C HAPTER 2 Section 2-6

S PECIAL F UNCTIONS Direct Variation Constant Function/Identity Function Step Function Greater Integer Function Absolute Value Function

D IRECT V ARIATION A linear function in the form y = mx + b and m  0. Note: X increases, 6, 7, 8 And Y increases. 12, 14, 16 Note: X decreases, 10, 5, 3 And Y decreases. 30, 15, 9

D IRECT V ARIATION Y varies directly as x means that y = kx where k is the constant of variation. (see any similarities to y = mx + b?) Another way of writing this is k =

Note: X decreases, -4, -16, -40 And Y decreases. -1,-4,-10 What is the constant of variation of the table above? Since y = kx we can say Therefore: -1/-4=k or k = ¼-4/-16=k or k = ¼ -10/-40=k or k = ¼Note k stays constant. y = ¼ x is the equation! Examples of Direct Variation:

No Yes No Tell if the following graph is a Direct Variation or not.

C ONSTANT F UNCTION Constant Function is a linear function of the form y = b, where b is a constant. It is also written as f ( x ) = b. The graph of a Constant Function is a horizontal line.

I DENTITY F UNCTION Identity Function is a linear function of the form y = x. It is also written as f ( x ) = x. It is called "Identity" because what comes out is identical to what goes in:

G REATEST I NTEGER F UNCTION When greatest integer acts on a number, the value that represents the result is the greatest integer that is less than or equal to the given number. There are several descriptors in that expression. First of all you are looking only for an integer. Secondly, that integer must be less than or equal to the given number and finally, of all of the integers that satisfy the first two criteria, you want the greatest one. The brackets which indicate that this operation is to be performed is as shown: ‘[ ]’.1 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.

f(x) = [x]xf(0) = [0] = 00 f(0.5) = [0.5] = f(0.7) = [0.7] = f(0.8) = [0.8] = f(0.9) = [0.9] = f(1) = [1] = 1 1 f(1.5) = [1.5] = f(1.6) = [1.6] = f(1.7) = [1.7] = f(1.8) = [1.8] = f(1.9) = [1.9] = f(2) = [2] = 2 2 f(-1) = [-1] = -1 f(-0.5) =[-0.5]= f(-0.9) =[-0.9]=

When all these points are strung together the graph looks something like this – 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 can’t really state the last (most right) x-value on each step because there is always another to the right of the last one you may name. So instead we describe the first x-value that is NOT on a given step. Example: (1,0)

5.Slope through closed points of each step = ab 1. Starting point: f(0) Five Steps to Construct Greatest Integer Graph b > 0 b < 0 2. Orientation of each step: 4. Vertical distance between Steps = |a| f(x) = a[bx - h] + k

S TEP F UNCTION A step function is a special type of function whose graph is a series of line segments. The graph of a step function looks like a series of small steps.

A BSOLUTE V ALUE F UNCTION Recall what the absolute value of a variable means What does y=|x| look like? *Hint: make a quick table

A BSOLUTE V ALUE F UNCTION

Lets graph y=|x| y=|x|+2 y=|x|-4 y=|x+3| y=|x-6| What do you think y=|x+3|-4 looks like? Create your formula

A BSOLUTE V ALUE F UNCTION y=|x-h|+v Where (h,v) is your vertex What could 'h' and 'v' stand for? Why is 'h' negative?

S PECIAL F UNCTIONS Direct Variation Constant Function/Identity Function Step Function Greater Integer Function Absolute Value Function