Algebraic Equations and Factoring

Solving Quadratic Equations by Factoring

Factoring is a method used to solve quadratic equations by expressing the equation as a product of its factors and setting each factor equal to zero.

• Quadratic Equation: An equation of the form .

• Factoring: Rewrite the equation so that one side equals zero, then factor the quadratic expression.

• Zero Product Property: If , then or .

• Example: Solve by moving all terms to one side and factoring.

Functions and Their Graphs

Identifying and Graphing Quadratic Functions

Quadratic functions are graphed as parabolas. The standard form is , where is the vertex.

• X-intercepts: Points where the graph crosses the x-axis ().

• Vertex: The turning point of the parabola.

• Example: For , set to find x-intercepts: or .

Transformations of Functions

Transformations shift, stretch, or reflect the graph of a function.

• Horizontal Shift: shifts right by units.

• Vertical Shift: shifts up by units.

• Example: is a horizontal stretch and shift of .

Graphing and Analyzing Polynomial Functions

Polynomial functions can be analyzed for their end behavior using the leading coefficient and degree.

• Leading Coefficient Test: Determines the end behavior of the graph.

• Even Degree, Negative Leading Coefficient: Both ends fall.

• Example: falls to the left and right.

Variation and Proportionality

Direct and Inverse Variation

Variation describes how one variable changes in relation to another.

• Direct Variation: (as increases, increases proportionally).

• Inverse Variation: (as increases, decreases proportionally).

• Combined Variation: varies directly as and inversely as : .

• Example: If varies directly as and inversely as the cube root of , .

Linear Functions and Applications

Modeling with Linear Equations

Linear equations can model real-world situations such as depreciation.

• Standard Form: where is the slope and is the y-intercept.

• Depreciation Example: A laptop worth $1000 per year: .

• Solving for a Value: Set and solve for to find when the laptop is worth $600$.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form .

• Growth: If , the function increases rapidly.

• Decay: If , the function decreases rapidly.

• Example: is an exponential decay function.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions.

• General Form: .

• Domain: .

• Vertical Asymptote: .

• Example: and .

Table: Properties of Logarithmic and Exponential Functions

Calculator Skills and Approximations

Evaluating Expressions

Some expressions require calculator approximation, especially with irrational numbers or exponents.

• Example: can be approximated to four decimal places using a scientific calculator.

Trigonometric Functions

Exact Values of Trigonometric Functions

Trigonometric functions have exact values for special angles.

• Example: .

Verifying and Correcting Equations

Properties of Exponents and Logarithms

Equations involving exponents and logarithms can be verified using their properties.

• Example: is false. The correct statement is .

• Logarithm Power Rule: .

Summary Table: Key Function Types and Their Properties

Additional info: The problems also include graph identification, domain and range analysis, and application of function transformations, all of which are core Precalculus skills.