41.1 Qualitative GraphsDefinition: Graph without scaling (tick marks and their numbers) on the axis.Medication CostspLinear CurvepricetTime
5Independent/Dependent Variables Price p is dependent on the year t.(p is dependent of t) p is a dependent variableYear t does not depend on the price p.(t is independent of p) t is an independent variable
6ExamplesYou are taking a patient’s pulse counting the beats b per minute m.The number of people n that can run m miles.Waiting inline in the cafeteria for lunch. Use n for number of people in front and w for the wait time.Answers:Independent - m, m, nDependent - b, n, w
7Independent/Dependent Variable (time) Independent most the time.Something happens over timetemperature over time, growth, population, etc.DependentTime it takes for something to happenwait time, time taken to cook, time taken to move a distance, etc
9InterceptsAverage Age of NursesmAm-interceptA-interceptAverage Agen-interceptparabolantYears Since 1900An intercept of a curve is the point where the curve intersects an axis (or axes).
10Graph Characteristics Increasing Curveupward from left to rightDecreasing Curvedownward from left to rightQuadrants
11ExamplesHeight of a ball thrown straight upHeart rate of a person on a treadmill as the pace is steadily increasedThe temperature in Celsius of boiled water placed in a freezerThe value of a car after a fixed amount of years
121.2 Graphing Linear Equations Solution, Satisfy, and Solution SetAn ordered pair (a,b) is a solution of an equation in x and y if the equation becomes a true statement when a substituted for x and b is substituted for y. We say (a,b) satisfies the equation.Solution set is the set of all solutions of the equation.
13Example y = 4x – 2 Find y if x = 2 y = 4(2) – 2 = 8 – 2 = 6 Thus the ordered pair (2,6) satisfies the equation y = 4x – 2 and (2,6) is a solution of the equation y = 4x – 2
14Continued Solution set of the equation y = 4x – 2 includes (2,6) x y Ordered pair (a,b) the independent variable isFirst (left) followed by the dependent variable inthe second position (right).
15Graphing Equations y = 5x + 2 Chose values, ex. 0, 1 Organize in table = 0 + 2= 2Solution: (0,2)y = 5(1) + 2= 5 + 2= 7Solution: (1,7)Chose values, ex. 0, 1Organize in tablex y71217
17(3, 17)A graph of an equation in two variables is a visual representation of the solutions of an equation.(2, 12)(1, 7)(0, 2)(-1,-3)(-2,-8)
18(1, 9) does not satisfy the equation (3, 17)(2, 12)(1, 9)(0, 2)(1, 9) does not satisfy the equation(-1,-3)(-2,-8)
19Linear Equation Forms Slope-Intercept Form Standard Form General Form y = mx + bm and b are both constants (m is the slope, b is the y-intercept)Standard FormAx + By = CA, B, C are integers (A and B are not both equal to zero)General FormAx + By + C = 0Point-Slope Form(y – y ) = m (x – x )Points (x, y) & (x ,y ), slope m1111
20General to Slope-Intercept Form 3y - 12x + 9 = 03y - 12x = 0 -93y – 12x = -93y - 12x + 12x = x3y = xy = xy = 4x - 3General to StandardTry to get y by itselfDivideSimplify
21Distributive Law First Distribute 2(2y-3) = 2(3x-4) + 2y Get y by itselfDivideSimplify2(2y-3) = 2(3x-4) + 2y4y – 6 = 6x – 8 +2y4y – 6 -2y = 6x – 8 + 2y -2y2y – 6 +6 = 6x – 8 +62y = 6x – 2y = 3x - 1
22Graphing Equations with Fraction Slopes y =¾x - 4Chose x values that are multiples of the denominator. Ex. 0, 4, 8, 12Make a chart and graphx y0 y = ¾ (0) – 4 = -44 y = ¾ (4) – 4 = -18 y = ¾ (8) – 4 = 2
23Intercepts of GraphsTo find the x-intercept, substitute 0 for y and solve for x. It will always be in the form (x,0).To find the y-intercept, substitute 0 for x and solve for y. It will always be in the form (0,y).y = - ½x +30 = - ½x + 30 -3 = - ½x -3-3 *(-2) = - ½x *(-2)6 = x~x-intercept (6,0)y = - ½ (0) +3y = 3~y-intercept (0,3)y-intercept (0,3)x-intercept (6,0)
24Vertical/Horizontal Lines If a and b are constants,Vertical lines are in the form x=aHorizontal lines are in the form y=by=6x=8
25How do we measure steepness? 1.3How do we measure steepness?
26Comparing SteepnessCalculate the ratio of vertical to horizontal distance.Patient 1 has his foot elevated 2 feet above the bed over a vertical distance of 2.5 feet. Patient 2 has his foot elevated 1.5 feet above the bed over a vertical distance of 2 feet. Which leg is elevated steeper?Patient 1’s foot is steeper because the ratio is slightly greater.Vertical DistanceHorizontal Distance21.52.5221.5= .8= .752.52
27SlopeLet (x , y ) and (x , y ) be two points on a line. The slope of the line is:vertical change rise y - yhorizontal change run x - x112221m = = =21Run is positive to the rightRun is negative to the leftRise is positive going upRise is negative going down
29Finding Slope of a Graph m = riserunm = 21m = 2m = y - yx - xm = 3 – 12 – 1m = 2 = 2121(2,3)212(1,1)1
30Investigating Slope of Horizontal/Vertical Lines 2 – Slope of a horizontal line4 – 2 2 is always 0.3 – 1 2 Slope of a vertical line1 – 1 0 is always undefined.m = = = 0m = = = Undefinedy = 2 (2,2) (4,2)x = 1 (1,1) (1,3)
31Parallel/Perpendicular Lines Parallel lines have the same slopem = mSlopes of perpendicular lines are opposite reciprocals1 1m 2l2Parallel lines never intersect21l1l2m =2 =2l11Perpendicular lines form 90 degree (right) angles
32Finding Slope of Linear Equations 1.4Finding Slope of Linear Equationsy = -3x - 2GraphriserunIf in slope-intercept form:m = slope, m = -3b = y-intercept, b = -21m = = = -3-31-3
33Using Slope and y-intercept to graph an equation Slope = 2 = rise 2run 1y-intercept (0, -2)Plot y-intercept (0, b)Use m = to plot 2nd point.Sketch line passing through the 2 points.riserun(0, -2)
34Vertical Change Property For a line y = mx + b, if the run in 1, then the rise is the slope m.m11mSlope Addition PropertyFor a line y = mx + b, if the independent variable increases by 1, then the dependent variable changes by slope m.y = -6x + 3 , x increases by 1, y changes by -6
35Identifying Parallel/Perpendicular Lines Are the lines 3y = 2x – 6 and 6y – 4x = 24 parallel, perpendicular or neither?3y = 2x – 6y = 2/3x - 36y – 4x + 4x = 18 +4x6y = 4x + 24y = 4/6x + 24/6y = 2/3x + 4Get y by itselfDivideSimplifyGet the equations in slope-intercept formSlope in both equations is 2/3, therefore the lines are parallel.
36Indentifying Linear Equations from points Set 1Set 2Set 3Set 4XY10246733-2411201234-31681735-41392336-5142737-6
37Answers Set 1 – slope -4 Set 2 – not a line Set 3 – slope 0
38Linear Equation Forms Slope-Intercept Form Standard Form General Form 1.5Slope-Intercept Formy = mx + bm and b are both constants (m is the slope, b is the y-intercept)Standard FormAx + By = CA, B, C are integers (A and B are not both equal to zero)General FormAx + By + C = 0Point-Slope Form(y – y ) = m (x – x )Points (x, y) & (x ,y ), slope m1111
39Find an equation given slope and a point Slope 4, point (6,2)Start with putting the slope into y=mx + by = 4x + bPlug the coordinates of the point in for x and y(2) = 4(6) + bSolve for b2 -24 = 24 + b -24-22 = by = 4x - 22
40Find an equation of the line using two points (3, 5) (4, -2)Find the slopey - y –x - x –Place slope into y = mx + by = -7x + bThen chose either point to plug in for x and y… (3, 5)(5) = -7(3) + bSolve for b5 +21 = b +2126 = b… y = -7x + 26m = = =2121
41Finding the approximate equation of a line (-4.56,-5.24) (7.72, -4.93)Find the slopey - y – (-5.24)x - x – (-4.56)Place slope into y = mx + by = .03x + bThen chose either point to plug in for x and y… (7.72, -4.93)(-4.93) = .03(7.72) + bSolve for b= b-5.16 = b… y = .03x – 5.16m = = = = = .032121
42Finding an equation of a line parallel to a given line y = 3x – 2 , point (1,6)m = 3, so slope is 3Parallel lines have similar slopes, so:y = 3x + bPlug in the coordinates for x and y(6) = 3(1) + b6 -3 = 3 + b -3b = 3y = 3x + 3 is parallel to y = 3x -2
43Finding an equation of a line perpendicular to a given line y = ½ x – 6, point (2,-2)Slopes of perpendicular lines are opposite reciprocals:flip the numerator and denominator and change the signor -2y = -2x + b(-2) = -2(2) + b-2 +4 = -4 + b +42 = b, y = -2x + 2 is perpendicular to y = ½ x - 61221
44Point-slope FormIf a non-vertical line has a slope m and contains the point (x ,y ), then the equation of the line is:(y - y ) = m (x - x )Given a point and the slope:m = 4 point (6,-3)y – (-3) = 4 (x - 6 )y = 4x – 24 -3y = 4x – 271111
45Finding an equation of a line using point-slope form Given two points (2,14) and (-8,-6)Find slope-6 –-8 –(y - y ) = 2(x - x )Substitute in one of the pointsy – (14) = 2(x – (2))y – = 2xy = 2x + 10m = = = 211
461.6 Functions Relation – set of ordered pairs Domain – set of all values of the independent variable (x-values)Range – set of all values of the dependent variable (y-values)Function – a relation in which each input leads to exactly one outputDomain = input, Range = output
47Equations by input/output x567y314 & 3x24y789RELATIONSRELATIONSinput(domain)output(range)input(domain)output(range)
48Is it a function? y = ± x y² = x y = x² x = 5 , y = ± 5 or y = 5 and y = -5input has 2 outputs, not a functiony² = xx = 16, y = ± 4 or y = 4 and y = -4y = x²x = 3, y = 3² = 9input has 1 output, function
49Is it a function? Set 1 Set 2 Set 3 Set 4 x y 4 1 5 6 10 9 -1 7 11 2 inputyoutput4156109-1711212813-214
50Answers Set 1 – not function, slope undefined, vertical line Set 2 – not a functionSet 3 – function, slope 0, horizontal lineSet 4 – function, slope -3, decreasing line
51Is it a function?When x =2y = 3 andy = -3, not afunction
52Vertical line testA relation is a function if and only if every vertical line intersects the graph of the relation at no more than one point.
53Functions from equations Is y =-3x -2 a function?Graph it!Any vertical linewould intersectonly once….Yes.
54Linear Function All non-vertical lines are functions. Linear function: a relation whose equation can be put iny=mx + b form (m and b are constants)
55Rule of Four Four ways to describe functions x y -2 -8 -1 -3 0 2 7 12 y = 5x + 2Four ways to describe functionsEquationGraphTableVerbally (input-output)x y71217For each input-output pair, the output is two morethan 5 times the input.
56Finding Domain and Range Domain = width = input = independent variable = x-valuesRange = height = output = dependent variable = y-valuesLeftmost point (-3,1)Rightmost point (4, 1)Domain -3 ≤ x ≤ 4Highest point (2, 3)Lowest point (-1,-2)Range -2 ≤ y ≤ 3
57Domain/Range Continued D = all real numbersR = y ≥ -1D = x ≥ 0R = y ≥ -1D = all real numbersR = all realnumbers