1. Modulus Function Knowledge Point: The modulus function, or absolute value, is a function that returns the magnitude of a number regardless of its sign. For example, the modulus of 3 is 3, and the modulus of -3 is also 3. Graphical Example: I will create a graph for the function 𝑦 = ∣ 𝑥 ∣ y=∣x∣, where 𝑥 x is a variable, to illustrate how the modulus function works. 2. Solving Equations with Modulus Knowledge Point: When solving equations involving modulus functions, we must consider both positive and negative scenarios. For instance, solving ∣ 𝑥 ∣ = 3 ∣x∣=3 gives 𝑥 = 3 x=3 or 𝑥 = − 3 x=−3. Graphical Example: I'll graphically demonstrate an equation like 𝑦 = ∣ 𝑥 − 2 ∣ y=∣x−2∣ and show how it behaves differently on either side of 𝑥 = 2 x=2. 3. Algebraic Identities Knowledge Point: Fundamental algebraic identities such as ( 𝑎 + 𝑏 ) 2 = 𝑎 2 + 2 𝑎 𝑏 + 𝑏 2 (a+b) 2 =a 2 +2ab+b 2 are essential for simplifying and solving equations. Graphical Example: We can generate a visual representation of a quadratic function and demonstrate how it expands using the identity.

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The modulus function, also known as the absolute value function, is a fundamental mathematical concept. It returns the magnitude of a number without considering its sign. The graph of y equals absolute value of x creates a distinctive V-shape, symmetric about the y-axis. When solving equations involving modulus functions, we must consider both positive and negative scenarios. For example, if the absolute value of x equals 3, then x can be either 3 or negative 3. The graph shows how the modulus function intersects with a horizontal line at two points, representing the two solutions. Algebraic identities are fundamental tools in mathematics. The square of a binomial, a plus b squared, equals a squared plus 2ab plus b squared. This identity helps us expand and simplify quadratic expressions. The graph shows a parabola representing this quadratic relationship. Complex modulus equations like the absolute value of 2x minus 3 equals 5 require systematic analysis. We solve by considering both positive and negative cases. The graph shows the V-shaped function intersecting the horizontal line at two points, giving us solutions x equals negative 1 and x equals 4. In summary, we've explored three fundamental mathematical concepts. The modulus function returns the magnitude of numbers, creating V-shaped graphs. When solving modulus equations, we must consider both positive and negative cases. Algebraic identities like the square of a binomial help us expand and simplify expressions. These concepts are essential building blocks for advanced mathematics.