Special Functions

• Direct Variation: A linear function in the form y=kx, where k 0

• Constant: A linear function in the form y=mx+b, where m = 0, therefore y = b

• Identity: A linear function in the form y=mx+b, where m = 1 and b = 0, therefore y = x

• Absolute Value: A function in the form y = |mx + b| + c (m 0)

• Greatest Integer: A function in the form y = [x]

Direct Variation Function: A linear function in the form y = kx, where k 0

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=2x

Constant Function: A linear function in the form y = mx + b, where m = 0, therefore y = b

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y = 3

Identity Function: A linear function in the form y = mx + b, where m = 1 and b = 0, therefore y = x

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=x

Ex #1: Graph y = |x| by completing a table of values:

x y-2 -1 0 1 2

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

Absolute Value Function: A function in the form y = |mx + b| + c (m 0)

y =|-2| = 2 y =|-1| = 1 y =|0| = 0 y =|1| = 1 y =|2| = 2

The vertex, or minimum point, is (0, 0).

Practice Problems

1. Identify each of the following as constant, identity, or direct variation functiona. f(x) = -½x b. g(x) = x

c. h(x) = 7 d. f(x) = 9x

Integers The integers (from the Latin integer, literally

"untouched", hence "whole“ Are natural numbers including 0 (0, 1, 2, 3, ...) and their

negatives (0, −1, −2, −3, ...) For example, 65, 7, and −756 are integers; 1.6 and 1½

are not integers. In other terms, integers are the numbers one can count

with items such as apples or fingers, and their negatives, as well as 0.

Symbol is “Z” which stands for Zahlen (German for numbers)

Greatest Integer Function: A function in the form y = [x]

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=[x]

Note: [x] means the greatest integer less than or equal to x. Forexample, the largest integer less than or equal to 3.5 is 3. The largestinteger less than or equal to -4.5 is -4.

The open circles mean that the particular point is not included

Greatest Integer Function [x] means the largest integer less than or equal to x

Examples: [8.2] = 8 [3.9] = 3 [5.0] = 5 [7.6] = 7

Example: [1.97] = 1

There are many integers less than 1.97; {1, 0, -1, -2, -3, -4, …} Of all of them, ‘1’ is the greatest.

Example: [-1.97] = -2

There are many integers less than -1.97; {-2, -3, -4, -5, -6, …} Of all of them, ‘-2’ is the greatest.

It may be helpful to visualize this function a little more clearly by using a number line.

0 1 2 3 4 5 6 7 8-1-2-3-4-5-6-7

Example: [6.31] = 6

6.31-6.31

Example: [-6.31] = -7

When you use this function, the answer is the integer on the immediate left on the number line. There is one exception. When the function acts on a number that is itself an integer. The answer is itself.

Example: [5] = 5 Example: [-5] = -5

Example: 06032

. Example: 3825

14

.

32

514

Let’s graph f(x) = [x]

To see what the graph looks like, it is necessary to determine some ordered pairs which can be determined with a table of values.

f(x) = [x]xf(0) = [0] = 00f(1) = [1] = 11f(2) = [2] = 22f(3) = [3] = 33

f(-1) = [-1] = -1-1f(-2) = [-2] = -2-2

If we only choose integer values for x then we will not really see the function manifest itself. To do this we need to choose non-integer values.

f(x) = [x]xf(0) = [0] =

00

f(0.5) = [0.5] = 0

0.5f(0.7) = [0.7] =

00.7

f(0.8) = [0.8] = 0

0.8f(0.9) = [0.9] =

00.9

f(1) = [1] = 11f(1.5) = [1.5] =

11.5

f(1.6) = [1.6] = 1

1.6f(1.7) = [1.7] =

11.7

f(1.8) = [1.8] = 1

1.8f(1.9) = [1.9] =

11.9

f(2) = [2] = 22

f(-1) = [-1] = -1-1

f(-0.5) =[-0.5]=-1

-0.5f(-0.9) =[-0.9]=-1

-0.9

y

x

When all these points are connected the graph looks something like a series of steps.For this reason it is sometimes called the ‘STEP FUNCTION’.

Notice that the left of each step begins with a closed (inclusive) point but the right of each step ends with an open (excluding point)We don’t include the last (most right) x-value on each step

Rather than place a long series of points on the graph, a line segment can be drawn for each step as shown to the right.

f(x) = [x] This is a rather tedious way to construct a graph and for this reason there is a more efficient way to construct it. Basically the greatest integer function can be presented with 4 parameters, as shown below.

f(x) = a[bx - h] + kBy observing the impact of these parameters, we can use them to predict the shape of the graph.

In these 3 examples, parameter ‘a’ is changed. As “a” increases, the distance between the steps increases.

f(x) = [x]

a = 1

f(x) = 2[x]a = 2

f(x) = 3[x]

a = 3

f(x) = a[bx - h] + k

Vertical distance between Steps = |a|

When ‘a’ is negative, notice that the slope of the steps is changed. Downstairs instead of upstairs.

Vertical distance between Steps = |a|

f(x) = -[x]

a = -1

f(x) = -2[x]

a = -2

Graph y= [x] + 2 by completing a table of values

x y -3 -2.75 -2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 0 1

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y x y -3 y= [-3]+2=-1 -2.75 y= [-2.75]+2=-1 -2.5 y= [-2.5]+2=-1 -2.25 y= [-2.25]+2=-1 -2 y= [-2]+2 =0 -1.75 y= [-1.75]+2=0 -1.5 y= [-1.5]+2=0 -1.25 y= [-1.25]+2=0 -1 y= [-1]+2=1 0 y= [0]+2=2 1 y= [1]+2=3

f(x) = [x]

b = 1

f(x) = [2x]

b = 2

xxf 21)(

21

b

As ‘b’ is increased from 1 to 2, each step gets shorter. Then as it is decreased to 0.5, the steps get longer.

bStepofLength 1

bStepofLength 1.3

5. Slope through closed points of each step = ab

1. Starting point: f(0)

Five Steps to Construct Greatest Integer Graph

b > 0b < 0

2. Orientation of each step:

4. Vertical distance between Steps = |a|

f(x) = a[bx - h] + k

f(x) = 3[-x – 3] + 5 a = 3; b = -1; h = 3; k = 51. Starting point: (0,-4)

f(0) = 3[-(0)-3]+5=3(-3)+5 = -

42. Orientation of each step:b <

0

111

1.3

b

StepofLength

5. Slope through closed points of each step

= ab

13

13

313

or

)(4. Vertical distance between Steps

=|3| = 3

= |a|

Practice problems

1. Identify each of the following as constant, identity, direct variation, absolute value, or greatest integer function

a. h(x) = [x – 6] e. f(x) = 3|-x + 1|b. f(x) = -½x f. g(x) = xc. g(x) = |2x| g. h(x) = [2 + 5x] d. h(x) = 7 h. f(x) = 9x

2. Graph the equation y = |x – 6| Hint: When completing the table of

values, you will need some bigger values for x, like x = 6, x = 7, x = 8

Answers

a) greatest integer function b) direct variation c) absolute value d) constant e) absolute value f) identity g) greatest integer function h) direct variation