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LECTURE 2Linear functions

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RANGE OF A FUNCTION THE RANGE OF THE FUNCTION F(X) IS THE ALL REAL NUMBERS Y FOR WHICH THERE IS SOME X WITH Y = F(X)

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

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GRAPHS OF FUNCTIONS AND RELATIONS 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 4)( 2 xxf 2522 yx

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THE VERTICAL LINE TEST 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

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EXAMPLE

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LINES IN THE PLANE 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.

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SLOPE, AND Y-INTERCEPT 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.

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FIGURE1THE STEEPNESS OR THE SLOPE OF THE LINE IS THE RATIO OF THE VERTICAL TO THE HORIZONTAL DISTANCE

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FIGURE2THE SLOPE INCREASES AS THE STEEPNESS OF THE LINE INCREASES

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EXAMPLEFind the slope and the y-intercept of the

straight line, then graph it,

2y – 4x = 8 x + y = 4

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SOLUTIONi) 2y – 4x = 8 y - 2x = 4 y = 2x + 4

m =2, c = 4

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Y = 2X + 4

3 2 1 1 2 3

2

2

4

6

8

10

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PRACTICAL MEANING OF THE SLOPE AND THE Y-INTERCEPT A manufacturing company determines

that the total cost in dollars of producing x units of a product is C = 25x + 3500.

Decide the practical significance of the y-intercept and the slope of the line given by the equation.

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PRACTICAL MEANING OF THE SLOPE AND THE Y-INTERCEPT

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PRACTICAL MEANING OF THE SLOPE AND THE Y-INTERCEPT 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.

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PRACTICAL MEANING OF THE SLOPE AND THE Y-INTERCEPT 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

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PRACTICAL MEANING OF THE SLOPE AND THE Y-INTERCEPT 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

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FINDING THE SLOPE OF A LINE The slope m of the line passing through

(x1, y1) and (x2, y2) is

Where 12

12

21

21

xx

yy

xx

yy

x

ym

21 xx

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EXAMPLEFind the slope of the line passing through

each pair of the following points,

(-2, 0) and (3, 1) (3, 5) and (2, 1)

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SOLUTION a)

b)

5

1

)2(3

01

12

12

xx

yym

41

4

23

15

12

12

xx

yym

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THE SLOPE OF A HORIZONTAL LINE 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.

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3 2 1 1 2 3

1

2

3

4

5

6

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EXAMPLE Find the slope of the line passing

through the points (1,3) and (2,3). Solution

01

0

12

33

12

12

xx

yym

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THE SLOPE OF A VERTICAL LINE For a vertical line x is constant, thus the

slope is not defined.

25

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EXAMPLE Find the slope of the line passing

through the points (7,2) and (7,3). Solution

Undefined

77

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12

12

xx

yym

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POINT – SLOPE FORM OF THE EQUATION OF A LINE The equation of the line with slope m

passing through the point (x1,y1) is given by

y – y1 = m(x-x1)

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EXAMPLE Find the equation of the line with a slope

3 and passes through the point (-4,5) Solution

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TWO POINTS FORM OF THE EQUATION OF A LINE The equation of a line passing through

the two points (x1,y1) and (x2,y2) is given by

m

xx

yy

xx

yy

12

12

1

1

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EXAMPLE Find the equation of line passing

through the two points (2,3), (-1, 1). Solution

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EXAMPLE: MAKING A LINEAR MODEL The sales per share for some company

were $25 in 2002 and $29 in 2007. Use this information to make a linear model that gives the sales per share at a year.

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SOLUTION: MAKING A LINEAR MODEL 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 =

0 2007 is equivalent to t = 5 Then we have two states for (t, S)

represented by the two points (0,25) and (5,29)

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SOLUTION: MAKING A LINEAR MODEL 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

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EXAMPLE: USING A LINEAR MODEL TO MAKE FUTURE PREDICTIONS In the last example, can you predict the

sales per share in the year 2011?

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SOLUTION: USING A LINEAR MODEL TO MAKE FUTURE PREDICTIONS

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

2.32$25)9(5

492011

t

SS

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POINTS TO REMEMBER 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