1. Grade 10 Mathematics
Graphs Application

2. Graphs, Linear Equations, and Functions
3-1 The Rectangular Coordinate System 3-2 The Slope of a Line 3-3 Linear Equations in two variables 3-5 Introduction to Functions

3. 3-1 The Rectangular Coordinate System
Plotting ordered pairs.
An “ordered pair” of numbers is a pair of numbers written within parenthesis in which the order of the numbers is important.
Example 1: (3,1), (-5,6), (0,0) are ordered pairs.
Note: The parenthesis used to represent an ordered pair are also used to represent an open interval. The context of the problem tells whether the symbols are ordered pairs or an open interval.

4. 3-1 The Rectangular Coordinate System
Graphing an ordered pair requires the use of graph paper and the use of two perpendicular number lines that intersect at their 0 points. The common 0 point is called the “origin”. The horizontal number line is referred to as the “x-axis” or “abscissa” and the vertical line is referred to as the “y-axis” or “ordinate”. In an ordered pair, the first number refers to the position of the point on the x-axis, and the second number refers to the position of the point on the y-axis.

5. 3-1 The Rectangular Coordinate System
Plotting ordered pairs.
The x-axis and the y-axis make up a “rectangular or Cartesian” coordinate system.
Points are graphed by moving the appropriate number of units in the x direction, than moving the appropriate number of units in the y direction. (point A has coordinates (3,1), the point was found by moving 3 units in the positive x direction, then 1 in the positive y direction)

6. 3-1 The Rectangular Coordinate System
The four regions of the graph are called quadrants. A point on the x-axis or y-axis does not belong to any quadrant (point E). The quadrants are numbered.

7. 3-1 The Rectangular Coordinate System
Finding ordered pairs that satisfy a given equation.
To find ordered pairs that satisfy an equation, select any number for one of the variables, substitute into the equation that value, and solve for the other variable.
Example 2: For 3x – 4y = 12, complete the table shown: Solution

8. 3-1 The Rectangular Coordinate System
Graphing lines: Since an equation in two variables is satisfied with an infinite number of ordered pairs, it is common practice to “graph” an equation to give a picture of the solution. Note: An equation like 2x +3y = 6 is called a “first degree” equation because it has no term with a variable to a power greater than 1. The graph of any first degree equation in two variable is a straight line

9. 3-1 The Rectangular Coordinate System
A linear equation in two variable can be written in the form: Ax + By = C, where A,B and C are real numbers (A and B both not zero). This form is called the “Standard Form”.

10. 3-1 The Rectangular Coordinate System
Graphing lines: Example 3: Draw the graph of 2x + 3y = 6 Step 1: Find a table of ordered pairs that satisfy the equation. Step 2: Plot the points on a rectangular coordinate system. Step 3: Draw the straight line that would pass through the points. Step 1 Step 2 Step 3

11. 3-1 The Rectangular Coordinate System
Finding Intercepts: In the equation of a line, let y = 0 to find the “x-intercept” and let x = 0 to find the “y-intercept”. Note: A linear equation with both x and y variables will have both x- and y-intercepts. Example 4: Find the intercepts and draw the graph of 2x –y = 4 x-intercept: Let y = 0 : 2x –0 = 4 2x = 4 x = 2 y-intercept: Let x = 0 : 2(0) – y = 4 -y = 4 y = -4 x-intercept is (2,0) y-intercept is (0,-4)

12. 3-1 The Rectangular Coordinate System
Finding Intercepts: In the equation of a line, let y = 0 to find the “x-intercept” and let x = 0 to find the “y-intercept”. Example 5: Find the intercepts and draw the graph of 4x –y = -3 x-intercept: Let y = 0 : 4x –0 = -3 4x = -3 x = -3/4 y-intercept: Let x = 0 : 4(0) – y = -3 -y = -3 y = 3

13. 3-1 The Rectangular Coordinate System
Recognizing equations of vertical and horizontal lines:
An equation with only the variable x will always intersect the x-axis
and thus will be vertical.
An equation with only the variable y will always intersect the y-axis
and thus will be horizontal.
Example 6: A) Draw the graph of y = 3
B) Draw the graph of x + 2 = x = -2
A) B)

14. 3-1 The Rectangular Coordinate System
Graphing a line that passes through the origin:
Some lines have both the x- and y-intercepts at the origin.
Note: An equation of the form Ax + By = 0 will always pass
through the origin. Find a multiple of the coefficients of x and y
and use that value to find a second ordered pair that satisfies the
equation.
Example 7:
A) Graph x + 2y = 0

15. 3-2 The Slope of a Line
Finding the slope of a line given two points on the line: The slope of the line through two distinct points (x1, y1) and (x2, y2) is: Note: Be careful to subtract the y-values and the x-values in the same order. Correct Incorrect

16. 3-2 The Slope of a Line
Finding the slope of a line given two points on the line:
Example 1) Find the slope of the line through the points
(2,-1) and (-5,3)

17. 3-2 The Slope of a Line
Finding the slope of a line given an equation of the line:
The slope can be found by solving the equation such that y is
solved for on the left side of the equal sign. This is called the
slope-intercept form of a line. The slope is the coefficient of x and
the other term is the y-intercept. The slope-intercept form is
y = mx + b
Example 2) Find the slope of the line given 3x – 4y = 12

18. 3-2 The Slope of a Line
Finding the slope of a line given an equation of the line:
Example 3) Find the slope of the line given y + 3 = 0
y = 0x - 3 The slope is 0
Example 4) Find the slope of the line given x + 6 = 0
Since it is not possible to solve for y, the slope is “Undefined”
Note: Being undefined should not be described as “no slope”
Example 5) Find the slope of the line given 3x + 4y = 9

19. 3-2 The Slope of a Line
Graph a line given its slope and a point on the line: Locate the first
point, then use the slope to find a second point.
Note: Graphing a line requires a minimum of two points. From the
first point, move a positive or negative change in y as indicated by
the value of the slope, then move a positive value of x.
Example 6) Graph the line given
slope = passing through (-1,4)
Note: change in y is +2

20. 3-2 The Slope of a Line
Graph a line given its slope and a point on the line: Locate the first
point, then use the slope to find a second point.
Example 7) Graph the line given
slope = -4 passing through (3,1)
Note: A positive slope indicates the line moves up from L to R
A negative slope indicates the line moves down from L to R

21. 3-2 The Slope of a Line
Using slope to determine whether two lines are parallel, perpendicular, or neither:
Two non-vertical lines having the same slope are parallel.
Two non-vertical lines whose slopes are negative reciprocals are perpendicular.
Example 8) Is the line through (-1,2) and (3,5) parallel to the line through (4,7) and (8,10)?

22. 3-2 The Slope of a Line
Using slope to determine whether two lines are parallel, perpendicular, or neither: Two non-vertical lines having the same slope are parallel. Two non-vertical lines whose slopes are negative reciprocals are perpendicular.

23. 3-2 The Slope of a Line
Example 9) Are the lines 3x + 5y = 6 and 5x - 3y = 2 parallel, perpendicular, or neither?

24. 3-2 The Slope of a Line
Solving Problems involving average rate of change: The slope gives the
average rate of change in y per unit change in x, where the value of y
depends on x.
Example 10) The graph shown approximates the percent of US households owing multiple pc’s in the years
Find the average rate of change between years 2000 and 1997.

25. 3-2 The Slope of a Line
Solving Problems involving average rate of change: The slope gives
the average rate of change in y per unit change in x, where the value
of y depends on x.
Example 11) In 1997, 36.4 % of high school students smoked. In 2001, 28.5 % smoked.
Find the average rate of change in percent per year.

26. 3-3 Linear Equations in Two Variables
Writing an equation of a line given its slope and y-intercept: The
slope can be found by solving the equation such that y is solved for
on the left side of the equal sign. This is called the slope-intercept
form of a line. The slope is the coefficient of x and the other term is
the y-intercept.
The slope-intercept form is y = mx + b, where m is the slope and b
is the y-intercept.
Example 1: Find an equation of the line with slope 2 and y-intercept (0,-3)
Since m = 2 and b = -3, y = 2x - 3

27. 3-3 Linear Equations in Two Variables
Graphing a line using its slope and y-
intercept:
Example 2: Graph the line using the slope and y-intercept: y = 3x – 6
Since b = -6, one point on the line is
(0,-6).
Locate the point and use the slope
(m = ) to locate a second point.
(0+1,-6+3)=
(1,-3)

28. 3-3 Linear Equations in Two Variables
Writing an equation of a line given its slope and a point on the line: The
“Point-Slope” form of the equation of a line with slope m and passing
through the point (x1,y1) is:
y - y1 = m(x - x1)
where m is the given slope and x1 and y1 are the respective values of the
given point.
Example 3: Find an equation of a line with slope and a given point (3,-4)

29. 3-3 Linear Equations in Two Variables
Writing an equation of a line given two points on the line: The
standard form for a line was defined as Ax + By = C.
Example 4: Find an equation of a line with passing through the points (-2,6) and (1,4). Write the answer in standard form.
Step 1: Find the slope:
Step 2: Use the point-slope
method:

30. 3-3 Linear Equations in Two Variables
Finding equations of Parallel or Perpendicular lines:
If parallel lines are required, the slopes are identical.
If perpendicular lines are required, use slopes that are negative
reciprocals of each other.
Example 5: Find an equation of a line passing through the point (-8,3) and parallel to 2x - 3y = 10.
Step 1: Find the slope Step 2: Use the point-slope method of the given line

31. 3-3 Linear Equations in Two Variables
Finding equations of Parallel or Perpendicular lines:
Example 6: Find an equation of a line passing through the point (-8,3) and perpendicular to 2x - 3y = 10.
Step 1: Find the slope Step 3: Use the point-slope method
of the given line
Step 2: Take the negative reciprocal
of the slope found

32. 3-3 Linear Equations in Two Variables

33. 3-3 Linear Equations in Two Variables
Writing an equation of a line that models real data: If the data
changes at a fairly constant rate, the rate of change is the
slope. An initial condition would be the y-intercept.
Example 7: Suppose there is a flat rate of R20 plus a charge of R10/minute to make a phone call. Write an equation that gives the cost y for a call of x minutes.
Note: The initial condition is the flat rate of R20 and the rate of change is R10/minute.
Solution: y = .10x + .20

34. 3-3 Linear Equations in Two Variables
Writing an equation of a line that models real data: If the data
changes at a fairly constant rate, the rate of change is the slope.
An initial condition would be the y-intercept.
Example 8: The percentage of mothers of children under 1 year old who participated in the US labor force is shown in the table. Find an equation that models the data.
Using (1980,38) and (1998,50)

35. 3-5 Introduction to Functions
Defining and Identifying Relations and Functions: If the value
of the variable y depends upon the value of the variable x,
then y is the dependent variable and x is the independent
variable.
Example 1: The amount of a paycheck depends upon the number of hours worked. Then an ordered pair (5,40) would indicate that if you worked 5 hours, you would be paid $40. Then (x,y) would show x as the independent variable and y as the dependent variable.
A relation is a “set of ordered pairs”.
{(5,40), (10,80), (20,160), (40,320)} is a relation.

36. 3-5 Introduction to Functions
Defining and Identifying Relations and Functions:
A function is a relation such that for each value of the independent
variable, there is one and only one value of the dependent variable.
Note: In a function, no two ordered pairs can have the same 1st component and different 2nd components.
Example 2: Determine whether the relation is a function {(-4,1), (-2,1), (-2,0)}
Solution: Not a function, since the independent variable has more than one dependent value.

37. 3-5 Introduction to Functions
Defining and Identifying Relations and Functions:
Since relations and functions are sets of ordered pairs, they can be represented as tables or graphs. It is common to describe the relation or function using a rule that explains the relationship between the independent and dependent variable.
Note: The rule may be given in words or given as an equation.
y = 2x + 4 where x is the independent variable and y is the
dependent variable

38. 3-5 Introduction to Functions
Domain and Range:
In a relation:
A) the set of all values of the independent variable (x) is the domain.
B) the set of all values of the dependent variable (y) is the range.
Example 3: Give the domain and range of each relation. Is the relation a function?
{(3,-1), (4,-2),(4,5), (6,8)}
Domain: {3,4,6}
Range: {-1,-2,5,8}
Not a function
Example 4: Give the domain and range of each relation. Is the relation a function?
Domain: {0,2,4,6}
Range: {4,8,12,16}

39. 3-5 Introduction to Functions
Domain and Range:
In a relation:
A) the set of all values of the independent variable (x) is the domain.
B) the set of all values of the dependent variable (y) is the range.
Example 4: Give the domain and range of each relation. Is the relation a function?
Domain:
{1994,1995,1996,1997,1998,1999}
Range:
{24,134 ; 33,786 ; 44,043 ; 55,312 ; 69,209 ; 86,047}
This is a function