p. 82

We will find that the “shifting technique” applies to ALL functions. If the additionor 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 verticalshift.

Absolute Value Function

0,

0,

xifx

xifxx

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

xasx

y

x

f(x)=|x|

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

x1

-2

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

2)

21

(2)( xxh

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

Ex: 4 Sketch the graph of 34)( 2 xxxf

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

1)2()( 2 xxf

(connect the points to show the parabola)

Now, we take into account the second function in this problem…the absolutevalue function

Recall that the absolute value measures the distance from zero, therefore, distancecannot be negative.

Any output value that is negative will now become positive.

y

x

3

3

Now, connect your points witha smooth curve.

Square Root Function

Parent function: xxf )( ),0[ fD

),0[ fR

The square root function is increasing on its entire domain. It has a minimumvalue of zero at x = 0.

y

x Connect your points

with a smooth curve.

Ex 5: Sketch the graph of 12)( xxg

This graph is the graph of the parent function shifted two units to theright and one unit down.

X-intercept: (3,0)

Ex 6: Sketch the graph of 12)( xxh

The leading coefficient must be 1 before the horizontal shift can be seen.

1)2()( xxh

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.

Greatest Integer Function

Denoted: xxf )( It is defined for a real number x to be thelargest integer that is less than or equal to x.

Ex: 033.0,22

251.1,12.1

The greatest integer function has a wide application…

Floor Function: used in computer science

Denoted: xxf )(

Ceiling Function:

Denoted: xxf )(

We round down

We round up

The greatest integer function has a range with gaps and its graph “jumps.”

The output of the greatest integer function is an integer…

2...12.... xthenxWhen

1...01 xthenx

0...10 xthenx …and so on…

The graph of the Greatest Integer Function xxf )(y

x Graph together

Again, the rules do not change for the shifting technique…

Ex: Sketch the graph of 23)( xxg

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)

15.0,1

5.0,)(

xxxifx

xxxifxxf