linear functions. big idea set-builder notation is more versatile than roster notation. in roster...

Linear Functions

Linear Functions

Big Idea

Set-builder notation is more versatile than roster notation. In roster notation you must be able to list ALL elements or unambiguously imply them (with …). In set-builder notation you only need to specify the rule that the elements must specify.

Goals• Translate between roster and set-builder notation in the description of a set.

1,2,3,...{ } ≡ x x is a whole number{ }

?,?{ } ≡ x x is a rational number between 0 and 1{ }

Linear Functions

Big Idea

Relationships between elements of two sets can be one-to-one (like a student ID). Relationships between the elements of two sets can be many-to-one (like gender or age).

Goals• Describe a mathematical relationship as a set of ordered pairs from the domain and range.

many-to-one

one-to-one

Linear Functions

Big Idea

The relationship between elements in a set can be described with a set of ordered pairs. In an ordered pair (a,b) the first of the pair (a) represents an element in one set and the second of the pair (b) represents an element in the second set.

Goals• Describe a mathematical relationship as a set of ordered pairs from the domain and range.

Linear FunctionsGoals• Describe a mathematical relationship as a set of ordered pairs from the domain and range.

Even Bigger Idea

The set of all possible first values among the ordered pairs of a relationship is called the domain. The set of all possible second values is called the range.

Here the domain is the set of all students that have been given an ID. The range is the set of all IDs.

Linear FunctionsGoals• Define a function in terms of a restriction on the number of ordered pairs having the same elements of the domain.

And The Biggest Idea

A function is a relationship in which no two ordered pairs have the same first value.

You can see how important it is to have just one name attached to just one ID. If two ordered pairs existed in which the first value was the same then that person would have two IDs.

Linear functions

HW Due 1/18 344.1, 345.2, 13, 15, 16, 19, 27, 29,31, 33, 34, 37, 346.39, 41

A rate is an example of a function. The cost of one can of soup is $1.39. So the cost of n cans is $1.39 x n.

Let C be the cost. Then in set-builder notation the set of ordered pairs is

{(n,C)|C=$1.39*n}

And in roster notation {(1,$1.39), (2, $2.78), (3, $3.87), …}

Linear functions

HW Due 1/18 344.1, 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, 346.39, 41

You can test to see if an ordered pair satisfies a relationship:

345.13. In y=5x, is the ordered pair (3,15) a solution?

345.15. In y=3x+1, is the ordered pair (7,22) a solution?

345.16 In 3x-2y=0, is (3,2) a solution?

345.19 In 3y>2x+1, is (4,3) a solution?

345.27 In 5x-2y<19, is (3,-2) a solution?

Linear functions

HW Due 1/18 344.1, 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, 346.39, 41

In a function there can only be one ordered pair that has a particular first value:

345.29 Is {(1,2),(2,1),(3,4),(4,3)} a function?

345.31 Is {(81,9),(81,-9),(25,5),(2-5,3)} a function?

Linear functions

HW Due 1/18 344.1, 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, 346.39, 41

The range is defined by the domain and the function:

345.33a. If {x|10<x<13 and x is an integer} what is the range of the relationship y=x/6?

345.33b. If {6x|x is a whole number} what is the range of the relationship y=x/6? (in roster notation the domain is {6,12,18,24, …}

Linear functions

HW Due 1/18 344.1, 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, 346.39, 41

The range is defined by the domain and the function:

345.33a. If {x|10<x<13 and x is an integer} what is the range of the relationship y=-x-2?

345.33b. If {6x|x is a whole number} what is the range of the relationship y=-x-2? (in roster notation the domain is {6,12,18,24, …}

That leaves three problems for you!

Linear functions

HW Due 1/18 344.1, 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, 346.39, 41

345.37 In an isosceles triangle whose perimeter is 54 centimeters, the length of the base in cm is x and the length of each leg in cm is y.

x + 2y=54cm

{(x, y) | x + 2y=54 and x is a real number}

(1,

53

2), (2,26), (3,

51

2),(10,22), (20,17), (40,7)

Some ordered pairs in the solution set

Linear functions

HW Due 1/18 344.1, 345.2, 13, 15, 16, 19, 27, 29, 31, 33, 34, 37, 346.39, 41

345.39 At the beginning of the day, the water in the pool was 2 feet deep. Water is added so that the depth increases by less than 0.4 foot per hour. What was the depth, y, after being filled for x hours?

y −2 ft< 0.4

foothour

• x

(1,2.1), (1,2.2), (1,2.3), (1,2.35), (1,2.355), (2,2.1), (2,2.6)

Some ordered pairs in the solution set

Linear functions345.39 An amusement park, rides are paid for with tokens.

Some rides require two tokens, and other require one. If x is the number of 2-token rides and y is the number of 1-token rides, what rides can be taken with 10 tokens?

2x + y≤10This is also the relationship when 10 feet of fence are used to enclose three sides of a rectangle pen attached to a garage where x is the number of feet perpendicular to the garage and y is the number of feet parallel to the garage.

(x,y)|2x+y≤10 and x and y are real numbers{ }All solutions:

Some pairs: (1,5), (1,8), (2,4), (2,5.678), (3,3), (3,3.3333), (3,4)

Linear functions

Big IdeaA linear equation in standard form is written Ax+By=C where A, B, and C are real numbers and A and B are not both zero.

GoalExpress a linear function symbolically in standard form and construct and interpret the graph.

The preceding example of a equality describing the relationship between the sides and base of an isosceles triangle whose perimeter is 54 cm, is a linear relationship in the standard form.; 2y+x=54

Linear functionsGoal

Express a linear function symbolically in standard form and construct and interpret the graph.

Big Idea To make a graph of an equation in standard form:• solve Ax+By=C for y: y=(C/B)-(A/B)x• select three values of x from the domain• evaluate the equation for y to get three ordered pairs• in the coordinate plane, locate these pairs• draw a straight line that passes through the three points



350.7 2y+x=7

y =3.5−0.5x

(−1,4),(0,3.5),(1,3)


350.7 2y+x=7 y =3.5−0.5x (−1,4),(0,3.5),(1,3)

Linear functionsGoalExpress a linear function symbolically in standard form and construct and interpret the graph.


Graph these on a single graph with the domain [-2.5,10]

2x + 3y=6 x −3y=9

2x + 3y=6 x −3y=9

HW Due 1/20 350.1, 3, 5, 6, 8, 13, 351.23, 25, 28, 29, 35, 36, 46, 48, 52, 53, 55, 353.13, 14, 15, 16, 18e, 354.19c, 19e, 24



Graph these on a single graph with the domain [-1,1]

5x −y=0 3x + y=0



Graph these on a single graph with the domain [-1,1]

2x + 3y=6 2x −2y=−4


Big Idea Standard form: Ax+By=C

Two cases of the standard form are especially important. These are when A=0 and the graph is a horizontal line. The other is when B=0 and the graph is a vertical line.

Graph these on a single graph with where 0≤x≤3 and 0≤y≤3

2x =6 y =3

Big Idea Standard form: Ax+By=C

Two cases of the standard form are especially important. These are when A=0 and the graph is a horizontal line. The other is when B=0 and the graph is a vertical line.

Linear functionsGoalCalculate the slope of a linear relationship using ordered pairs.

Big Idea The slope of a linear relationship is the ratio of the change in the y-values and the change in the x-values. A pair of ordered pairs can be used to calculate the slope.

slope =

difference in y-valuesdifference in x-values

=ΔyΔx

For two ordered pairs

Example: Given (1,3) and (2,4), the slope =(4-3)/(2-1)=1


Big Idea In a linear relationship if the dependent variable, y, increases as the independent variable, x, increases, then the slope is positive.


Big Idea In a linear relationship if the dependent variable, y, doesn’t change as the independent variable, x, increases then the slope is 0; the graph is parallel to the x-axis.

In a linear relation if the independent variable, x, doesn’t change as the dependent variable, y, increases then the slope is undefined; the graph is parallel to the y-axis.

HW Due 1/21360.1, 361.3-8, 362.9, 13, 14, 19, 26, 29, 30, 31, 363.33, 35


HW Due 1/21360.1, 361.3-8, 362.9, 13, 14, 19, 26, 29, 30, 31, 363.33, 35

362.18 Draw a line with a slope of 2 that goes through the ordered pair (0,0).


HW Due 1/21360.1, 361.3-8, 362.9, 13, 14, 19, 26, 29, 30, 31, 363.33, 35

362.26 Draw a line with a slope of -1/3 that goes through the ordered pair (-2,3).


HW Due 1/21360.1, 361.3-8, 362.9, 13, 14, 19, 26, 29, 30, 31, 363.33, 35

362.31 Plot the points (3,-2), (9,-2), (7,4), and (1,4). What is the shape? What are the slopes of the sides?

Linear functionsBig IdeaIf two lines are parallel they have the same slope. If two lines have the same slope they are parallel.

If two lines are perpendicular their slopes are negative reciprocals. If the slopes of two lines are negative reciprocals they are are perpendicular.

If a = -1/b then b is the negative reciprocal of a and a•b=-1.

HW Due 1/24365.2, 3, 5, 12, 366.13

Linear functionsGoalExpress a linear function symbolically in intercept form and construct and interpret the graph.

Big IdeaIf the slope of a linear equation is neither undefined (vertical line) or 0 (horizontal line) then the graph of the equation crosses the y-axis at the y-intercept (where x=0) and crosses the x-axis at the x-intercept (where y=0).


Big IdeaThe intercept form of a linear equation is obtained by dividing the standard form of a linear equation, Ax+By=C, by C. The x and y-intercepts are seen in the ratios formed with x and y.

Ax + By=C ⇒

xC / A

+y

C / B=1

x-intercept y-intercept


Ax + By=C ⇒

xC / A

+y

C / B=1


Dividing by a fraction, C/A, is the same thing as multiplying by the inverse of the fraction. The inverse of C/A is A/C. So

x

(C / A)=

AC

x


Example

−x+ 2y=2⇒

x2 / (−1)

+y

2 / 2=

x−2

+y1=1

In standard form this linear equation is transformed to intercept form:


Linear functionsGoalExpress a linear function symbolically in slope-intercept form and construct and interpret the graph.

Big IdeaThe slope-intercept form of a linear equation is obtained by solving the standard form of a linear equation, Ax+By=C, for y.

In the slope-intercept form the coefficient of x is the slope and the ratio C/B is the y-intercept.

Ax + By=C ⇒ y=−

AB

⎛

⎝⎜⎞

⎠⎟x+

CB

⎛

⎝⎜⎞

⎠⎟

slope y-intercept

Linear functionsGoalTranslate among standard, intercept, and slope-intercept forms of a linear equation.

Express the linear equation in standard form in intercept and slope-intercept forms:

2x + y=5

x

5 / 2+

y5 / 1

=x

(5 / 2)+

y5=1

Standard form

Intercept form

Slope-intercept form y =−2x+5



2x −5y=−10Standard form

Intercept form

Slope-intercept form



2x −5y=−10Standard form

Intercept form


x

(−10) / 2+

y(−10) / (−5)

=x

(−5)+

y2=1

−5y=−10−2x⇒ y=

25

⎛

⎝⎜⎞

⎠⎟x+ 2


Express the linear equation in slope-intercept form in standard and intercept forms:

Standard form

Intercept form

Slope-intercept form y =3x+ 7



Standard form

Intercept form


y =3x+ 7

y =3x+ 7 ⇒ −3x+ y=7

⇒

x7 / (−3)

+y

7 / 1=

x(−7 / 3)

+y7=1


Express the linear equation in slope-intercept form in standard and intercept forms:

Standard form

Intercept form

Slope-intercept form y =−2x+5



Standard form

Intercept form


2x + y=5

x

(5 / 2)+

y(5 / 1)

=x

(5 / 2)+

y5=1

y =−2x+5

Linear functionsMaking a quick plot of a slope-intercept form

y =−2x+5• plot the y-intercept• draw a line through the y-intercept with the slope

Linear functionsMaking a quick plot of an intercept form

• plot the y-intercept• plot the x-intercept• draw a line to connect them

x

(5 / 2)+

y5=1

HW Due 1/25369.1, 2, 9, 14, 17, 18, 20, 23, 24

Linear functionsFinding the linear equation through a pair of points:

HW Due 1/25369.1, 2, 9, 14, 17, 18, 20, 23, 24

351.52 What is the linear equation with solutions (-2,3) and (1,-3)? Is the ordered pair (0,-1) a solution?

ΔyΔx

=−3−31−(−2)

=−2• calculate the slope

y =−2x+b• use the slope-intercept form to find the intercept

−3=−2 +b⇒ b=−1

Linear functionsGoalDescribe the effect of translation, reflection, and scaling on the graph of a linear function.

Translation:When a value is added to right hand side of a linear equation the line is shifted up (positive value added) or down (negative value added).


Reflection:When the x value is replace by its opposite the line is reflected.


Reflection:When the x value is replaced by its opposite the line is reflected over the y-axis.


Scaling:When the x value is replaced by a constant multiplied by x the graph is contracted (the constant is less than 1 and greater than 0) or expanded (the constant is greater than 1). Replacing x by x/2 is equivalent to replacing m by m/2.


HW Due 1/26373.7, 11, 14, 21, 22, 23, 24, 25, 26, 27

Describe how y=x has been translated, reflected and/or scaled and sketch a graph

y =x−1y=−x−1

y=13

x


HW Due 1/26373.7, 11, 14, 21, 22, 23, 24, 25, 26, 27

Indicate if y=x has been translated, reflected and/or scaled and sketch a graph

y =x−1y=−x−1

y=13

x

translate

translate & reflect

scale

Linear functionsGoalDescribe direction variation as a slope-intercept form of a linear equation in which the y-intercept is 0.

Big Ideay=mx is a direct variation. The slope of the line is the constant of variation.

HW Due 1/27376.3, 377.6, 12, 13, 15, 16, 378.18

ExampleA printer can print 160 characters (y, the dependent variable) in 10 seconds (x, the independent variable).

y =

16 characterssecond

x

Linear functionsGoalSolve narrative problems by applying the concept of the slope.

linear functions. big idea set-builder notation is more versatile than roster notation. in roster...

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