.

Permutation

An ordered arrangement: ${}_nP_r = \dfrac{n!}{(n-r)!}$.

Combination

An unordered selection: ${}_nC_r = \dfrac{n!}{r!(n-r)!}$.

Factorial

$n! = n \cdot (n-1) \cdots 2 \cdot 1$; $0! = 1$.

Expected value

The long-run average of a random variable: $\sum x_i P(x_i)$.

Sample vs. population

A population is the whole group; a sample is a subset used to estimate it.

Random sampling

Each member has an equal chance of selection; reduces bias and supports inference.

Correlation

A measure of linear association; $r$ ranges from $-1$ to $1$.

Correlation vs. causation

A strong correlation does not prove one variable causes the other.

Line of best fit

A linear model minimizing squared residuals; used to predict from data.

Two-way table

Displays counts for two categorical variables; used for joint and conditional probabilities.

Margin of error

The range around a sample estimate within which the true value likely falls.

Histogram

A bar graph of frequency over equal numeric intervals (bins).

Box plot (five-number summary)

Shows minimum, $Q_1$, median, $Q_3$, and maximum.

Theoretical vs. experimental probability

Theoretical is computed from the model; experimental is observed from trials.

Law of large numbers

As trials increase, the experimental probability approaches the theoretical probability.

Percentile

The value below which a given percent of data falls; the 90th percentile beats 90% of values.

Bias in a study

A systematic error that skews results away from the truth, e.g. from non-random sampling.

Fundamental counting principle

If one event has $m$ outcomes and another $n$, together they have $m \cdot n$.

Quartiles

Values splitting ordered data into four equal parts: $Q_1$, $Q_2$ (median), $Q_3$.