Equations and Inequalities

Solving Linear and Quadratic Equations

Solving equations is a fundamental skill in algebra. Linear equations have the form ax + b = 0, while quadratic equations have the form ax^2 + bx + c = 0. Solutions can be found by isolating the variable or using the quadratic formula.

• Linear Equation Example: →

• Quadratic Formula:

• Example: → or

Solving Equations with Logarithms and Exponents

Logarithmic and exponential equations require knowledge of their properties. For example, to solve , rewrite as .

• Example: →

• Exact Logarithm Value:

Graphs and Functions

Graphing Trigonometric Functions

Trigonometric functions such as cosine can be transformed by changing amplitude, period, and phase. The function has an amplitude of 0.4 and is reflected across the x-axis due to the negative sign.

• Amplitude: 0.4

• Period:

• Intercepts: and

• Key Points: At , ; at ,

Example: The graph below shows for from $0. The negative sign inverts the cosine curve.

Graphing Logarithmic Functions

Logarithmic functions have the form . The graph of increases slowly for large and is undefined for .

• Domain:

• Range: All real numbers

• Example:

Polynomials and Rational Functions

Simplifying Expressions with Exponents and Radicals

Expressions with exponents and radicals can be simplified using exponent rules and rationalizing denominators.

• Positive Exponents:

• Example:

• Radical Simplification:

• Rationalizing Denominator: →

Inverse, Exponential, and Logarithmic Functions

Solving Logarithmic and Exponential Equations

To solve equations involving logarithms and exponentials, use properties such as and .

• Example: →

• Common Logarithm:

Systems and Matrices

Solving Systems of Equations

Systems of equations can be solved using substitution, elimination, or matrix methods. For example, finding the intersection of two lines or solving for unknowns in a circuit.

• Example: →

• Resistance Calculation:

Conic Sections

Equations of Circles, Ellipses, and Parabolas

Conic sections include circles, ellipses, and parabolas. Their equations depend on the center, radius, and other parameters.

• Circle:

• Ellipse:

• Parabola: or

• Example: Parabola with vertex (0,0) and directrix

Geometry and Trigonometry Applications

Distance, Area, and Triangle Solutions

Geometry and trigonometry are used to find distances, areas, and solve triangles using the Law of Sines and Law of Cosines.

• Distance Formula:

• Area of Circle:

• Law of Sines:

• Law of Cosines:

• Example: Find angle between routes using Law of Cosines

Additional info:

• Some answers and steps were inferred from context and standard algebraic methods.

• Graph and table for included for visual reinforcement.