Linear functions 1

From the figure, we can find that the values of f(x) are in the interval [−1,∞[ 2

 The graph of a relation is given by all points which satisfy the relation. The graph of a function f(x) is the graph of the relation f(x) = y.  For Examples 3

 From the graph of a relation we can determine if this relation is a function or not. As we know a function assigns every x to exactly one y. So a graph of a relation is a graph of a function if on every vertical line there is at most one point of the graph.  In the previous slide The first graph is a graph of a function, since on every vertical line there  is at most one point of the graph.  The second graph is not a graph of a function, if you draw a vertical line at x=1 4

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 The simplest mathematical model for relating two variables is the linear equation y=m x + b  It is called linear because its graph is a line. 6

 By letting x=0, we see that the y-intercept is y=b.  The quantity m is the steepness or the slope of the line. 7

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Find the slope and the y-intercept of the straight line, then graph it,  2y – 4x = 8  x + y = 4 10

i) 2y – 4x = 8 y - 2x = 4 y = 2x + 4  m =2, c = 4 11

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 A manufacturing company determines that the total cost in dollars of producing x units of a product is C = 25x  Decide the practical significance of the y- intercept and the slope of the line given by the equation. 13

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15  The y-intercept is actually here the C- intercept. It is just the value of C, the cost, when x, the units produced, equals zero. Thus the y-intercept here is the cost when no units are produced, i.e. the Fixed Cost. In this example the fixed cost is $3500.

 The slope m = 25. It represents the additional cost for each unit produced. So if you produce one unit the cost will increase by $25. If you produce 2 units the cost increase by $50,…, and if you produce x units, the cost increase by $25x. Economists call the cost per unit the Marginal cost 16

 The above straight line represents the cost as a function of the produced units. It starts when x = 0 C = the fixed cost $3500, and for each increase by 1 unit for x, C increases by the marginal cost $25. Models of the form y = m x +c are called Linear Models 17

 The slope m of the line passing through (x 1, y 1 ) and (x 2, y 2 ) is  Where 18

Find the slope of the line passing through each pair of the following points,  (-2, 0) and (3, 1)  (3, 5) and (2, 1) 19

 a)  b) 20

 For a horizontal line y is constant.  So the equation of a horizontal line is y = C m = 0.  Thus the slope of a horizontal line is zero. 21

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23  Find the slope of the line passing through the points (1,3) and (2,3).  Solution

24  For a vertical line x is constant, thus the slope is not defined.

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 Find the slope of the line passing through the points (7,2) and (7,3).  Solution Undefined 26

 The equation of the line with slope m passing through the point (x 1,y 1 ) is given by y – y 1 = m(x-x 1 ) 27

28  Find the equation of the line with a slope 3 and passes through the point (-4,5)  Solution

 The equation of a line passing through the two points (x 1,y 1 ) and (x 2,y 2 ) is given by 29

 Find the equation of line passing through the two points (2,3), (-1, 1).  Solution 30

 The sales per share for some company were $25 in 2002 and $29 in Use this information to make a linear model that gives the sales per share at a year. 31

 Let the time in years be represented by t.  Let the sales per share be represented by S.  Let the year 2002 be represented by t = is equivalent to t = 5  Then we have two states for (t, S) represented by the two points (0,25) and (5,29) 32

 The linear relation between S and t is thus represented by the equation of the line passing through the 2 points (0,25) and (5,29)  This is given by 33

 In the last example, can you predict the sales per share in the year 2011? 34

35  Since the year 2002 is represented by t = 0, then 2011 is represented by t = 2011 – 2002 = 9  Substitute by t = 9 in the equation

36  Lines in the Plane.  Slope.  y- intercept.  Practical meaning of slope and y-intercept  Point-Slope equation of a straight Line.  Two Points equation of a straight Line.  Making Linear Models  Using Linear Models to make predictions