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2.6 Day 1 Notes: Special Functions

This lesson covers the concept of piecewise-defined functions and absolute value functions, including how to graph and evaluate them, identify the domain and range, and understand shifts in the graphs.

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2.6 Day 1 Notes: Special Functions

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  1. 2.6 Day 1 Notes:Special Functions

  2. A function that is written using two or more expressions is called a piecewise-defined function. On the graph of a piecewise-defined function, a dot indicates that the point is included in the graph. A circle indicates that the point is not included in the graph.

  3. *Why we use them: sometimes the output of functions depends on the values being put into them. The circled part is the domain of the function.

  4. Example 1: Evaluate the function for the given value of x. a) f(0) b) f(3) c) f(6)

  5. Example 2:Evaluate the function for the given value of x. a) f(0) b) f(–7)

  6. Steps to graph a Piecewise Function: Step 1: Use the domain to see where your line will split. You can draw a dotted line here to show the break. Step 2: Make a table for each “piece.” - Plug in the x-value from the domain to find the corresponding y-value. - Then plug in 2 x-values in the direction that the domain indicates to go. - Draw the line in the correct direction. Step 3: Repeat the same process for each piece. Step 4: Be sure your graph passes the Vertical Line Test to be sure you graphed a function.

  7. Example 3:Graph Identify the domain and range.

  8. Example 4:Graph Identify the domain and range.

  9. A piecewise-linear function only contains a single expression. A common piecewise-linear function is the step function. The graph of a step function consists of line segments. Example 5: Find the value of: a) b) c)

  10. Example 6: The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Which graph represents this situation? a) b) c) d)

  11. 2.6 Day 2 Notes:Special Functions

  12. Another piecewise-linear function is the absolute value function. An absolute value function is a function that contains an algebraic expression within absolute value symbols.

  13. Shifts of an absolute value graph. General equation: Vertex: If a is negative, it opens down. If a is positive, it opens up. If it’s stretchedvertically. (narrower) If it’s compressed vertically. (wider) (+) means left () means right (+) means up () means down

  14. Example 7: Graph y = |x| + 1. Identify the domain and range. Explain each shift.

  15. Example 8: Graph y = |x+2|. Identify the domain and range. Explain each shift.

  16. Example 9: Graph . Identify the domain and range. Explain each shift.

  17. Example 10: Identify the function shown by the graph. Explain each shift. a) y = |x| – 1 b) y = |x – 1| – 1 c) y = |x – 1| d) y = |x + 1| – 1

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