AK STAR Math Grade 9 cheat sheet

For any nonzero base a, a^m · a^n = a^(m+n), (a^m)^n = a^(mn), and a^m / a^n = a^(m−n).

A number is rational if it can be written as p/q with integers p and q and q ≠ 0; otherwise it is irrational.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiplying shifts the decimal or changes the exponent.

To simplify a radical, factor out the largest perfect-square factor: √(ab) = √a · √b when a, b ≥ 0.

To rationalize a denominator with one square-root term, multiply numerator and denominator by that radical.

A unit rate compares a quantity to one unit of another quantity; convert units so matching units cancel.

The absolute value |x| is the distance from x to 0 on the number line, so |x| ≥ 0 and |−x| = |x|.

Slope between (x1,y1) and (x2,y2) is m = (y2−y1)/(x2−x1), where x2 ≠ x1.

Slope-intercept form y = mx + b gives slope m and y-intercept b; standard form is Ax + By = C.

For a system, the solution is the ordered pair satisfying both equations; parallel lines give no solution, identical lines give infinitely many.

Solving an inequality: adding or subtracting keeps the symbol; multiplying or dividing by a negative reverses it.

Quadratic formula: x = [−b ± √(b²−4ac)]/(2a) for ax² + bx + c = 0, a ≠ 0.

Discriminant b²−4ac: positive gives two real roots, zero gives one, negative gives no real roots.

Zero product property: if (x−r)(x−s)=0, then x=r or x=s.

Difference of squares: a²−b²=(a+b)(a−b); perfect-square trinomial: a²±2ab+b²=(a±b)².

A relation is a function if each input has exactly one output; the vertical line test detects functions on a graph.

Function notation f(x) means the output for input x; evaluate by substituting the value for x.

Domain is the set of allowed inputs; range is the set of outputs.

Average rate of change between x=a and x=b is [f(b)−f(a)]/(b−a).

Linear functions y=mx+b change at a constant rate; their graphs are straight lines.

Exponential functions y=a·b^x change by a constant factor; growth when b>1, decay when 0<b<1.

A graph crosses the y-axis at f(0); it crosses the x-axis at solutions of f(x)=0.

Transformations: f(x)+k shifts up k; f(x−h) shifts right h; −f(x) reflects over the x-axis; a·f(x) stretches vertically when |a|>1.

Pythagorean theorem: for a right triangle with legs a,b and hypotenuse c, a²+b²=c².

Distance formula: d=√[(x2−x1)²+(y2−y1)²].

Midpoint formula: ((x1+x2)/2, (y1+y2)/2).

Rigid motions-translations, rotations, reflections-preserve size and shape, so images are congruent.

Dilations preserve shape but change size; the image is similar, with side lengths scaled by the scale factor.

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume formulas: cylinder V=πr²h; cone V=(1/3)πr²h; sphere V=(4/3)πr³.

Surface area is the total area of all faces; for a rectangular prism, SA=2(lw+lh+wh).

Mean = sum of values ÷ number of values; median is the middle value when ordered; mode is most frequent.

Range = maximum − minimum; interquartile range IQR = Q3 − Q1 and measures the spread of the middle half.

A line of best fit models a scatter plot; its slope estimates change in y per unit change in x.

Correlation coefficient r ranges from −1 to 1; r near 0 indicates a weak linear relationship, not necessarily no relationship.

Correlation does not imply causation; a lurking variable may explain an association.

Two-way tables show joint frequencies; conditional relative frequency compares a cell to its row or column total.

Probability of an event = favorable outcomes ÷ total equally likely outcomes.

For independent events, P(A and B)=P(A)·P(B); for mutually exclusive events, P(A or B)=P(A)+P(B).