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Understanding Absolute Value Equations and Piecewise Functions in Math II

This lesson explores the relationship between absolute value equations and piecewise functions, as outlined in Math II Unit MM2A1. We'll discuss how piecewise functions can represent different equations for various parts of a domain, demonstrating their application in real-world scenarios, such as ticket pricing. Additionally, we'll learn how to express absolute value functions as piecewise formats and practice transforming equations into their piecewise representations. Join us for an engaging session that ties together essential concepts in mathematics!

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Understanding Absolute Value Equations and Piecewise Functions in Math II

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  1. Daily Check Graph the following equations.

  2. Math II UNIT QUESTION: How are absolute value equations similar to piecewise functions? Standard: MM2A1 Today’s Question: How are absolute value equations similar to piecewise functions? Standard: MM2A1.a,b

  3. Absolute Value as Piecewise Functions Section 2.5

  4. Piecewise Functions • Piecewise functions are functions that can be represented by more than one equation, with each equation corresponding to a different part of the domain. • Piecewise functions do not always have to be line segments. The “pieces” could be pieces of any type of graph. • This type of function is often used to represent real-life problems like ticket prices.

  5. Example x + 1, if x < 1 2, if 1 ≤ x ≤ 3 (x-3)2 + 2, if x > 3 f (x) =

  6. Absolute Value as Piecewise • We usually write an absolute value function as f (x)= x , but since absolute value is a measure of distance and distance is always positive, it also can be written as follows: -x, if x < 0 x, if x ≥ 0 f (x) =

  7. Writing Abs. Value as Piecewise • For I x – h I ≥ 0, simplify the equation given by distributing and combining like terms. • For I x – h I < 0, substitute –(x – h) in place of I x - h I. Then, simplify.

  8. Example • Write y = 2 Ix – 4I – 10 as a piecewise function. • For (x-4) ≥ 0 2(x – 4) – 10 = 2x – 8 – 10 = 2x – 18 (when x ≥ 4) • For (x-4) < 0 2[-(x-4)] – 10 = 2(-x + 4) – 10 = -2x + 8 – 10 = -2x – 2 (when x < 4))

  9. Graphs of Both y=-2x-2 y=2x-18

  10. EOCT Practice A

  11. EOCT Practice C

  12. Writing Abs. Value as Piecewise • Using a graph

  13. Writing Abs. Value as Piecewise • Try this one...

  14. Practice • Worksheet

  15. Homework • Worksheet

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