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Learn how to find the equation of a line through given points and explore various types of linear functions. This guide covers the direct variation function, constant function, identity function, absolute value function, and greatest integer function. We illustrate how each function is represented mathematically and provide examples, including finding equations and graphing techniques. You'll need a solid grasp of linear equations, slopes, and intercepts to fully appreciate these concepts.

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  1. Do Now • Find the equation of a line through the points (7, -2) and (3, -1).

  2. Do Now: • Find the equation of a line through the points (7, -2) and (3, -1). • y = - ¼ x – ¼

  3. 2-6: Special Functions • Direct Variation: A linear function in the form y = kx, where k 0 • Constant: A linear function in the form y = b • Identity: A linear function in the form y = x • Absolute Value: A function in the form y = |mx + b| + c (m 0) • Greatest Integer: A function in the form y = [x]

  4. y 6 4 y=2x 2 –6 –4 –2 2 4 6 x –2 –4 –6 Direct Variation Function: A linear function in the form y = kx, where k 0.

  5. y = 3 Constant Function: A linear function in the form y = b.

  6. y=x Identity Function: A linear function in the form y = x.

  7. y=|x - 2|-1 Absolute Value Function: A function in the form y = a|x - b| + c (m 0) Example #1 The vertex, or minimum point, is (2, -1).

  8. y = -|x + 1| Absolute Value Function: A function of the form y = A|x - B| + C (m 0) Example #2 The vertex, or maximum point, is (-1, 0).

  9. Absolute Value Functions Graph y = |x| - 3 The vertex, or minimum point, is (0, -3).

  10. y 6 4 2 y=[x] –6 –4 –2 2 4 6 x –2 –4 –6 Greatest Integer Function: A function in the form y = [x] Note: [x] means the greatest integer less than or equal to x. For example, the largest integer less than or equal to -3.5 is -4.

  11. y 6 4 2 –6 –4 –2 2 4 6 x –2 –4 –6 Greatest Integer Function: A function in the form y = [x] Graph y= [x] + 2

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