Special Functions Direct Variation A linear function in
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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 y 6 4 y=2 x 2 – 6 – 4 – 2 2 – 4 – 6 4 6 x
Constant Function: A linear function in the form y = mx + b, where m = 0, therefore y = b y=3
Identity Function: A linear function in the form y = mx + b, where m = 1 and b = 0, therefore y = x y=x
Absolute Value Function: A function in the form y = |mx + b| + c (m 0) Ex #1: Graph y = |x| by completing a table of values: x -2 -1 0 1 2 y 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 function a. f(x) = -½x b. g(x) = x c. h(x) = 7 d. f(x) = 9 x
Integers 4 The integers (from the Latin integer, literally "untouched", hence "whole“ 4 Are natural numbers including 0 (0, 1, 2, 3, . . . ) and their negatives (0, − 1, − 2, − 3, . . . ) 4 For example, 65, 7, and − 756 are integers; 1. 6 and 1½ are not integers. 4 In other terms, integers are the numbers one can count with items such as apples or fingers, and their negatives, as well as 0. 4 Symbol is “Z” which stands for Zahlen (German for numbers)
Greatest Integer Function: A function in the form y = [x] y less than or equal to x. For Note: [x] means the greatest integer example, the largest integer less 6 than or equal to 3. 5 is 3. The largest integer less than or equal to -4. 5 is -4. 4 2 – 6 – 4 – 2 y=[x] 2 4 6 x – 2 – 4 – 6 The open circles mean that the particular point is not included
Greatest Integer Function 4 [x] means the largest integer less than or equal to x 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. Examples: [8. 2] = 8 [5. 0] = 5 [3. 9] = 3 [7. 6] = 7
It may be helpful to visualize this function a little more clearly by using a number line. -6. 31 -7 -6 6. 31 -5 -4 -3 -2 Example: [6. 31] = 6 -1 0 1 2 3 4 5 6 7 8 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:
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. x 0 1 f(x) = [x] f(0) = [0] = 0 f(1) = [1] = 1 2 3 -1 -2 f(2) = [2] = 2 f(3) = [3] = 3 f(-1) = [-1] = -1 f(-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 noninteger values.
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) xvalue 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] + k By observing the impact of these parameters, we can use them to predict the shape of the graph.
f(x) = [x] f(x) = a[bx - h] + k f(x) = 2[x] a=1 In these 3 examples, a = 2 parameter ‘a’ is changed. As “a” increases, the distance between the steps increases. a=3 f(x) = 3[x] Vertical distance between Steps = |a|
f(x) = -[x] a = -1 f(x) = -2[x] a = -2 When ‘a’ is negative, notice that the slope of the steps is changed. Downstairs instead of upstairs. Vertical distance between Steps = |a|
Graph y= [x] + 2 by completing a table of values x -3 -2. 75 -2. 25 -2 -1. 75 -1. 25 -1 0 1 y y= [-3]+2=-1 y= [-2. 75]+2=-1 y= [-2. 25]+2=-1 y= [-2]+2 =0 y= [-1. 75]+2=0 y= [-1. 25]+2=0 y= [-1]+2=1 y= [0]+2=2 y= [1]+2=3 y 6 4 2 – 6 – 4 – 2 – 4 – 6 2 4 6 x
f(x) = [x] b=1 f(x) = [2 x] b=2 As ‘b’ is increased from 1 to 2, each step gets shorter. Then as it is decreased to 0. 5, the steps get longer.
Five Steps to Construct Greatest Integer Graph f(x) = a[bx - h] + k 1. Starting point: f(0) 2. Orientation of each step: b > 0 b < 0 4. Vertical distance between Steps = |a| 5. Slope through closed points of each step = ab
f(x) = 3[-x – 3] + 5 a = 3; b = -1; h = 3; k = 5 1. Starting point: (0, -4) f(0) = 3[-(0)-3]+5 =3(-3)+5 = -4 2. Orientation of each step: b < 0 4. Vertical distance between Steps = |a| =|3| = 3 5. Slope through closed points of each step = ab
Practice problems 1. Identify each of the following as constant, identity, direct variation, absolute value, or greatest integer function a. b. c. d. 2. h(x) = [x – 6] f(x) = -½x g(x) = |2 x| h(x) = 7 e. f. g. h. f(x) = 3|-x + 1| g(x) = x h(x) = [2 + 5 x] f(x) = 9 x 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 4 a) greatest integer function 4 b) direct variation 4 c) absolute value 4 d) constant 4 e) absolute value 4 f) identity 4 g) greatest integer function 4 h) direct variation
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