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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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slide1

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?
slide9

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.
slide18
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
slide20
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.
slide25
Interpolation
    • Estimates values that are between two or more known data values.
  • Extrapolation
    • Estimates values that are not between two known data values.
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.
slide33

2.3

Linear Equations

slide34
Page 118 #18
  • Page 118 #20
slide35
Page 118 #22
  • Page 118 #24
slide36
Page 118 #30
  • Page 119 #36
slide38
Page 119 #60
  • Page 119 #64
slide44

2.4

Linear Inequalities

slide45
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 ≥)

slide46
Interval Notation
    • Open interval ( )
    • Half-open interval ( ], [ )
    • Closed interval [ ]
    • Infinite intervals
slide48
Follow the equation rules when solving inequalities.
  • When you multiply or divide both sides by a negative flip the inequality sign.
slide49
Page 134 #14
  • Page 135 #18
slide50
Page 135 #22
  • Page 135 #28
slide51
Page 135 #38
  • Page 135 #40
slide58

2.5

Piecewise-Defined Functions

slide63
Greatest Integer Function
    • [[x]] is the greatest integer less than or equal to x.

(Always round down.)

[[1.2]] = 1

[[1.9]] = 1

slide67
Solving Absolute Value Equations and Inequalities
    • Break it into 2 parts
    • Flip the second symbol and sign
slide68
Page 154 #40
  • Page 154 #42
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Page 155 #68
  • Page 155 #74
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