PSAT Math Master

Linear Algebra

Equations · Inequalities · Systems

📌Key Concepts & Formulas to Memorize

Standard form of a linear equation: $ax + b = c$ → solve by isolating $x$
Slope formula: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept form: $y = mx + b$ where $m$ = slope, $b$ = y-intercept
Point-slope form: $y - y_1 = m(x - x_1)$
Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals
Systems of equations: substitution or elimination method

$y = mx + b$ → slope: $m$, y-intercept: $(0, b)$

⚡ Memorize ThisIf a system has no solution, lines are parallel (same slope, different intercept). Infinite solutions → same line.

Warm-up Example

A line passes through $(2, 5)$ and $(6, 13)$. What is the equation of the line?

Solution: Slope $m = \dfrac{13-5}{6-2} = \dfrac{8}{4} = 2$. Using point-slope: $y - 5 = 2(x - 2)$ → $y = 2x + 1$.

Problem Solving & Ratios

Ratios · Percentages · Rates · Proportions

📌Key Concepts & Formulas to Memorize

Percent Change $= \dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100$

⚡ Memorize This"Of" means multiply. "Is" means equals. "What percent of 80 is 20?" → $\dfrac{20}{80} \times 100 = 25\%$

Warm-up Example

A shirt's price decreased from \$40 to \$28. What is the percent decrease?

Solution: $\dfrac{40-28}{40} \times 100 = \dfrac{12}{40} \times 100 = 30\%$

Advanced Mathematics

Quadratics · Polynomials · Functions · Exponents

📌Key Concepts & Formulas to Memorize

Quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for $ax^2 + bx + c = 0$
Vertex form: $y = a(x-h)^2 + k$; vertex at $(h, k)$
Discriminant: $D = b^2 - 4ac$. D > 0: 2 real roots; D = 0: 1 root; D < 0: no real roots
FOIL: $(a+b)(c+d) = ac + ad + bc + bd$
Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
Exponent rules: $x^a \cdot x^b = x^{a+b}$, $\dfrac{x^a}{x^b} = x^{a-b}$, $(x^a)^b = x^{ab}$
Function composition: $(f \circ g)(x) = f(g(x))$

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

⚡ Memorize ThisPerfect square: $(a+b)^2 = a^2 + 2ab + b^2$. Perfect square trinomial → one repeated root!

Warm-up Example

Solve: $x^2 - 5x + 6 = 0$

Solution: Factor: $(x-2)(x-3) = 0$ → $x = 2$ or $x = 3$

Data Analysis & Statistics

Mean · Median · Probability · Scatterplots

📌Key Concepts & Formulas to Memorize

Mean $= \dfrac{\text{Sum of values}}{\text{Number of values}}$
Median: middle value when data is sorted in order
Mode: most frequently occurring value
Range = Maximum − Minimum
Probability: $P(\text{event}) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$
Standard deviation: measures spread; larger SD = more spread out
Scatterplot: positive correlation → slope up; negative → slope down

Mean $= \dfrac{\sum x_i}{n}$ | $P(A) = \dfrac{\text{Favorable}}{n(\text{Total})}$

⚡ Memorize ThisOutliers affect the MEAN more than the median. If asked which is more "resistant," always choose MEDIAN.

Warm-up Example

Data set: {3, 7, 7, 10, 13}. Find the mean and median.

Solution: Mean $= \dfrac{3+7+7+10+13}{5} = \dfrac{40}{5} = 8$. Median (middle value) $= 7$.

📝 PSAT Math Free-Response Exam