3.1 The Cartesian Coordinate Plane y axis x axis Quadrant I (+, +)Quadrant II (-, +) Quadrant III (-, -)Quadrant IV (+, -) Origin (0, 0) (-6,-3) (5,-2) When distinct points are plotted as above the graph is called a scatter plot – ‘points that are scattered about’ y - $$ in thousands x Yrs Graphs represent trends in data. For example: x – number of years in business y – thousands of dollars of profit Equation : x - 2y = 6 (0,-3) (6,0) y intercept x intercept A point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y) Is the point (12, 2) on the line? Is the point (10, 2) on the line?

X and Y intercepts y x (0,-3) (6,0) y intercept x intercept Equation: x – 2y = 6 The y intercept happens where y is something & x = 0: (0, ____) Let x = 0 and solve for y: (0) – 2y = 6 => y = -3 The x intercept happens where x is something & y = 0: (____, 0) Let y = 0 and solve for x: x – 2(0) = 6 => x =

3.2 Functions/Relations The table above establishes a relation between the year and the cost of tuition at a public college. For each year there is a cost, forming a set of ordered pairs. A relation is a set of ordered pairs (x, y). The relation above can be written as 4 ordered pairs as follows: S = {(1997, 3111), (1998, 3247), (1999, 3356), (2000, 3510)} x y x y x y x y Domain – the set of all x-values. D = {1977, 1998, 1999, 2000} Range – the set of all y-values. R = {3111, 3247, 3356, 3510} Year Cost 1997 $ $ $ $3510 independent variable (x) dependent variable (y) The cost depends on the year. Year(x) Cost(y) Thinking Exercise: Draw a ‘line’ in the x/y axes. What is the Domain & Range?

Functions & Linear Data Modeling y – Profit in thousands of $$ (Dependent Var) x - Years in business (Independent Var) (0,-3) (6,0) y intercept x intercept Equation: y = ½ x – 3 Function: f(x) = ½ x – 3 x y = f(x) 0 -3 f(0) = ½(0)-3= f(2) = ½(2)-3= f(6) = ½(6)-3=0 8 1 f(8) = ½(8)-3=1 Input x Function f Output y=f(x) A function has exactly one output value (y) for each valid input (x). Use the vertical line test to see if an equation is a function. If it touches 1 point at a time then FUNCTION If it touches more than 1 point at a time then NOT A FUNCTION.

Diagrams of Functions Function: f(x) = ½ x – 3 x y = f(x) 0 -3 f(0) = ½(0)-3= f(2) = ½(2)-3= f(6) = ½(6)-3=0 8 1 f(8) = ½(8)-3= f f A functionNOT a function

Graphing Linear Functions The graph of a linear function is a line. A linear function is of the form f(x) = mx + b, where m and b are constants. y = 3x + 2 y = 3x + 5x y = -2x –3 y = (2/3)x -1 y = 4 6x + 3y = 12 (Standard Form) y x x y=3x+2 x y=2/3x – All of these equations are linear functions. Note: The y in each can be replaced by f(x)

Things to know: 1.Find slope from graph 2.Find a point using slope 3.Find slope using 2 points 4.Understand slope between 2 points is always the same on the same line Slope Slope = 5 – 2 = Slope = 1 – (-1) = 2 3 – 0 3 y = mx + b m = slope b = y intercept x y=3x+2 x y=2/3x – Slope is the ratio of RISE (How High) y 2 – y 1  y (Change in y) RUN (How Far) x 2 – x 1  x (Change in x) y x =

The Possibilities for a Line’s Slope (m) Positive Increasing slope x y m > 0 Line rises from left to right. Zero Slope x y m = 0 Line is horizontal. m is undefined Undefined Slope x y Line is vertical. Negative/Decreasing slope x y m < 0 Line falls from left to right. Example: y = 2 Example: x = 3 Example: y = ½ x + 2 Example: y = -½ x + 1 Things to know: 1.Identify the type of slope given a graph. 2.Given a slope, understand what the graph would look like and draw it. 3.Find the equation of a horizontal or vertical line given a graph. 4.Graph a horizontal or vertical line given an equation 5.Estimate the point of the y-intercept or x-intercept from a graph. Question: If 2 lines are parallel do you know anything about their slopes?

Linear Equation Forms (2 Vars) Standard FormAx + By = CA, B, C are real numbers. A & B are not both 0. Example: 6x + 3y = 12 Slope Intercept Formy = mx + bm is the slope b is the y intercept Example: y = - ½ x - 2 Point Slope Formy – y 1 = m(x – x 1 ) Example: Write the linear equation through point P(-1, 4) with slope 3 y – y 1 = m(x – x 1 ) y – 4 = 3(x - - 1) y – 4 = 3(x + 1) Things to know: 1.Find Slope & y-intercept 2.Graph using slope & y-intercept Things to know: 1.Change from point slope to/from other forms. 2.Find the x or y-intercept of any linear equation Things to know: 1.Graph using x/y chart 2.Know this makes a line graph.

Parallel and Perpendicular Lines & Slopes PARALLEL Vertical lines are parallel Non-vertical lines are parallel if and only if they have the same slope y = ¾ x + 2 y = ¾ x -8 Same Slope PERPENDICULAR Any horizontal line and vertical line are perpendicular If the slopes of 2 lines have a product of –1 and/or are negative reciprocals of each other then the lines are perpendicular. y = ¾ x + 2 y = - 4/3 x - 5 Negative reciprocal slopes 3 -4 = -12 = Product is -1 Things to know: 1.Identify parallel/non-parallel lines. Things to know: 1.Identify (non) perpendicular lines. 2.Find the equation of a line parallel or perpendicular to another line through a point or through a y-intercept.

Practice Problems 1.Find the slope of a line passing through (-1, 2) and (3, 8) 2.Graph the line passing through (1, 2) with slope of - ½ 3.Is the point (2, -1) on the line specified by: y = -2(x-1) + 3 ? 4.Parallel, Perpendicular or Neither? 3y = 9x + 3 and 6y + 2x = 6 5.Find the equation of a line parallel to y = 4x + 2 through the point (-1,5) 6.Find the equation of a line perpendicular to y = - ¾ x –8 through point (2, 7) 7.Find a line parallel to x = 7; Find a line perpendicular to x = 7 Find a line parallel to y = 2; Find a line perpendicular to y = 2

3.7 Linear Inequalities Graph: x + y < 4x y Test Point (0,0)  < TRUE y x Step 1: Graph the line Step 2: Decide: Dashed or Solid Step 3: Choose a test point on 1 side Step 4: Plug in test point & check Step 5: Shade the TRUE side. Graph the intersection (  )of 2 inequalities 2x – y < 2 x + 2y > 6 x y y x Test and Shade See in-class example Question: How would you graph the union?

Calculator Skills for Chapter3 1.Setup X/Y Axes (Cartesian Coordinate plane) with [Window] 2.Graph a y = ___________ equation 3.View table values associated with a graph 4.Locate X and Y intercepts graphically using [Trace Button] 5.Locate X and Y intercepts using the table values 6.[Quit] out of graphical mode to Normal calculator mode.