Graphs, Linear Equations, and Functions

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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

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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.

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.

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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)

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.

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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:SolutionX Y

00

-12-4

Given: Equation Solution3x - 4y =12

x = 0 3(0) -4y = 12 y = -3y = 0 3x - 4(0) =12 x = 4

y = -12 3x - 4(-12) = 12 x = -12x = -4 3(-4) -4y = 12 y = -6

X Y0 -34 0

-12 -12-4 -6

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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 degreeequation 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.

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.

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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

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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 = 4x-intercept: Let y = 0 : 2x 0 = 4 2x = 4 x = 2y-intercept: Let x = 0 : 2(0) y = 4 -y = 4 y = -4

x-intercept is (2,0)y-intercept is (0,-4)

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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 = -3x-intercept: Let y = 0 : 4x 0 = -3 4x = -3 x = -3/4y-intercept: Let x = 0 : 4(0) y = -3 -y = -3 y = 3

3x-intercept is (- ,0)4

y-intercept is (0,3)

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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 = 3B) Draw the graph of x + 2 = 0 x = -2

A) B)

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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

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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

12 1

2 12

( x )

x

y yrise change in yslope mrun change in x x

x= =

= =

2 1 1 2

2 1 1 2

y y y yorx x x x

2 1 1 2

1 2 2 1

y y y yorx x x x

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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)

2 1

2 1

3 ( 1) 4 4= ( 5) 2 7 7

y yriseslope mrun x x

= = = = =

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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 = 123 4 12

4 3 123 34

x yy x

y x

= = +

=

3 The slope is 4

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Finding the slope of a line given an equation of the line:

Example 3) Find the slope of the line given y + 3 = 0y = 0x - 3 The slope is 0

Example 4) Find the slope of the line given x + 6 = 0Since 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

3 4 94 3 9

3 94 4

x yy x

y x

+ == +

= +

3 The slope is -4

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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

23

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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 RA negative slope indicates the line moves down from L to R

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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 reciprocalsare perpendicular.

Example 8) Is the line through (-1,2) and (3,5) parallel to the line through (4,7) and (8,10)?

1 2

For line 1: For line 2:5 2 3 10 7 3

Y3 (ES

1) 4 8 4 4m m =

= = =

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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 9) Are the lines 3x + 5y = 6 and 5x - 3y = 2 parallel, perpendicular, or neither?

For line 1: For line 2:3 5 6 5 - 3 25 3 6 3 5 2

3 6 5 2

3 5 is the negative reciprocal of 5

3

5 5 3

3

x y x yy x y x

y x

Perpendicula

y x

r

+ = == + = +

=

+

=

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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 pcs in the years 1997-2001.Find the average rate of change between years 2000 and 1997.

Use the ordered pairs: (1997,10) and (2000,20.8)

20.8 10 10.8200

%0 1997 3

3.6myr

= = =

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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.

Use the ordered pairs: (1997,36.4) and (2001,28.5)

28.5 36.4 7.92001

%1.9751997 4

myr

= = =

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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

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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)

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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)

251 1( )

2( 4) ( 3)5

2 645 5

2 6 20 2 265 5 5 5 5

5 2 26 or in standard form 2 5 26y x

y y m x x

y x

y x

y x x

x y

=

=

+ =

= =

= =

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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:

1 1

4 6 2 21 ( 2) 3 3

( )26 ( ( 2))32 463 3

2 4 18 2 143 3

23 3 3

3 2 14 or 3 14

m

y y m x x

y x

y x

y x x

x x yy

= = =

=

=

=

= + =

+ += =

+

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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

2 3 103 2 10

2 10 3

3

23

x yy x

y mx

= = +

= =

1 1

3 2 25 or

( )23 ( ( 8))32 1633 3

-2 3 25

2 16 9 2 253 3 3 3 3

y y m x x

y x

y x

y x

y x x y

x

=

=

= + +

+

= + + =

=

=

+

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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 methodof the given line

Step 2: Take the negative reciprocalof the slope found

2 3 103 2 10

2 10 3

3

23

x yy x

y mx

= = +

= =

1 1( )

3 ( ( 8))

33 1

32

2 3 18 or 3 18

22

92

2

3

y y m x x

y x

y x

y x

y x x y

=

=

=

=

= + = 12 3

3 2

mm = =

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3-3 Linear Equations in Two VariablesForms of Linear Equations

Equation Description When to Use

Y = mx + bSlope-Intercept Form

slope is m y-intercept is (0,b)

Given an equation, the slope and y-intercept can be easily identified and used to graph

y - y1 = m(x-x1)Point-Slope Form

slope is m line passes through (x1,y1)

This form is ideal to use when given the slope of a line and one point on the line or given

two points on the line.

Ax + By = C

Standard Form (A,B, and C are integers, A 0)

Slope is -(A/B) x-intercept is (C/A,0) y-intercept is (0,C/B)

X- and y-intercepts can be found quickly

y = bHorizontal line

slope is 0 y-intercept is (0,b)

Graph intersects only the y axis, is parallel to the x-axis

x = aVertical line

slope is undefined x-intercept is (a,0)

Graph intersects only the x axis, is parallel to the y-axis

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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 $.20 plus a charge of $.10/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 $.20 and the rate of change is $.10/minute.

Solution: y = .10x + .20

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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)

Year Percent

1980 381984 471988 511992 541998 59

59 38 211998 1980

1.1

181.1

6767

38

m

my x

= =

= +=

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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.

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3-5 Introduction to FunctionsDefining 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.

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3-5 Introduction to FunctionsDefining 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

x y0 42 84 126 16

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3-5 Introduction to FunctionsDomain 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}

x y0 42 84 126 16

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3-5 Introduction to FunctionsDomain 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: {24134,33786,44043,55312,69209,86047}This is a function

YearSubscribers (thousands)

1994 24,1341995 33,7861996 44,0431997 55,3121998 69,2091999 86,047

Cell Phone Users

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3-5 Introduction to FunctionsDomain 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 5: Give the domain and range of each relation.

Domain: (-, ) Domain: [-4, 4]Range: (-, 4] Range: [-6, 6]

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3-5 Introduction to FunctionsAgreement on Domain:

Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.

The function

has all real numbers except x = 0Note: In general, the domain of a function defined by an algebraic expression is all real numbers except those numbers that lead to division by zero or an even root of a negative number.

The function is not defined for values <

1yx

=

3 2y x= 23

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3-5 Introduction to FunctionsIdentifying functions defined by graphs and equations:Vertical Line Test:If every vertical line intersects the graph of a relation in no more than one point, the relation represents a function .

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Identifying functions defined by graphs and equations:Vertical Line Test:If every vertical line intersects the graph of a relation in no more than one point, the relation represents a function.

Function Not a Function

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Identifying functions defined by graphs and equations:

Example 5: Decide whether the equations shown define a function and give the domain

Function Function Not a Function

Domain: Domain: Domain:

3 2y x= 51

yx

=

2y x=

2 ,3

( ) ( ),1 1, U [ )0,

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Using Function Notation:When a rule or equation is defined such that y is dependent on x, the Function Notation y = f(x) is used and is read as y = f of x where the letter f stands for function.

Note: The symbol f(x) does not indicate that f is multiplied by x, but represents the y-value for the indicated x-value.

If y = 9x -5, then f(x) = 9x -5 and f(2) = 9(2) -5 = 13and f(0) = 9(0) -5 = -5

Example 6: 3 5( )

2: ( 3)

3( 3) 5 9 5 14( 3) 72 2 2

3 5( )2

( )

xf x

Find f and f

f

rf

r

r

+ + = = = =

=

+

+

=

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Using Function Notation:When a rule or equation is defined such that y is dependent on x, the Function Notation y = f(x) is used and is read as y = f of x where the letter f stands for function.

Example 7: f(x) = 5x -1 find: f(m + 2)

f(m + 2) = 5(m + 2) -1 = 5m + 10 - 1f(m + 2) = 5m + 9

Example 8: Rewrite the equation given and find f(1) and f(a)2

2

2

2

4 3: (1) ( )

4 33

43( )

4

x yFind f and f a

y xxy

xf x

=

= +

=

=

2

2

(1) 3 2 1(

: (

1)4 4 2

3

1)

4

( )

( )

Find f and f a

f

af a

= = =

=

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Identifying Linear Functions:A function that can be defined by f(x) = mx + b for real numbers m and b is a Linear Function

Note: The domain of a linear function is (-, ). The range is (-, ).

Note: Remember that m represents the slope of a line and (0,b) is the y-intercept.

A function that can be defined by f(x) = b is called a Constant Function which has a graph that is a horizontal line.

Note: The range of a constant function is {b}.

Graphs, Linear Equations, and Functions

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