Algebra Review Quiz

Quadratic Equations

• A quadratic equation is a polynomial equation of degree 2, written in the form: ax^2 + bx + c = 0
• Where a, b, and c are constants, and a ≠ 0
• The quadratic formula is used to solve quadratic equations: x = (-b ± √(b^2 - 4ac)) / 2a
• Quadratic equations can have:
  • Two distinct real roots
  • One repeated real root
  • Two complex roots
• Graphs of quadratic equations are parabolas that open upward or downward

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range)
• Domain: the set of all input values
• Range: the set of all possible output values
• Function notation: f(x) = output
• Types of functions:
  • Linear functions: f(x) = mx + b
  • Quadratic functions: f(x) = ax^2 + bx + c
  • Exponential functions: f(x) = a^x
  • Polynomial functions: f(x) = a_n x^n + a_(n-1) x^(n-1) +... + a_1 x + a_0
• Function operations:
  • Composition: (f ∘ g)(x) = f(g(x))
  • Inverse: f^(-1)(x) is the inverse of f(x)

Graphing

• Graphing involves plotting points on a coordinate plane to visualize relationships between variables
• Graphing functions:
  • Identify the x-intercepts (roots) by setting y = 0
  • Identify the y-intercept by setting x = 0
  • Identify the axis of symmetry (vertical line) for quadratic functions
  • Identify the vertex (turning point) for quadratic functions
• Graphing quadratic functions:
  • Parabolas open upward or downward
  • The vertex is the minimum or maximum point
  • The axis of symmetry is the vertical line that passes through the vertex

Polynomials

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication
• Degree of a polynomial: the highest power of the variable
• Types of polynomials:
  • Monomial: a single term
  • Binomial: two terms
  • Trinomial: three terms
• Polynomial operations:
  • Addition and subtraction: combine like terms
  • Multiplication: distribute each term to every term in the other polynomial
• Factoring polynomials:
  • Greatest common factor (GCF)
  • Difference of squares: a^2 - b^2 = (a + b)(a - b)
  • Sum and difference: a^2 + b^2 = (a + b)(a - b)

Systems of Equations

• A system of equations is a set of two or more equations with variables
• Solving systems of equations:
  • Substitution method: solve one equation for one variable, then substitute into the other equation
  • Elimination method: add or subtract equations to eliminate one variable, then solve for the other variable
• Types of systems:
  • Independent systems: have a unique solution
  • Dependent systems: have infinitely many solutions
  • Inconsistent systems: have no solution
• Graphing systems of equations:
  • Identify the intersection point(s) of the graphs
  • Identify the number of solutions (none, one, or infinitely many)