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# 2.1 - PowerPoint PPT Presentation

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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Presentation Transcript

### 2.1

Linear Functions and Models

• 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

• 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

• 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

• 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.

• 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

• 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.

• 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?

### 2.2

Equations of Lines

• 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.

• 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

• 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:

• y = mx + b

• m = slope

• b= y-intercept

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

• 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.

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

• -3x – 5y = 15

• (2/3)y – x = 1

• 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.

• 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

Examples the lines are parallel, perpendicular or neither.

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 the lines are parallel, perpendicular or neither.

• 2x – 3y = 9, (-9, 7)

• 4x + 5y = 16, (-2, 3)

Write the Equation Perpendicular to a given line the lines are parallel, perpendicular or neither.

• 3x + 4y = 8, (7, 3)

• 2x – 8y = 10, (1, 0)

Example the lines are parallel, perpendicular or neither.

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

• Interpolation the lines are parallel, perpendicular or neither.

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

• Extrapolation

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

• Page 101 #54 the lines are parallel, perpendicular or neither.

Direct Variation the lines are parallel, perpendicular or neither.

• 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.

• Page 102 #80 the lines are parallel, perpendicular or neither.

• Page 103 #88 the lines are parallel, perpendicular or neither.

• Page 03 #90 the lines are parallel, perpendicular or neither.

### 2.3 the lines are parallel, perpendicular or neither.

Linear Equations

• Page 118 #18 the lines are parallel, perpendicular or neither.

• Page 118 #20

• Page 118 #22 the lines are parallel, perpendicular or neither.

• Page 118 #24

• Page 118 #30 the lines are parallel, perpendicular or neither.

• Page 119 #36

• Page 119 #46 the lines are parallel, perpendicular or neither.

• Page 119 #60 the lines are parallel, perpendicular or neither.

• Page 119 #64

• Page 120 #80 the lines are parallel, perpendicular or neither.

• Page 121 #86 the lines are parallel, perpendicular or neither.

• Page 121 #88 the lines are parallel, perpendicular or neither.

• Page 122 #98 the lines are parallel, perpendicular or neither.

### 2.4 the lines are parallel, perpendicular or neither.

Linear Inequalities

• Interval Notation cab be written in the form

• Open interval ( )

• Half-open interval ( ], [ )

• Closed interval [ ]

• Infinite intervals

### 2.5 cab be written in the form

Piecewise-Defined Functions

Page 152 #2 cab be written in the form

• Greatest Integer Function cab be written in the form

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

(Always round down.)

[[1.2]] = 1

[[1.9]] = 1