1. العلاقات والدوال
أ. ريما عباس
ريض 152

2. 2-1 Relations and Functions
A set of ordered pairs forms a relation.
Example: {(2, 4) (0, 3) (4, -2) (-1, -8)}
The domain is the set of all the first coordinates (x-coordinate) and the range is the set of all the second coordinates (y-coordinate).
Example: domain: {2, 0, 4, -1} and range: {4, 3, -2, -8}
Mapping shows how each member of the domain and range are paired.
Example: 7

3. 2-1 Relations and Functions (cont.)
A function is a relation where an element from the domain is paired with only one element from the range.
Example (from mapping example): The first is a function, but the second is not because the 1 is paired with both the 3 and the 0.
If you can draw a vertical line everywhere through the graph of a relation and that line only intersects the graph at one point, then you have a function.

4. 2-1 Relations and Functions (cont.)
A discrete function consists of individual points that are not connected.
When the domain of a function can be graphed with a smooth line or curve, then the function is called continuous.

5. 2-1 Practice Find the domain and range of the following:
{(3, 6) (-1, 5) (0, -2)}
{(4, 1) (1, 0) (3, 1) (1, -2)}
Are the following functions? If yes, are they discrete or continuous?
{(2, 2) (3, 6) (-2, 0) (0, 5)} c. y = 8x2 + 4
{(9, 3) (8, -1) (9, 0) (9, 1) (0, -4)}
Answers: 1)a) domain: {3, -1, 0} range: {6, 5, -2} b) domain: {4, 1, 3} range: {1, 0, -2} 2)a) function; discrete b) not a function c) function; continuous

6. 2-2 Linear Equations
A linear equation is an equation whose graph is a straight line. The standard form of a linear equation is: Ax + By = C, where A, B, and C are all integers and A and B cannot both be 0.
Linear functions have the form f(x) = mx + b, where m and b are real numbers.
A constant function has a graph that is a straight, horizontal line. The equation has the form f(x) = b

7. 2-2 Linear Equations (cont.)
The point on the graph where the line crosses the y-axis is called the y-intercept.
Example: find the y-intercept of 4x – 3y = 6
4(0) – 3y = 6  substitute 0 for x
y = -2, so the graph crosses the y-axis at the point (0, -2)
The point on the graph where the line crosses the x-axis is called the x-intercept.
Example: find the x-intercept of 3x + 5y = 9
3x + 5(0) = 9  substitute 0 for y
x = 3, so the graph crosses the x-axis at the point (3, 0)

8. 2-2 Practice
Determine if the following are linear equations. If so, write the equation in standard form and determine A, B, and C.
4x + 3y = 10 c. 5 – 3y = 8x
x2 + y = 2 d. 1/x + 4y = -5
Find the x- and y-intercepts of the following:
4x – 3y = -12
½ y + 2 = ½ x
2)a) x-intercept: -3 y-intercept: 4 b) x-intercept: 4 y-intercept: -4
Answers: 1)a) yes; A = 4, B = 3, C = 10 b) no c) yes; A = 8, B = 3, C = 5 d) no

9. 2-3 Slope The slope of a line is the change in y over the change in x.
If a line passes through the points (x1, y1) and (x2, y2), then the slope is given by
m = y2 – y1 , where x1 ≠ x2.
x2 – x1
In an equation with the from y = mx + b, m is the slope and b is the y-intercept.
Two lines with the same slope are parallel.
If the product of the slopes of two lines is -1, then the lines are perpendicular.

10. 2-3 Practice Find the slope of the following:
(3.5, -2) (0, -16) e. 12x + 3y – 6 = 0
y = 3x + b f. y = -7
Determine whether the following lines are perpendicular or parallel by finding the slope.
(4, -2) (6, 0), (7, 3) (6, 2)
y = 2x – 3, (6, 6) (4, 7)
Answers: 1)a) -2 b) 4 c) 3 d) 0 e) -4 f) 0 2)a) 1; parallel b) -1; perpendicular

11. 2-4 Writing Linear Equations
The form y = mx + b is called slope-intercept form, where m is the slope and b is the y-intercept.
The point-slope form of the equation of a line is y – y1 = m(x – x1). Here (x1, y1) are the coordinates of any point found on that line.

12. 2-4 Writing Linear Equations
Example: Find the slope-intercept form of the equation passing through the point (-3, 5) with a slope of 2.
y = mx + b
5 = (2)(-3) + b
5 = -6 + b
b = 11
y = 2x + 11

13. 2-4 Writing Linear Equations
Example: Find the point-slope form of the equation of a line that passes through the points (1, -5) and (0, 4).
m = y2 – y1 y – y1 = m(x – x1)
x2 – x1 y – (-5) = (-1)(x – 1)
m = y + 5 = -x + 1
0 – 1 y = -x – 4
m = 9 -1
m = -1

14. 2-4 Practice Find the slope-intercept form of the following:
a line passing through the point (0, 5) with a slope of -7
a line passing through the points (-2, 4) and (3, 14)
Find the point-slope form of the following:
a line passing through the point (-2, 6) with a slope of 3
a line passing through the points (0, -9) and (-2, 1)
Answers: 1)a) y = -7x + 5 b) y = 2x + 8 2)a) y = 3x + 12 b) y = -5x -9

15. 2-5 Modeling Real-World Data Using Scatter Plots
Plotting points that do not form a straight line forms a scatter plot.
The line that best represents the points is the best-fit line.
A prediction equation uses points on the scatter plot to approximate through calculation the equation of the best-fit line.

16. 2-5 Practice
Plot the following data. Approximate the best-fit line by creating a prediction equation.
Person
ACT Score 1 15 2 19 3 21 4 28 5 30 6 35
Answers: 1) y = 4x + 11

17. 2-6 Special Functions
Whenever a linear function has the form y = mx + b and b = 0 and m ≠ 0, it is called a direction variation.
A constant function is a linear function in the form y = mx + b where m = 0.
An identity function is a linear function in the form y = mx + b where m = 1 and b = 0.

18. 2-6 Special Functions
Step functions are functions depicted in graphs with open circles which mean that the particular point is not included.
Example:

19. 2-6 Special Functions
A type of step function is the greatest integer function which is symbolized as [x] and means “the greatest integer not greater than x.”
Examples: [8.2] = [3.9] = [5.0] = [7.6] = 7
An absolute value function is the graph of the function that represents an absolute value.
Examples: |-4| = |-9| = 9

20. 2-6 Practice
Identify each of the following as constant, identity, direct variation, absolute value, or greatest integer function
h(x) = [x – 6] e. f(x) = 3|-x + 1|
f(x) = -½ x f. g(x) = x
g(x) = |2x| g. h(x) = [2 + 5x]
h(x) = 7 h. f(x) = 9x
Graph the equation y = |x – 6|

21. 2-6 Answers
Answers: 1)a) greatest integer function b) direct variation c) absolute value d) constant e) absolute value f) identity g) greatest integer function h) direct variation )

22. 2-7 Linear Inequalities Example: Graph 2y – 8x ≥ 4
Graph the “equals” part of the equation.
2y – 8x = 4
2y = 8x + 4
y = 4x + 2
x-intercept
0 = 4x + 2
-2 = 4x
-1/2 = x
y-intercept
y = 4(0) +2
y = 2

23. 2-7 Linear Inequalities
Use “test points” to determine which side of the line should be shaded. (2y – 8x ≥ 4)
(-2, 2)
2(2) – 8(-2) ≥ 4
4 – (-16) ≥ 4
20 ≥ 4  true
(0, 0)
2(0) – 8(0) ≥ 4
0 – 0 ≥ 4
0 ≥ 4  false
So we shade the side of the line that includes the “true” point, (-2, 2)

24. 2-7 Linear Inequalities Example: Graph 12 < -3y – 9x
Graph the line.
12 ≠ -3y – 9x
3y ≠ -9x – 12
y ≠ -3x – 4
x-intercept
0 = -3x – 4
4 = -3x
-4/3 = x
y-intercept
y = 3(0) – 4
y = -4

25. 2-7 Linear Inequalities
Use “test points” to determine which side of the line should be shaded. (12 < -3y – 9x)
(-3, -3)
12 < -3(-3) – 9(-3)
12 <
12 < 36  true
(0, 0)
12 < -3(0) – 9(0)
12 < 0 – 0
12 < 0  false
So we shade the side of the line that includes the “true” point, (-3, -3)

26. 2-7 Problems Graph each inequality. 2x > y – 4 e. 2y ≥ 6|x|
5 ≥ y f. 42x > 7y
4 < -2y g. |x| < y + 2
y ≤ |x| h. x – 4 ≤ 8y

27. 2-7 Answers
1)a) b)
c) d)

28. 2-7 Answers
1)e) f)
g) h)