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  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 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 (x 1,y 1 ) 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.

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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.

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2.3 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

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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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2.5 Piecewise-Defined Functions

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

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