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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

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 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
• 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
• 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
• 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.
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?

### 2.2

Equations of Lines

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
• 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
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:
• 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
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.
Examples
• Locate the x- and y-intercepts on the following lines.
• -3x – 5y = 15
• (2/3)y – x = 1
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.
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 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
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

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
• 2x – 3y = 9, (-9, 7)
• 4x + 5y = 16, (-2, 3)
Write the Equation Perpendicular to a given line
• 3x + 4y = 8, (7, 3)
• 2x – 8y = 10, (1, 0)
Example
• Show that the points (1, 1), (3,4) and (4,-1) from the vertices of a right triangle.
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
• 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.

Linear Equations

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### 2.4

Linear Inequalities

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 ≥)

Interval Notation
• Open interval ( )
• Half-open interval ( ], [ )
• Closed interval [ ]
• Infinite intervals
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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### 2.5

Piecewise-Defined Functions

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

(Always round down.)

[[1.2]] = 1

[[1.9]] = 1

Solving Absolute Value Equations and Inequalities
• Break it into 2 parts
• Flip the second symbol and sign
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