2. Ch1: Graphs y axis x axis Quadrant I (+, +)Quadrant II (-, +) Quadrant III (-, -)Quadrant IV (+, -) Origin (0, 0) 2 4 6 -6 -2 (-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 : y = ½ x – 3 (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)

3. 1.1 Distance & Midpoint y x Origin (0, 0) 2 4 6 -6 -2 (-6,-3) The Distance Formula To find the distance between 2 points (x 1, y 1 ) and (x 2, y 2 ) d =  (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 The Midpoint Formula To find the coordinates of the midpoint (M) of a segment given segment endpoints of (x 1, y 1 ) and (x 2, y 2 ) x 1 + x 2, y 1 + y 2 2 2 M A point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y) (5,-2) Things to know: 1.Find distance or midpoint given 2 points 2.Given midpoint and 1 point, find the other point

4. 1.1 & 1.2 Linear Equations The graph of a linear equation is a line. A linear function is of the form y = 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 y x x y=3x+2 x y=2/3x –1 0 2 0 -1 1 5 3 1 All of these equations are linear. Three of them are graphed above.

5. X and Y intercepts y x (0,-3) (6,0) y intercept x intercept Equation: y = ½ x – 3 The y intercept happens where y is something & x = 0: (0, ____) Let x = 0 and solve for y: y = ½ (0) – 3 = -3 The x intercept happens where x is something & y = 0: (____, 0) Let y = 0 and solve for x: 0 = ½ x – 3 => 3 = ½ x => x = 6 -3 6

6. 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 = 3 1 - 0 Slope = 1 – (-1) = 2 3 – 0 3 y = mx + b m = slope b = y intercept x y=3x+2 x y=2/3x –1 0 2 0 -1 1 5 3 1 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 =

7. The Possibilities for a Line’s Slope (m) Positive 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 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?

8. 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 3.Application meaning of of slope & intercepts 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.

9. 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 = -1 4 3 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.

10. 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 the equation of a line passing through the points (-2, 1) and (3, 7) 8. Graph (using an x/y chart – plotting points) and find intercepts of any equation such as: y = 2x + 5 or y = x 2 – 4

11. Symmetry and Odd/Even Functions Y-Axis Symmetry  even functions f (-x) = f (x) For every point (x,y), the point (-x, y) is also on the graph. Test for symmetry: Replace x by –x in equation. Check for equivalent equation. Origin Symmetry  odd functions f (-x) = -f (x) For every point (x, y), the point (-x, -y) is also on the graph. Test for symmetry: Replace x by –x, y by –y in equation. Check for equivalent equation. X-Axis Symmetry (For every point (x, y), the point (x, -y) is also on the graph.) Test for symmetry: Replace y by –y in equation. Check for equivalent equation. y = x 3 Origin Symmetry y = x 2 Y-axis Symmetry (EVEN) x = y 2 X-axis Symmetry Symmetry Test -y = (-x) 3 -y = -x 3 y = x 3 Symmetry Test y = (-x) 2 y = x 2 Symmetry Test x = (-y) 2 X = y 2 ODD) Try these without Using a graph: y = 3x 2 – 2 y = x 2 + 2x + 1

12. A Rational Function Graph & Symmetry y = 1 x x y -2 -1/2 -1/2 -2 0 Undefined ½ 2 1 2 ½ Intercepts: No intercepts exist If y = 0, there is no solution for x. If x = 0, y is undefined The line x = 0 is called a vertical asymptote. The line y = 0 is called a horizontal asymptote. Symmetry: y = 1/-x => No y-axis symmetry -y = 1/-x => y = 1/x => origin symmetry -y = 1/x => y = -1/x => no x-axis symmetry

13. 1.3 Functions and Graphs 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 $3111 1998 $3247 1999 $3356 2000 $3510 independent variable (x) dependent variable (y) The cost depends on the year. Year(x) Cost(y) 19973111 19983247 19993356 20003510 Thinking Exercise: Draw a ‘line’ in the x/y axes. What is the Domain & Range?

14. 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=-3 2 -2 f(2) = ½(2)-3=-2 6 0 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.

15. Diagrams of Functions Function: f(x) = ½ x – 3 x y = f(x) 0 -3 f(0) = ½(0)-3=-3 2 -2 f(2) = ½(2)-3=-2 6 0 f(6) = ½(6)-3=0 8 1 f(8) = ½(8)-3=1 0 2 6 -3-3 8 -2-2 0 1 f 123123 f A functionNOT a function 4545 456456 A function is a correspondence fro the domain to the range such that each element in the domain corresponds to exactly one element in the range.

16. How to Determine if an equation is a function Example 1: x 2 + y = 4 y = 4 – x 2 For every value of x there Is exactly 1 value for y, so This equation IS A FUNCTION. Graphically: Use the vertical line test Symbolically/Algebraically: Solve for y to see if there is only 1 y-value. Example 2: x 2 + y 2 = 4 y 2 = 4 – x 2 y = 4 – x 2 or y = - 4 – x 2 For every value of x there are 2 possible values for y, so This equation IS NOT A FUNCTION.

17. Are these graphs functions? Use the vertical line test to tell if the following are functions: y = x 3 Origin Symmetry y = x 2 Y-axis Symmetry x = y 2 X-axis Symmetry

18. More on Evaluation of Functions f(x) = x 2 + 3x + 5 Evaluate: f(2) f(2) = (2) 2 + 3(2) + 5 f(2) = 4 + 6 + 5 f(2) = 15 Evaluate: f(x + 3) f(x + 3) = (x + 3) 2 + 3 (x + 3) + 5 f(x + 3) = (x + 3)(x + 3) + 3x + 9 + 5 f(x + 3) = (x 2 + 3x + 3x + 9) + 3x + 14 f(x + 3) = (x 2 + 6x + 9) + 3x + 14 f(x + 3) = x 2 + 9x + 23 Evaluate: f(-x) f (-x) = ( -x) 2 + 3( -x) + 5 f (-x) = x 2 - 3x + 5

19. More on Domain of Functions A function ’ s domain is the largest set of real numbers for which the value f(x) is a real number. So, a function’s domain is the set of all real numbers MINUS the following conditions: specific conditions/restrictions placed on the function Bounds relating to real-life data modeling (Example: y = 7x, where y is dog years and x is dog ’ s age) values that cause division by zero values that result in an even root of a negative number What is the domain the following functions: 1.f(x) = 6x2. g(x) = 3x + 12 3. h(x) = 2x + 1 x 2 – 9

20. Slope & Average Rate of Change y = x 2 - 4x + 4 y - $$ in thousands x Yrs (0,-3) (6,0) y = ½ x – 3 The slope of a line may be interpreted as the rate of change. The rate of change for a line is constant (the same for any 2 points) y 2 – y 1 x 2 – x 1 Non-linear equations do not have a constant rate of change. But you can Find the average rate of change from x 1 to x 2 along a secant to the graph. f(x 2 ) – f(x 1 ) x 2 – x 1 See Page 38-39 for more examples.

21. Definition of a Difference Quotient The average rate of change for f(x) is called the “difference quotient” and is defined below. (This is an important concept in calculus – it becomes the mathematical definition of the derivative you will learn about next semester. Example: Find the difference quotient for : f(x) = 2x 2 -3 f(x + h) = 2(x + h ) 2 - 3 = 2x 2 + 4xh + 2h 2 -3 – 2x 2 + 3 h = 2(x + h)(x + h) -3 = 4xh + 2h 2 = 2(x 2 + 2xh + h 2 ) -3 h = 2x 2 + 4xh + 2h 2 -3 = 4x + 2h So, the difference quotient is: 2x 2 + 4xh + 2h 2 -3 – (2x 2 -3) h

22. 1.4 Increasing, Decreasing, and Constant Functions Constant f (x 1 ) = f (x 2 ) (x 1, f (x 1 )) (x 2, f (x 2 )) Increasing f (x 1 ) < f (x 2 ) (x 1, f (x 1 )) (x 2, f (x 2 )) Decreasing f (x 1 ) > f (x 2 ) (x 1, f (x 1 )) (x 2, f (x 2 )) A function is increasing on an interval if for any x 1, and x 2 in the interval, where x 1 < x 2, then f (x 1 ) < f (x 2 ). A function is decreasing on an interval if for any x 1, and x 2 in the interval, where x 1 f (x 2 ). A function is constant on an interval if for any x 1, and x 2 in the interval, where x 1 < x 2, then f (x 1 ) = f (x 2 ).

23. Observations Decreasing on the interval (-oo, 0) Increasing on the interval (0, 2) Decreasing on the interval (2, oo). -5-4-3-212345 5 4 3 1 -2 -3 -4 -5 -4-3-212345 5 4 3 2 1 -2 -3 -4 -5 a.b. More Examples Observations a. Two pieces (a piecewise function) b. Constant on the interval (-oo, 0). c. Increasing on the interval (0, oo). Challenge Yourself: What might be the definition of the piecewise function for this graph? (You will learn about these Later. Can you guess what it might be?)

24. Relative (local) Min & Max 0 x y 180360 90270 1 f(x) = sin (x)x y 0  /2 1  0 3  /2 -1 2  0 2 -2 The point at which a function changes its increasing or decreasing behavior is called a relative minimum or relative maximum. (90, f(90)) f(90), or 1, is a local max (270, f(270)) f(270), or -1, is a local min A function value f(a) is a relative maximum of f if there exists an open interval about a such that f(a) > f(x) for all x in the open interval. A function value f(b) is a relative minimum of f if there exists an open interval about b such that f(b) < f(x) for all x in the open interval.

25. 1.4 Library of Functions/Common Graphs y = c x y = x x y = x 2 x y = x 3 x y = x x y = |x| x y = 1/x y = x 1/3 x

26. Step Function Application Example y = int(x) or y = [[x]](Greatest Integer Function) f(x) = int(x) y – Tax (+) or Refund (-) in thousands of $$ x – Income in $10,000’s Find: 1)f (1.06) 2)f (1/3) 3)f (-2.3) What other applications of the step function can you think of?

27. Piecewise Functions f(x) = x 2 + 3 if x < 0 5x + 3 if x>=0 f(-5) = (-5) 2 + 3 = 25 + 3 = 28 f(6) = 5(6) + 3 = 33 A function that is defined by two (or more) equations over a specified domain is called a piecewise function. See Page 247 for more examples

28. 1.5 Transformation of Functions A transformation of a graph is a change in its position, shape or size. For a given function, y = f(x) y = f(x) +c [shift up c] y = f(x) – c [shift down c] y = f(x + c) [shift left c] y = f(x – c) [shift right c] y = -f(x) [flip over x-axis] y = f(-x) [flip over y-axis] y = cf(x) [multiply y value by c] [if c > 1, stretch vertically] [if 0 < c < 1, shrink vertically] Example function: y = x 2 Graph: y = x 2 + 4 y = x 2 - 4 y = (x+4) 2 y = (x – 4) 2 y = -x 2 y = (-x) 2 y = ½ x 2 Can you graph : y = ½ (x + 2) 3 + 2

29. More Transformation Practice Suppose that the x-intercepts of the graph of y = f(x) are -5 and 3 (a)What are the x-intercepts of the graph of y = f(x + 2) (b)What are the x-intercepts of the graph of y = f(x – 2) (c)What are the x-intercepts of the graph y = 4f(x) (d)What are the x-intercepts of the graph of y = f(-x)

30. 1.6 Sum, Difference, Product, and Quotient of Functions Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows: Sum: (f + g)(x) = f (x)+g(x) Difference:(f – g)(x) = f (x) – g(x) Product:(f g)(x) = f (x) g(x) Quotient:(f / g)(x) = f (x)/g(x), provided g(x) does not equal 0 Example: Let f(x) = 2x+1 and g(x) = x 2 -2. f+g = 2x+1 + x 2 -2 = x 2 +2x-1 f-g = (2x+1) - (x 2 -2)= -x 2 +2x+3 fg = (2x+1)(x 2 -2) = 2x 3 +x 2 -4x-2 f/g = (2x+1)/(x 2 -2)

31. Adding & Subtracting Functions If f(x) and g(x) are functions, then: (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) Examples: f(x) = 2x + 1 and g(x) = -3x – 7 Method1 Method1 (f + g)(4) = 2(4) + 1 + -3(4) – 7 (f – g)(6) = 2(6) + 1 – [-3(6) – 7] = 8 + 1 + -12 – 7 = 12 + 1 - [-18 – 7] = 9 + -19 = 13 - [-25] = -10= 13 + 25 Method2 Method2= 38 (f + g)(4) = 2x + 1 + -3x – 7 (f - g)(6) = 2x + 1 - [-3x – 7] = -x – 6 = 2x + 1 + 3x + 7 = -4 – 6 = 5x + 8 = - 10 = 5(6) + 8 = 30 + 8 = 38 Adding/subtracting also extends to non-linear functions you will see in a subsequent chapter.

32. The Composition of Functions f o g - composition of the function f with g is is defined by the equation (f o g)(x) = f (g(x)). f (x) = 3x – 4 and g(x) = x 2 + 6 (f o g)(x) = f (g(x)) = 3g(x) – 4 = 3(x 2 + 6) – 4 = 3x 2 + 18 – 4 = 3x 2 + 14 (g o f)(x) = g(f (x)) = (f (x)) 2 + 6 = (3x – 4) 2 + 6 = 9x 2 – 24x + 16 + 6 = 9x 2 – 24x + 22

33. 1.7 Inverse Function If f (g(x)) = x for every x in the domain of g and g(f (x)) = x for every x in the domain of f. Then the function g is the inverse of the function f denoted by f -1 and the function f is the inverse of the function g denoted by g -1 The domain of f is equal to the range of f -1, and vice versa. (x,y) in f => (y, x) in f --1 Examples: Verifying inverses (Are f & g inverses?) f (x) = 5x and g(x) = x/5. f(x)= 3x + 2 g(x) = x - 2 3 f (g(x)) = 5(g(x)) = 5 x = x f (g(x)) = 3(g(x))+2 5 = 3 x-2 + 2 g( f (x)) = f(x) = 5x = x 3 5 5 = x – 2 + 2 = x f (g(x)) = x and g( f (x)) = x g(f(x) = f(x) – 2 = 3x + 2 – 2 = x 3 3 Thus they are inverses. Thus they are inverses f(x) = 5x f -1 (x)=x/5 f(x ) = 3x + 2 f -1 (x) = (x-2)/3

34. Example: Find the inverse of f (x) = 7x – 5. Step 1: Replace f (x) by y: y = 7x – 5 Step 2: Interchange x and y : x = 7y – 5. Step 3: Solve for y. : x + 5 = 7y x + 5 = y 7 Step 4 Replace y by f -1 (x). x + 5 7 f -1 (x) = How to Find the Inverse of a Function

35. The Horizontal Line Test For Inverse Functions A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point f(x) = x 2 +3x-1 NO Inverse Function y x f(x) = x + 4 YES this has an inverse Function

36. Graphing the Inverse of a Function y x 1.Draw any function (f) that passes the horizontal line test. 2. To graph the inverse (f --1 ) reverse each (x, y) point on f, graphing (y, x). Create your own Example: