Questions and Answers
What is the correct way to write the equation y = 2x + 3 in standard form?
Which of the following equations represents a linear function?
What is the slope of the line represented by the equation y = -2x + 4?
What is the y-intercept of the line represented by the equation 2x - 3y = 6?
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Which of the following is a solution to the equation 2x + y = 5?
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What is the solution to the system of equations x + y = 5 and x - y = 1?
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Which of the following represents a quadratic equation?
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What is the domain of the function f(x) = √(x - 2)?
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Study Notes
Algebra
Definitions
- Algebra: a branch of mathematics that deals with variables and their relationships, often expressed through symbols, equations, and functions.
- Variable: a symbol that represents a value that can change.
- Constant: a value that does not change.
Equations and Inequalities
- Equation: a statement that says two expressions are equal, often written as
ax + by = cwherea,b, andcare constants. - Inequality: a statement that says one expression is greater than, less than, or equal to another, often written as
ax + by > c,ax + by < c, orax + by ≥ c. - Linear Equation: an equation in which the highest power of the variable is 1, e.g.
2x + 3 = 5. - Quadratic Equation: an equation in which the highest power of the variable is 2, e.g.
x^2 + 4x + 4 = 0.
Functions
- Function: a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
- Domain: the set of values that can be input into a function.
- Range: the set of values that a function can output.
- Linear Function: a function in which the output is directly proportional to the input, e.g.
f(x) = 2x + 3. - Quadratic Function: a function in which the output is proportional to the square of the input, e.g.
f(x) = x^2 + 4x + 4.
Graphing
- Cartesian Plane: a grid system used to graph functions, with the x-axis representing the input and the y-axis representing the output.
- X-intercept: the point at which a function crosses the x-axis.
- Y-intercept: the point at which a function crosses the y-axis.
- Slope: a measure of how steep a line is, calculated as rise over run (m = Δy / Δx).
Systems of Equations
- System of Equations: a set of two or more equations that must be true at the same time.
- Substitution Method: a method for solving systems of equations by substituting one equation into another.
- Elimination Method: a method for solving systems of equations by adding or subtracting equations to eliminate one variable.
Important Concepts
- Distributive Property: a property that states that multiplication can be distributed over addition, e.g.
a(b + c) = ab + ac. - FOIL Method: a method for multiplying two binomials, e.g.
(a + b)(c + d) = ac + ad + bc + bd. - Factoring: the process of expressing an expression as a product of simpler expressions, e.g.
x^2 + 5x + 6 = (x + 3)(x + 2).
Algebra
Definitions
- Algebra is a branch of mathematics dealing with variables and their relationships, expressed through symbols, equations, and functions.
- A variable is a symbol representing a value that can change.
- A constant is a value that does not change.
Equations and Inequalities
- An equation is a statement saying two expressions are equal, often written as
ax + by = cwherea,b, andcare constants. - An inequality is a statement saying one expression is greater than, less than, or equal to another.
- A linear equation has the highest power of the variable as 1, e.g.
2x + 3 = 5. - A quadratic equation has the highest power of the variable as 2, e.g.
x^2 + 4x + 4 = 0.
Functions
- A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
- The domain is the set of values that can be input into a function.
- The range is the set of values a function can output.
- A linear function has output directly proportional to the input, e.g.
f(x) = 2x + 3. - A quadratic function has output proportional to the square of the input, e.g.
f(x) = x^2 + 4x + 4.
Graphing
- The Cartesian Plane is a grid system used to graph functions, with x-axis representing input and y-axis representing output.
- X-intercept is the point at which a function crosses the x-axis.
- Y-intercept is the point at which a function crosses the y-axis.
- Slope measures how steep a line is, calculated as rise over run (m = Δy / Δx).
Systems of Equations
- A system of equations is a set of two or more equations that must be true at the same time.
- The substitution method solves systems of equations by substituting one equation into another.
- The elimination method solves systems of equations by adding or subtracting equations to eliminate one variable.
Important Concepts
- The distributive property states that multiplication can be distributed over addition, e.g.
a(b + c) = ab + ac. - The FOIL method multiplies two binomials, e.g.
(a + b)(c + d) = ac + ad + bc + bd. - Factoring expresses an expression as a product of simpler expressions, e.g.
x^2 + 5x + 6 = (x + 3)(x + 2).
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Description
Test your understanding of basic algebra concepts such as variables, constants, equations, and inequalities. Learn the definitions and properties of these algebraic expressions.
