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Questions and Answers
What is the characteristic of an independent system of equations?
Which of the following is a quadratic equation in standard form?
What is the composition of functions f(x) = x^2 and g(x) = 2x + 1?
What is the modulus of the complex number 3 + 4i?
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What is the graph of the inequality 2x + 3y > 6?
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What is the type of asymptote that occurs at x-values where the function is undefined?
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What is the number of solutions of a quadratic equation with real and rational coefficients?
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What is the conjugate of the complex number 2 + 3i?
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What is the quadratic formula for solving quadratic equations?
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What is the type of function that has a domain and range of all real numbers?
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Study Notes
Systems of Equations
- A system of equations is a set of two or more equations that must be true at the same time
- Can be solved using substitution, elimination, or graphical methods
- Types of systems:
- Independent systems: have a unique solution
- Dependent systems: have infinitely many solutions
- Inconsistent systems: have no solution
Quadratic Equations
- A quadratic equation is a polynomial equation of degree two
- Standard form: ax^2 + bx + c = 0, where a, b, and c are constants
- Can be solved using:
- Factoring
- Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
- Types of solutions:
- Real and rational
- Real and irrational
- Complex
Functions
- A function is a relation between a set of inputs (domain) and a set of possible outputs (range)
- Notation: f(x) = output
- Types of functions:
- Linear: f(x) = mx + b
- Quadratic: f(x) = ax^2 + bx + c
- Exponential: f(x) = a^x
- Logarithmic: f(x) = loga(x)
- Properties:
- Domain and range
- Even and odd functions
- Composition of functions
Complex Numbers
- A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (i^2 = -1)
- Operations:
- Addition and subtraction
- Multiplication and division
- Properties:
- Conjugate: a - bi
- Modulus: |a + bi| = √(a^2 + b^2)
- Argument: arg(a + bi) = tan^(-1)(b/a)
Inequalities
- An inequality is a statement that one expression is greater than, less than, or equal to another
- Notation:
- > (greater than)
- < (less than)
- ≥ (greater than or equal to)
- ≤ (less than or equal to)
- Solving inequalities:
- Add or subtract the same value to both sides
- Multiply or divide both sides by a positive value
- Graphing inequalities:
- Shade above or below the line
- Solid line for ≥ or ≤, dashed line for > or <
Asymptotes
- A vertical asymptote is a vertical line that the graph of a function approaches as x approaches a certain value
- A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity
- Types of asymptotes:
- Vertical asymptotes: occur at x-values where the function is undefined
- Horizontal asymptotes: occur as x approaches infinity or negative infinity
Logarithmic Equations
- A logarithmic equation is an equation involving logarithms
- Types of logarithms:
- Natural logarithm (ln)
- Common logarithm (log)
- Properties:
- Logarithmic identity: loga(x) = y ⇔ a^y = x
- Logarithmic rules:
- Product rule: loga(xy) = loga(x) + loga(y)
- Quotient rule: loga(x/y) = loga(x) - loga(y)
- Power rule: loga(x^y) = y loga(x)
Compound Interest with e
- Compound interest is interest calculated on both the principal and any accrued interest
- Formula: A = P(1 + r/n)^(nt), where A is the amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years
- The number e:
- Approximately equal to 2.718
- Used as the base for continuous compounding: A = Pe^(rt)
Systems of Equations
- A system of equations is a set of two or more equations that must be true at the same time
- Can be solved using substitution, elimination, or graphical methods
- Independent systems have a unique solution
- Dependent systems have infinitely many solutions
- Inconsistent systems have no solution
Quadratic Equations
- A quadratic equation is a polynomial equation of degree two
- Standard form: ax^2 + bx + c = 0, where a, b, and c are constants
- Can be solved using factoring and the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
- Types of solutions: real and rational, real and irrational, and complex
Functions
- A function is a relation between a set of inputs (domain) and a set of possible outputs (range)
- Notation: f(x) = output
- Types of functions: linear, quadratic, exponential, and logarithmic
- Properties: domain and range, even and odd functions, and composition of functions
Complex Numbers
- A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (i^2 = -1)
- Operations: addition, subtraction, multiplication, and division
- Properties: conjugate, modulus, and argument
- Conjugate: a - bi
- Modulus: |a + bi| = √(a^2 + b^2)
- Argument: arg(a + bi) = tan^(-1)(b/a)
Inequalities
- An inequality is a statement that one expression is greater than, less than, or equal to another
- Notation: >, <, ≥, and ≤
- Solving inequalities: add or subtract the same value to both sides, and multiply or divide both sides by a positive value
- Graphing inequalities: shade above or below the line, and use solid or dashed lines for different types of inequalities
Asymptotes
- A vertical asymptote is a vertical line that the graph of a function approaches as x approaches a certain value
- A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity
- Types of asymptotes: vertical and horizontal
- Vertical asymptotes occur at x-values where the function is undefined
- Horizontal asymptotes occur as x approaches infinity or negative infinity
Logarithmic Equations
- A logarithmic equation is an equation involving logarithms
- Types of logarithms: natural logarithm (ln) and common logarithm (log)
- Logarithmic identity: loga(x) = y ⇔ a^y = x
- Logarithmic rules: product, quotient, and power rules
Compound Interest with e
- Compound interest is interest calculated on both the principal and any accrued interest
- Formula: A = P(1 + r/n)^(nt), where A is the amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years
- The number e: approximately equal to 2.718, used as the base for continuous compounding: A = Pe^(rt)
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Description
Test your understanding of systems of equations and quadratic equations, including types of systems and methods for solving them.
