Your FREE Praxis 5165 Flashcards 2026 – 200+ Cards

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Lines with equal slopes; they never intersect.

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

Slope 0; equation $y = c$.

Undefined slope; equation $x = c$.

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

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

Add or subtract the equations to cancel a variable.

The lines are parallel — same slope, different intercept.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find a common denominator, combine numerators, then simplify.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Two pairs of equal angles make triangles similar.

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

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

$A = lw$ (length times width).

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

$A = \pi r^2$.

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

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

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

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

$V = lwh$.

$V = \pi r^2 h$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The most frequently occurring value in a data set.

The difference between the largest and smallest values.

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

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

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

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

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

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

A symmetric bell curve; mean = median = mode.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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