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### 2.1

### 2.2

### 2.3

### 2.4

### 2.5

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?

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

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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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

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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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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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