PreACT Math Review (Version 1)

Essential PreACT math formulas, rules, and vocabulary covering numbers, quantity, algebra, functions, geometry, statistics, and essential skills.

Product Rule of Exponents

When multiplying powers with the same base, add their exponents: $x^a \times x^b = x^{a+b}$. Dividing subtracts the exponents, and power of a power multiplies them: $(x^a)^b = x^{ab}$.

Negative and Fractional Exponents

$x^{-a} = \frac{1}{x^a}$ (a negative exponent indicates a reciprocal) and $x^{\frac{1}{n}} = \sqrt[n]{x}$ (a fractional exponent indicates the $n\text{th}$ root of $x$).

Imaginary Unit ($i$)

The unit where $i = \sqrt{-1}$ and $i^2 = -1$. Powers of $i$ cycle every $4$: $i$, $-1$, $-i$, $1$.

Scalar Multiplication of a Matrix

The operation of multiplying every entry in a matrix by a single number (scalar), resulting in a matrix that keeps the same dimensions.

Quadratic Formula

The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ used to solve $ax^2 + bx + c = 0$. The expression $\frac{-b}{2a}$ alone gives the $x$-coordinate of the parabola vertex.

Factoring $x^2 + bx + c$

Finding two numbers that multiply to $c$ and add to $b$, yielding the binomial factors $(x + p)(x + q)$.

Slope and Slope-Intercept Form

Slope is defined as $m = \frac{y_2 - y_1}{x_2 - x_1}$. Slope-intercept form is $y = mx + b$, where $b$ is the $y$-intercept.

Linear System Solutions

A system of linear equations has no solution if lines have the same slope but different intercepts (parallel); infinitely many solutions if lines have the same slope and same intercept; and exactly one solution if lines have different slopes.

Factor Theorem

If $(x - a)$ is a factor of a polynomial $p(x)$, then $p(a) = 0$, meaning $a$ is a zero of $p(x)$. Dividing out $(x - a)$ reveals the remaining factors.

Composite Function $f(g(x))$

A combination of functions evaluated by working inside out: first evaluate $g(x)$, and then substitute that result into $f$.

Arithmetic Sequence Formulas

The $n\text{th}$ term is given by $a_n = a_1 + (n - 1)d$, where $d$ is the common difference. The sum of the first $n$ terms is $\frac{n(a_1 + a_n)}{2}$.

Exponential Growth and Decay Formula

$y = a(1 \pm r)^t$, where $a$ is the starting amount and $r$ is the rate written as a decimal; use $+$ for growth and $-$ for decay.

Logarithm

$\log_b(x) = y$ means $b^y = x$. A logarithm determines what power of base $b$ equals $x$ (e.g., $\log_2(8) = 3$ because $2^3 = 8$).

Pythagorean Theorem

The relation $a^2 + b^2 = c^2$, which applies strictly to right triangles, where $c$ is the hypotenuse.

Special Right Triangles

Triangles with specific side ratios: $30^\circ\text{-}60^\circ\text{-}90^\circ$ has side ratios $1 : \sqrt{3} : 2$, and $45^\circ\text{-}45^\circ\text{-}90^\circ$ has side ratios $1 : 1 : \sqrt{2}$.

SOH-CAH-TOA

The basic trigonometric definitions: $\sin = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan = \frac{\text{opposite}}{\text{adjacent}}$, along with the identity $\sin^2 + \cos^2 = 1$.

Triangle and Polygon Angle Facts

The interior angles of a triangle sum to $180^\circ$, the largest angle is opposite the longest side, and isosceles triangles have two equal base angles. The interior angle sum of an $n$-sided polygon is $(n - 2) \times 180^\circ$.

Circle Formulas

Area is $A = \pi r^2$, circumference is $C = 2\pi r$, and the standard equation is $(x - h)^2 + (y - k)^2 = r^2$ with center $(h, k)$. A regular hexagon inscribed in a circle has each side equal to the radius $r$.

Distance and Midpoint Formulas

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ and $\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Prism and Cylinder Volumes

Volume of a prism is $lwh$; volume of a cylinder is $\pi r^2 h$ (base area $\times$ height). Cones and pyramids have a volume equal to $\frac{1}{3}$ of their corresponding cylinder or prism.

Missing Value from a Known Mean

A technique where $\text{Total} = \text{mean} \times \text{number of values}$, and the missing value is found by subtracting all known values from the total.

Median from a Frequency Table

The middle value determined by counting through cumulative frequencies; for an even count $n$, take the average of the $\left(\frac{n}{2}\right)\text{th}$ and $\left(\frac{n}{2} + 1\right)\text{th}$ values.

Expected Value

The sum of each $(\text{value} \times \text{probability})$, where all probabilities must sum to $1$.

Mutually Exclusive Events

Events where $P(A\text{ and }B) = 0$, simplifying the addition rule to $P(A\text{ or }B) = P(A) + P(B)$. In general, $P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and }B)$.

Percent Change

$\frac{\text{new} - \text{old}}{\text{old}} \times 100$, calculated by always dividing the difference between the values by the original (old) amount.