How do you solve the absolute value equation |ax + b| = c where c >= 0?

Split into two linear equations: ax + b = c or ax + b = -c, then solve each equation independently.

What geometric interpretations correspond to a linear 2x2 system with one solution, infinitely many solutions, or no solution?

One solution: lines intersect at a single point; Infinitely many: lines are identical (coincident); No solution: lines are parallel and distinct.

State the slope-intercept form and point-slope form of a linear equation.

Slope-intercept form: y = mx + b (where m is slope, b is y-intercept); Point-slope form: y - y1 = m(x - x1).

What is the Euclidean distance formula between two Cartesian points (x1, y1) and (x2, y2)?

d = √((x2 - x1)^2 + (y2 - y1)^2).

Define an injective (one-to-one) function and describe the test used to verify it graphically.

A function where distinct domain elements produce distinct outputs (f(a) = f(b) implies a = b); verified by the Horizontal Line Test (no horizontal line intersects the graph more than once).

Define the composite function (f ∘ g)(x) and state how its domain is determined.

(f ∘ g)(x) = f(g(x)); its domain consists of all x in the domain of g such that g(x) is in the domain of f.

How does the technique of completing the square convert y = ax^2 + bx + c into vertex form?

Factor out 'a', add and subtract (b/(2a))^2 inside the parentheses to form a perfect square trinomial: y = a(x - h)^2 + k where h = -b/(2a).

How do the substitution and elimination methods differ when solving a system of two linear equations?

Substitution solves one equation for one variable and substitutes it into the other; elimination multiplies equations by constants to cancel a variable when added together.

How are negative exponents and fractional exponents defined mathematically?

Negative exponent: x^(-n) = 1 / x^n (for x != 0); Fractional exponent: x^(m/n) = n-th root of (x^m) = (n-th root of x)^m.

What graphical test determines whether a curve in the xy-plane represents a function y = f(x)?

The Vertical Line Test: every vertical line intersects the graph at most once.

How do you identify vertical and horizontal asymptotes of a rational function R(x) = P(x) / Q(x)?

Vertical asymptotes occur at real zeros of Q(x) not shared by P(x); horizontal asymptotes are determined by comparing degrees of P(x) and Q(x) (y = 0 if deg P < deg Q; ratio of leading coefficients if deg P = deg Q).

What is the mathematical condition relating the slopes m1 and m2 of two non-vertical perpendicular lines?

The slopes are negative reciprocals: m1 * m2 = -1, or m2 = -1 / m1.

What is the fundamental equivalence defining the logarithm log_b(y) = x?

log_b(y) = x is equivalent to b^x = y, where the base b > 0 and b != 1, and the argument y > 0.

What are the coordinates of the vertex of the parabola given by y = ax^2 + bx + c?

Vertex (h, k) where h = -b / (2a) and k = c - b^2 / (4a).

State the three core logarithm properties for product, quotient, and power of an argument.

1. log_b(u * v) = log_b(u) + log_b(v) 2. log_b(u / v) = log_b(u) - log_b(v) 3. log_b(u^k) = k * log_b(u).

What is the formula for the coordinates of the midpoint between points (x1, y1) and (x2, y2)?

M = ((x1 + x2) / 2, (y1 + y2) / 2).

What property of equality states that adding or multiplying the same quantity to both sides preserves the equation?

The addition property of equality (if a = b, then a + c = b + c) and multiplication property of equality (if a = b, then a*c = b*c for c != 0).

What is the discriminant of a quadratic equation, and how does its sign determine the nature of the roots?

Δ = b^2 - 4ac. If Δ > 0: two distinct real roots; if Δ = 0: one repeated real root; if Δ < 0: two complex conjugate roots.

What does the Polynomial Remainder Theorem state regarding division of P(x) by (x - c)?

When a polynomial P(x) is divided by the linear divisor (x - c), the remainder is equal to the value P(c).

How do the transformations y = f(x - c) and y = f(x) + c alter the graph of y = f(x) for c > 0?

y = f(x - c) shifts the graph horizontally to the right by c units; y = f(x) + c shifts the graph vertically upward by c units.

State the algebraic factorization formulas for the sum and difference of two cubes.

Sum: a^3 + b^3 = (a + b)(a^2 - ab + b^2); Difference: a^3 - b^3 = (a - b)(a^2 + ab + b^2).

What does the Factor Theorem state regarding roots of a polynomial P(x)?

A linear term (x - c) is a factor of the polynomial P(x) if and only if P(c) = 0 (c is a root of P(x)).

Why is it essential to check for extraneous solutions when solving radical or rational equations?

Squaring both sides or multiplying by polynomial denominators can introduce false solutions that violate the original equation's domain or equality.

What algebraic conditions define an even function versus an odd function?

Even function: f(-x) = f(x) (symmetric across the y-axis); Odd function: f(-x) = -f(x) (symmetric about the origin).

What is the relationship between the graph of a function f(x) and the graph of its inverse f^-1(x)?

The graph of f^-1(x) is the reflection of the graph of f(x) across the line y = x, with domain and range swapped.

State the exponent product rule, quotient rule, and power of a power rule for real numbers.

Product rule: x^a * x^b = x^(a+b); Quotient rule: x^a / x^b = x^(a-b); Power of a power: (x^a)^b = x^(a*b).

What is the solution set to the absolute value inequality |x - a| 0?

The bounded open interval -b < x - a < b, which simplifies to a - b < x < a + b.

Write the logarithmic change-of-base formula expressing log_b(a) in terms of base c.

log_b(a) = log_c(a) / log_c(b) = ln(a) / ln(b).

What is the definition of a mathematical function from set X to set Y?

A relation that assigns exactly one element of the codomain Y to each element of the domain X.

State the standard form of a quadratic equation and the quadratic formula used to solve it.

Standard form: ax^2 + bx + c = 0 (with a != 0); Quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

Cálculo — Básico (A1/A2) 4: Algebra, Functions, Exponents & Coordinate Geometry

[Español] Cálculo: Polynomial factoring, linear/quadratic equations, function domain and range, exponent rules, logarithms, and Cartesian coordinate geometry

Level: basic
Languages: es → es
Cards: 32
Markdown: Cálculo — Básico (A1/A2) 4: Algebra, Functions, Exponents & Coordinate Geometry (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How do you solve the absolute value equation \|ax + b\| = c where c >= 0?	Split into two linear equations: ax + b = c or ax + b = -c, then solve each equation independently.
2	What geometric interpretations correspond to a linear 2x2 system with one solution, infinitely many solutions, or no solution?	One solution: lines intersect at a single point; Infinitely many: lines are identical (coincident); No solution: lines are parallel and distinct.
3	State the slope-intercept form and point-slope form of a linear equation.	Slope-intercept form: y = mx + b (where m is slope, b is y-intercept); Point-slope form: y - y1 = m(x - x1).
4	What is the Euclidean distance formula between two Cartesian points (x1, y1) and (x2, y2)?	d = √((x2 - x1)^2 + (y2 - y1)^2).
5	Define an injective (one-to-one) function and describe the test used to verify it graphically.	A function where distinct domain elements produce distinct outputs (f(a) = f(b) implies a = b); verified by the Horizontal Line Test (no horizontal line intersects the graph more than once).
6	Define the composite function (f ∘ g)(x) and state how its domain is determined.	(f ∘ g)(x) = f(g(x)); its domain consists of all x in the domain of g such that g(x) is in the domain of f.
7	How does the technique of completing the square convert y = ax^2 + bx + c into vertex form?	Factor out 'a', add and subtract (b/(2a))^2 inside the parentheses to form a perfect square trinomial: y = a(x - h)^2 + k where h = -b/(2a).
8	How do the substitution and elimination methods differ when solving a system of two linear equations?	Substitution solves one equation for one variable and substitutes it into the other; elimination multiplies equations by constants to cancel a variable when added together.
9	How are negative exponents and fractional exponents defined mathematically?	Negative exponent: x^(-n) = 1 / x^n (for x != 0); Fractional exponent: x^(m/n) = n-th root of (x^m) = (n-th root of x)^m.
10	What graphical test determines whether a curve in the xy-plane represents a function y = f(x)?	The Vertical Line Test: every vertical line intersects the graph at most once.
11	How do you identify vertical and horizontal asymptotes of a rational function R(x) = P(x) / Q(x)?	Vertical asymptotes occur at real zeros of Q(x) not shared by P(x); horizontal asymptotes are determined by comparing degrees of P(x) and Q(x) (y = 0 if deg P < deg Q; ratio of leading coefficients if deg P = deg Q).
12	What is the mathematical condition relating the slopes m1 and m2 of two non-vertical perpendicular lines?	The slopes are negative reciprocals: m1 * m2 = -1, or m2 = -1 / m1.
13	What is the fundamental equivalence defining the logarithm log_b(y) = x?	log_b(y) = x is equivalent to b^x = y, where the base b > 0 and b != 1, and the argument y > 0.
14	What are the coordinates of the vertex of the parabola given by y = ax^2 + bx + c?	Vertex (h, k) where h = -b / (2a) and k = c - b^2 / (4a).
15	State the three core logarithm properties for product, quotient, and power of an argument.	1. log_b(u * v) = log_b(u) + log_b(v) 2. log_b(u / v) = log_b(u) - log_b(v) 3. log_b(u^k) = k * log_b(u).
16	What is the formula for the coordinates of the midpoint between points (x1, y1) and (x2, y2)?	M = ((x1 + x2) / 2, (y1 + y2) / 2).
17	What property of equality states that adding or multiplying the same quantity to both sides preserves the equation?	The addition property of equality (if a = b, then a + c = b + c) and multiplication property of equality (if a = b, then ac = bc for c != 0).
18	What is the algebraic factorization formula for the difference of two squares?	a^2 - b^2 = (a - b)(a + b).
19	What is the discriminant of a quadratic equation, and how does its sign determine the nature of the roots?	Δ = b^2 - 4ac. If Δ > 0: two distinct real roots; if Δ = 0: one repeated real root; if Δ < 0: two complex conjugate roots.
20	What does the Polynomial Remainder Theorem state regarding division of P(x) by (x - c)?	When a polynomial P(x) is divided by the linear divisor (x - c), the remainder is equal to the value P(c).
21	How do the transformations y = f(x - c) and y = f(x) + c alter the graph of y = f(x) for c > 0?	y = f(x - c) shifts the graph horizontally to the right by c units; y = f(x) + c shifts the graph vertically upward by c units.
22	State the standard equation of a circle with center (h, k) and radius r in the Cartesian coordinate plane.	(x - h)^2 + (y - k)^2 = r^2.
23	State the algebraic factorization formulas for the sum and difference of two cubes.	Sum: a^3 + b^3 = (a + b)(a^2 - ab + b^2); Difference: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
24	What does the Factor Theorem state regarding roots of a polynomial P(x)?	A linear term (x - c) is a factor of the polynomial P(x) if and only if P(c) = 0 (c is a root of P(x)).
25	Why is it essential to check for extraneous solutions when solving radical or rational equations?	Squaring both sides or multiplying by polynomial denominators can introduce false solutions that violate the original equation's domain or equality.
26	What algebraic conditions define an even function versus an odd function?	Even function: f(-x) = f(x) (symmetric across the y-axis); Odd function: f(-x) = -f(x) (symmetric about the origin).
27	What is the relationship between the graph of a function f(x) and the graph of its inverse f^-1(x)?	The graph of f^-1(x) is the reflection of the graph of f(x) across the line y = x, with domain and range swapped.
28	State the exponent product rule, quotient rule, and power of a power rule for real numbers.	Product rule: x^a * x^b = x^(a+b); Quotient rule: x^a / x^b = x^(a-b); Power of a power: (x^a)^b = x^(a*b).
29	What is the solution set to the absolute value inequality \|x - a\| < b where b > 0?	The bounded open interval -b < x - a < b, which simplifies to a - b < x < a + b.
30	Write the logarithmic change-of-base formula expressing log_b(a) in terms of base c.	log_b(a) = log_c(a) / log_c(b) = ln(a) / ln(b).
31	What is the definition of a mathematical function from set X to set Y?	A relation that assigns exactly one element of the codomain Y to each element of the domain X.
32	State the standard form of a quadratic equation and the quadratic formula used to solve it.	Standard form: ax^2 + bx + c = 0 (with a != 0); Quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).