Algebra I (US High School)
US High School Algebra 1 deck aligned with Common Core (CCSS.MATH.CONTENT.HSA/HSF). Covers linear equations and inequalities, systems, exponents and exponential functions, polynomials and factoring, quadratics, radicals, rational expressions, statistics, and sequences.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 451 kort
Innehåll
- A linear equation in one variable has the standard form ax + b = 0, where a and b are constants and a ≠ 0. It has exactly one solution: x = -b/a.
- The slope-intercept form of a line is y = mx + b, where m is the slope (rate of change) and b is the y-intercept (the y-value when x = 0).
- The point-slope form of a line is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a known point on the line. Use this when you have a point and slope.
- The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, A is non-negative, and A and B are not both zero.
- Slope is calculated as m = (y₂ - y₁) / (x₂ - x₁), the change in y divided by the change in x between any two points (x₁, y₁) and (x₂, y₂) on the line.
- A horizontal line has slope 0 and the equation y = c. A vertical line has undefined slope and the equation x = c. Vertical lines are not functions.
- Two non-vertical lines are parallel if and only if they have the same slope (m₁ = m₂) and different y-intercepts.
- Two non-vertical lines are perpendicular if and only if the product of their slopes equals -1, that is, m₁ · m₂ = -1. Their slopes are negative reciprocals.
- The x-intercept of a line is found by setting y = 0 and solving for x. The y-intercept is found by setting x = 0 and solving for y.
- The Addition Property of Equality states that if a = b, then a + c = b + c. You can add (or subtract) the same value to both sides of an equation without changing its solutions.
- The Multiplication Property of Equality states that if a = b, then ac = bc. Multiplying or dividing both sides by the same non-zero number preserves equality.
- The Distributive Property states that a(b + c) = ab + ac. It allows you to remove parentheses by multiplying each term inside by the factor outside.
- When solving an inequality, multiplying or dividing both sides by a negative number reverses the direction of the inequality symbol. For example, -2x < 6 becomes x > -3.
- On a number line, an open circle (⚬) indicates a strict inequality (<, >); a closed circle (⬤) indicates a non-strict inequality (≤, ≥) where the endpoint is included.
- A compound inequality with AND (conjunction) like -3 < x < 5 has solutions that satisfy BOTH conditions; the solution is the intersection of the two solution sets.
- A compound inequality with OR (disjunction) like x < -2 OR x > 5 has solutions that satisfy AT LEAST ONE condition; the solution is the union of the two sets.
- The absolute value of a number x, written |x|, is its distance from zero on the number line. |x| is always non-negative: |x| ≥ 0 for all real x.
- To solve |x| < a (where a > 0), rewrite as -a < x < a. To solve |x| > a, rewrite as x < -a OR x > a. If a < 0, |x| < a has no solution.
- A literal equation contains more than one variable. To solve for one variable, isolate it using the same operations as a regular equation. Example: Solve A = lw for w gives w = A/l.
- A linear equation has no solution when the variable terms cancel and the constants are unequal (e.g., 2x + 3 = 2x + 5 gives 3 = 5, false). The graph would be two parallel lines.
- A linear equation has infinitely many solutions when both sides simplify to identical expressions (e.g., 2(x + 3) = 2x + 6). The equation is called an identity.
- A positive slope rises from left to right; a negative slope falls from left to right; a zero slope is horizontal; an undefined slope is vertical.
- When graphing a linear inequality in two variables (e.g., y > 2x + 1), use a dashed line for strict inequalities (<, >) and a solid line for ≤, ≥. Shade the half-plane containing solutions.
- To choose the half-plane to shade, pick a test point not on the line (often (0, 0) if the line doesn't pass through the origin). If the inequality is true, shade that side; otherwise shade the other.
- A system of linear equations is a set of two or more linear equations in the same variables. The solution is the set of values that make all equations true simultaneously.
- A system of two linear equations has exactly one solution if the lines intersect (different slopes), no solution if the lines are parallel (same slope, different intercepts), or infinitely many if the lines coincide.
- A consistent system has at least one solution; an inconsistent system has no solution. A system is independent if it has exactly one solution and dependent if it has infinitely many.
- The substitution method solves a system by isolating one variable in one equation and substituting the expression into the other equation. Useful when a variable already has coefficient 1.
- The elimination method (also called addition or linear combination) adds or subtracts equations after multiplying so that one variable cancels. Useful when no variable is already isolated.
- When solving by elimination, multiplying equations by constants to create opposite coefficients allows one variable to cancel when the equations are added. For example, multiply 2x + 3y = 7 by 2 and x + y = 3 by -3 to eliminate y.
- Graphing a system finds the intersection point visually. It is accurate only when solutions are integers; for non-integer solutions, use substitution or elimination algebraically.
- When solving a system algebraically yields a false statement like 0 = 5, the system has no solution. When it yields a true statement like 0 = 0, the system has infinitely many solutions.
- A system of linear inequalities has a solution set that is the intersection of the half-planes for each inequality. The solution region is typically a polygon or unbounded region.
- The Product of Powers Rule: when multiplying powers with the same base, add the exponents: xᵃ · xᵇ = xᵃ⁺ᵇ. Example: 2³ · 2⁴ = 2⁷.
- The Quotient of Powers Rule: when dividing powers with the same base, subtract the exponents: xᵃ / xᵇ = xᵃ⁻ᵇ (for x ≠ 0). Example: 5⁷ / 5³ = 5⁴.
- The Power of a Power Rule: (xᵃ)ᵇ = xᵃᵇ. When raising a power to a power, multiply the exponents. Example: (3²)⁴ = 3⁸.
- The Power of a Product Rule: (xy)ⁿ = xⁿyⁿ. Each factor inside is raised to the power. Example: (2x)³ = 8x³.
- The Power of a Quotient Rule: (x/y)ⁿ = xⁿ / yⁿ, for y ≠ 0. Example: (3/4)² = 9/16.
- The Zero Exponent Rule: any non-zero number raised to the zero power equals 1. That is, x⁰ = 1 for x ≠ 0. The expression 0⁰ is undefined (indeterminate).
- The Negative Exponent Rule: x⁻ⁿ = 1/xⁿ (for x ≠ 0). A negative exponent indicates a reciprocal. Example: 2⁻³ = 1/8.
- Scientific notation expresses a number as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. Example: 3{,}200 = 3.2 × 10³, and 0.00075 = 7.5 × 10⁻⁴.
- An exponential function has the form f(x) = a · bˣ, where a ≠ 0, b > 0, and b ≠ 1. The base b is the growth/decay factor and a is the initial value (when x = 0).
- In f(x) = a · bˣ, the function represents exponential growth when b > 1 and exponential decay when 0 < b < 1. Growth and decay rates are written as b = 1 + r or b = 1 - r, respectively.
- An exponential growth model: A = P(1 + r)ᵗ, where P is the initial amount, r is the growth rate (as a decimal), and t is time. Used for population growth and compound interest.
- An exponential decay model: A = P(1 - r)ᵗ, where P is the initial amount, r is the decay rate (as a decimal), and t is time. Used for radioactive decay, depreciation, and cooling.
- The graph of f(x) = bˣ (with b > 1) passes through (0, 1), increases from left to right, and has a horizontal asymptote at y = 0. It is always positive.
- The graph of f(x) = bˣ (with 0 < b < 1) passes through (0, 1), decreases from left to right, and has a horizontal asymptote at y = 0. It is always positive.
- Compound interest formula: A = P(1 + r/n)^(nt), where P is principal, r is annual rate, n is number of times compounded per year, and t is time in years.
- A monomial is a single-term algebraic expression with no negative or fractional exponents on its variables (e.g., 5x³, -7, 4xy). Constants are monomials of degree 0.
- A polynomial is a sum of monomials called terms. A polynomial with one term is a monomial, with two a binomial, with three a trinomial.