Algebra Review Quiz

Study Notes

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)

Description

Test your understanding of algebra concepts, including quadratic equations, functions, graphing, polynomials, and systems of equations. This quiz covers various topics, from solving quadratic equations to graphing functions and solving systems of equations.