MCAS Grade 10 Math cheat sheet

Real numbers include rational and irrational numbers; a rational number can be written as a/b where a and b are integers and b ≠ 0.

Decimal rule: rational numbers have terminating or repeating decimals; irrational numbers have decimals that neither terminate nor repeat.

Exponent rules: a^m · a^n = a^(m+n), (a^m)^n = a^(mn), (ab)^n = a^n b^n, and a^0 = 1 for a ≠ 0.

Negative and fractional exponents: a^(-n) = 1/a^n and a^(1/n) is the nth root of a; a^(m/n) is the nth root of a^m.

Radical rules: √(ab) = √a · √b for nonnegative a and b; √(a/b) = √a/√b for b > 0; rationalize denominators by multiplying by a form of 1.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents and dividing subtracts exponents.

Percent change = ((new value − old value)/old value) × 100%; a positive result is an increase and a negative result is a decrease.

In unit conversions, set up conversion factors so unwanted units cancel; a rate such as 60 km/h can be treated as a fraction equal to 1 when converted correctly.

Linear forms: y = mx + b, Ax + By = C, and y − y1 = m(x − x1); slope m = (y2 − y1)/(x2 − x1).

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1, so their slopes are negative reciprocals.

A system’s solution satisfies all equations; substitution isolates one variable, while elimination adds or subtracts equations to cancel one variable.

For a linear system, intersecting lines give one solution, parallel distinct lines give none, and coincident lines give infinitely many.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives two, one, or no real roots when positive, zero, or negative.

The graph of y = ax² + bx + c is a parabola with vertex x-coordinate −b/(2a); it opens upward when a > 0 and downward when a < 0.

Factoring: remove the GCF first; a² − b² = (a − b)(a + b); x² + (p + q)x + pq = (x + p)(x + q); check by expanding.

Inequalities: multiplying or dividing by a negative number reverses the inequality symbol; |x| < a means −a < x < a, while |x| > a means x < −a or x > a.

Exponential models y = a(1 + r)^t show growth when r > 0 and decay when −1 < r < 0; a rational expression is undefined where its denominator equals zero.

A function assigns exactly one output to each input; the vertical line test checks whether a graph represents a function.

Domain restrictions come from denominators that cannot be zero, even roots of negative numbers, and real-world limits on inputs.

Average rate of change on [a, b] = (f(b) − f(a))/(b − a); it is the slope of the secant line through the two points.

Transformations: f(x) + k shifts vertically; f(x − h) shifts horizontally; a·f(x) stretches or compresses vertically; −f(x) reflects across the x-axis.

Linear functions f(x) = mx + b have constant rate of change m; quadratic functions form parabolas; exponential functions a·b^x have a constant ratio.

Vertex form f(x) = a(x − h)² + k has vertex (h, k), axis of symmetry x = h, and a maximum or minimum at the vertex.

An inverse function reverses the original; find it by swapping x and y and solving for y, and its graph reflects across y = x.

Composition (f ∘ g)(x) = f(g(x)); evaluate g first, and exclude inputs where g(x) is not in the domain of f.

Key graph features include x-intercepts where f(x) = 0, the y-intercept f(0), increasing and decreasing intervals, and end behavior.

Pythagorean theorem: in a right triangle, a² + b² = c² where c is the hypotenuse; its converse proves a triangle is right when a² + b² = c².

Special right triangles: 45-45-90 has sides x, x, x√2; 30-60-90 has sides x, x√3, 2x opposite 30°, 60°, and 90°.

Distance = √((x2 − x1)² + (y2 − y1)²); midpoint = ((x1 + x2)/2, (y1 + y2)/2).

Similar polygons have congruent corresponding angles and proportional corresponding sides; scale factor k gives area ratio k² and volume ratio k³.

Congruence criteria: SSS, SAS, ASA, AAS, and HL for right triangles; AAA and SSA do not guarantee congruence.

Circle formulas: C = 2πr and A = πr²; arc length = (central angle/360) · 2πr; sector area = (central angle/360) · πr².

An inscribed angle measures half its intercepted arc; an angle inscribed in a semicircle is 90°; a tangent is perpendicular to the radius at the point of tangency.

Volume: prism or cylinder V = Bh; pyramid or cone V = (1/3)Bh; sphere V = (4/3)πr³; for a cylinder or cone, B = πr².

Right-triangle trig: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent; circle equation: (x − h)² + (y − k)² = r².

Mean = sum of values/n; median is the middle value after ordering; mode is the most frequent value; range = maximum − minimum.

Standard deviation measures typical distance from the mean; a larger standard deviation means greater spread, and adding a constant shifts the mean but not the spread.

IQR = Q3 − Q1; a value is a potential outlier if it is below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR); box plots show median and quartiles.

Probability P(A) is between 0 and 1; complement rule: P(not A) = 1 − P(A); the total probability of all outcomes is 1.

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); mutually exclusive events have P(A and B) = 0.

Conditional probability: P(A|B) = P(A and B)/P(B) when P(B) > 0; events are independent when P(A|B) = P(A).

For independent events, P(A and B) = P(A) · P(B); for dependent events without replacement, the denominator changes after each selection.

In a two-way table, a cell gives joint frequency, row and column totals give marginal frequencies, and conditional probability uses a row or column total as the denominator.

Correlation measures association, not causation; a line of best fit predicts y from x, and residual = actual value − predicted value.