1. 22 – The Absolute Value Function No Calculator
Piecewise Investigations
22 – The Absolute Value Function No Calculator

2. But……. The Absolute Value Function
The absolute value function is a two-piece piecewise function:
Piece 1 – The expression without absolute value when the expression is greater than or equal to zero.
Piece 2 – The opposite of the expression without absolute value when the expression is less than zero.
But…….

3. This is not how we usually write a piecewise function…..
We simplify each piece and the inequalities so we can work with them.
We will call this a ‘simplified piecewise function.’

4. Rewrite each of the following as a simplified piecewise function.

5. Rewrite each of the following as a simplified piecewise function.

6. Rewrite each of the following as a simplified piecewise function.

7. Type of Transformation
Transformations Review (See Notes Section 15)
Replacement Function
Type of Transformation
Effect on Graph of f(x)
f(x) + k
vertical
translation
If k > 0, translate up
If k < 0, translate down
f(x + k)
horizontal translation
If k > 0, translate left
If k < 0, translate right
kf(x)
stretch/compress
If 0 < k < 1, vertical compression
If k > 1, vertical stretch
If -1 < k < 0, reflect over x-axis, vertical compression
If k < -1, reflect over x-axis, vertical stretch
If k = -1, reflect over x-axis
f(kx)
horizontal
If 0 < k < 1, horizontal stretch
If k > 1, horizontal compression
If -1 < k < 0, reflect over y-axis, horizontal stretch
If k < -1, reflect over y-axis vertical compression
If k = -1, reflect over y-axis

8. Vertical translation down 2
Special Note:
Just because an equation has an absolute value in it, this does not mean the answer is always positive.

9. Horizontal translation left 3

10. Vertical Stretch – Factor of 2

11. Horizontal Translation – 3 right
Vertical Translation – 2 up

12. Horizontal Translation – 2 left
Vertical Compression – Factor of ½
Vertical Translation – 4 down

13. Reflect negative values over x-axis.

14. Step 1 – Reflect negative values over x-axis.

15. Step 1 – Reflect negative values over x-axis.
Step 2 – Vertical translation down 3.

16. Absolute Value Equations