1. p. 82

2. We will find that the “shifting technique” applies to ALL functions. If the addition or subtraction occurs prior to the function occurring, then it is a horizontal shift. If the addition or subtraction occurs after the function occurs, then it is a vertical shift. Absolute Value Function The absolute value function is defined by f(x) = |x|. If x ≥ 0 then the graph coincides with the line y = x. If x < 0 then the graph coincides with the line y = -x

3. y x f(x)=|x| Ex 1: Use the graph of y = |x| to sketch the graph of g(x) = |x – 1| - 2 y x 1 -2

4. Ex 2: Sketch the graph of h(x) = -|2x – 1| + 2 Note: Follow the order of operations to make the changes to your new graph. x y 4 things will happen to the graph of the parent function to get our new graph: The output of f(x) = |x| will be multiplied by a factor of 2 The graph will be shifted 1/2 unit to the right The graph will be shifted 2 units up. The graph will be reflected over the x-axis

5. Ex 3: Sketch the graph of f(x)= |2x – 6| + 1 Solution: Before you can see horizontal and vertical shifts, the leading coefficient must be 1. f(x)=| 2(x-3) | + 1 x y 3 1 y = |x| f(x)=|2x – 6| + 1

6. Ex: 4 Sketch the graph of Solution:We can see that we have a composition of functions (one function within another) Just like the order of operations we work from the inside out. Factor the quadratic to find the x-intercepts of the parabola. f(x) = | (x - 3)(x – 1) |x-int: (3, 0), (1, 0) Graph the parabola: you may need to complete the square to find the vertex. y x (connect the points to show the parabola)

7. Now, we take into account the second function in this problem…the absolute value function Recall that the absolute value measures the distance from zero, therefore, distance cannot be negative. Any output value that is negative will now become positive. y x 3 3 Now, connect your points with a smooth curve.

8. Square Root Function Parent function: The square root function is increasing on its entire domain. It has a minimum value of zero at x = 0. y x Connect your points with a smooth curve.

9. Ex 5: Sketch the graph of This graph is the graph of the parent function shifted two units to the right and one unit down. X-intercept: (3,0) Ex 6: Sketch the graph of The leading coefficient must be 1 before the horizontal shift can be seen. This graph is the graph from the last example reflected over the y-axis. Note: When the negative remains inside the function it is a reflection over the y-axis, if the negative is outside the function it is a reflection over the x-axis.

10. Greatest Integer Function Denoted: It is defined for a real number x to be the largest integer that is less than or equal to x. Ex: The greatest integer function has a wide application… Floor Function: used in computer science Denoted: Ceiling Function: Denoted: We round down We round up

11. The greatest integer function has a range with gaps and its graph “jumps.” The output of the greatest integer function is an integer… …and so on… The graph of the Greatest Integer Function y xGraph together

12. Again, the rules do not change for the shifting technique… Ex: Sketch the graph of Calculators (graphing) permit you to set the number of decimal places that you want your answer to round to. Such a function would look like… (computer science)