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2.1. Linear Functions and Models. Graphing Discrete Data. We can graph the points and see if they are linear. Enter the data from example 1 into L1 and L2 Stat Edit

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2 1

2.1

Linear Functions and Models


Graphing discrete data

Graphing Discrete Data

  • We can graph the points and see if they are linear.

  • Enter the data from example 1 into L1 and L2

    • Stat

    • Edit

    • Enter the data…if there is already data in your lists you can more the cursor to the very top (on L1) and press clear enter. This will clear the list.

    • Turn Stat plot on

    • Press 2nd y= (stat plot)

    • Select 1

    • Select on

    • Type: first option, this is the scatter plot

    • Graph the scatter plot

    • Zoom

    • Zoomstat – this changes the window to values that are best for your set of data, without us having to pick the best window


Graphing discrete data1

Graphing Discrete Data

  • Check by graphing the equation as well

    • Put the equation into y1

    • Press trace or graph

    • The line should go through all of the points

    • When you are done using the stat plot option go to stat plot and choose 4:plots off enter. This will shut off the statplots.

    • To change your window back to standard –10-10 press zoom, zoom standard


The regression line

The Regression Line

  • The Regression Line

    • Also called the least-squares fit

    • Approximate model for functions

  • Uses of the regression line

    • Finding slope…tells us how the data values are changing.

    • Analyzing trends

    • Predicting the future (not always accurate)

    • Shows linear trend of the data


Linear regression line

Linear Regression Line

  • Enter data into L1 and L2

  • Plot the scatter plot

  • Go to stat

  • Calc

  • LinReg(a + bx)

  • Enter

  • Enter

  • a = slope, b = y-intercept

  • Graph with the points to see how close it is.


Linear equations

Linear Equations

  • If the equation has a constant rate of change then it is a line.

  • Linear functions are formulas for graphing straight lines.

    • Slope intercept form: y = mx + b

    • Standard form: ax + by = c, a and b are both integers, a>0


Writing equations given slope and initial value

Writing equations- given slope and initial value

  • The initial value is when x = 0, which happens to be the y-intercept (b)

    • Use y = mx + b

  • Initial value of 35, slope of ½

  • A phone company charges a flat fee of $29.99 plus $0.05 a minute.


Example

Example

  • A local school is going on a field trip. The cost is $130 for the bus and an additional $2 per child.

    • Write a formula for the linear function that models the cost for n children.

    • How much is it for 15 children to attend?


2 1

2.2

Equations of Lines


Point slope form

Point-Slope Form

  • Given point, (a, b), and slope, m, the equation can be found using the formula

    y – b = m(x – a).

    This is called the point-slope form of the line.


Examples

Examples

  • Find the equation of the line passing through the given point with the given slope. Write your answer in point slope form.

    • (6, 12), m = –1/3

    • (1, -4), m = 1/3


Examples1

Examples

  • Find the equation of the line passing through the given points. Use the first point as (x1,y1) and write your answer in point slope form.

    • (-2,3), (1,0)

    • (-1,2), (-2,-3)


Slope intercept form of a line

Slope-Intercept Form of a Line

  • Slope Intercept form of a line:

    • y = mx + b

      • m = slope

      • b= y-intercept


Writing equations given slope and a point

Writing equations- given slope and a point

  • Find the equation of each line in point slope form and in slope-intercept form.

    • (2,3) m = ½

    • (-3, 5) m = 2

    • (-8, 7) m = -3/2


Intercepts

Intercepts

  • Horizontal Intercept- where the line crosses the x axis

    • This can be found by letting y = 0

  • Vertical Intercept- where the line crosses the y axis

    • This can be found by letting x = 0.


Examples2

Examples

  • Locate the x- and y-intercepts on the following lines.

    • -3x – 5y = 15

    • (2/3)y – x = 1


Horizontal and vertical lines

Horizontal and Vertical Lines

  • An equation of the horizontal line with y-intercept b is y = b.

  • An equation of the vertical line with x-intercept k is x = k.


2 1

  • Find the equation of the line satisfying the conditions:

    • Vertical passing through (1.95, 10.7)

    • Horizontal passing through (1.95, 10.7)


Parallel and perpendicular lines

Parallel and Perpendicular Lines

  • Parallel lines have the same slope. (They are changing at the same rate)

  • Slopes of perpendicular lines are negative reciprocals of one other.

    • Ex: ½ and -2; 2/3 and -3/2; -5 and 1/5


2 1

  • For each of the following pairs of lines, determine whether the lines are parallel, perpendicular or neither.

    a. 3x + 2y = 4 and 6x + 4y = 9

    b. 5x – 7y = 3 and 4x – 3y = 8


Examples3

Examples

c. 5x – 7y = 15 and 15y - 21x = 7

d. ax = by = c and akx +aky = d, (a ≠ 0)


Write the equation parallel to a given line

Write the Equation Parallel to a given line

  • 2x – 3y = 9, (-9, 7)

  • 4x + 5y = 16, (-2, 3)


Write the equation perpendicular to a given line

Write the Equation Perpendicular to a given line

  • 3x + 4y = 8, (7, 3)

  • 2x – 8y = 10, (1, 0)


Example1

Example

  • Show that the points (1, 1), (3,4) and (4,-1) from the vertices of a right triangle.


2 1

  • Interpolation

    • Estimates values that are between two or more known data values.

  • Extrapolation

    • Estimates values that are not between two known data values.


2 1

  • Page 101 #54


Direct variation

Direct Variation

  • Let x and y denote two quantities. Then y is directly proportional to x, or y varies directly with x, if there exists a nonzero number k such that y = kx.

  • k is called the constant of proportionality or the constant of variation.


2 1

  • Page 104 # 100


2 1

  • Page 102 #80


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  • Page 103 #88


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  • Page 03 #90


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  • Page 104 #104


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2.3

Linear Equations


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  • Page 118 #18

  • Page 118 #20


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  • Page 118 #22

  • Page 118 #24


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  • Page 118 #30

  • Page 119 #36


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  • Page 119 #46


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  • Page 119 #60

  • Page 119 #64


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  • Page 120 #80


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  • Page 121 #86


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  • Page 121 #88


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  • Page 122 #98


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  • Page 122 #100


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2.4

Linear Inequalities


2 1

  • A linear inequality in one variable is an inequality that cab be written in the form

    ax + b >0 where a ≠ 0.

    (The symbol > can be replaced by <, ≤, or ≥)


2 1

  • Interval Notation

    • Open interval ( )

    • Half-open interval ( ], [ )

    • Closed interval [ ]

    • Infinite intervals


2 1

  • Page 134 #4

  • #6

  • #8

  • #10


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  • Follow the equation rules when solving inequalities.

  • When you multiply or divide both sides by a negative flip the inequality sign.


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  • Page 134 #14

  • Page 135 #18


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  • Page 135 #22

  • Page 135 #28


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  • Page 135 #38

  • Page 135 #40


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  • Page 135 #60


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  • Page 136 #70


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  • Page 136 #86


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  • Page 137 #88


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  • Page 137 #92


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  • Page 138 #98


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2.5

Piecewise-Defined Functions


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Page 152 #2


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  • Page 153 #6


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  • Page 153 #8


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  • Page 153 #12


2 1

  • Greatest Integer Function

    • [[x]] is the greatest integer less than or equal to x.

      (Always round down.)

      [[1.2]] = 1

      [[1.9]] = 1


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  • Page 154 #24


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  • Page 154 #28


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  • Page 154 #38


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  • Solving Absolute Value Equations and Inequalities

    • Break it into 2 parts

    • Flip the second symbol and sign


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  • Page 154 #40

  • Page 154 #42


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  • Page 155 #56


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  • Page 155 #60


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  • Page 155 #68

  • Page 155 #74


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  • Page 155 #80


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  • Page 156 #100


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  • Page 156 #104


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