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##### Lecture 2

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**Lecture 2**Linear functions**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,∞[**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**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**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.**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.**Figure1The steepness or the slope of the line is the ratio**of the vertical to the horizontal distance**Figure2The slope increases as the steepness of the line**increases**Example**Find the slope and the y-intercept of the straight line, then graph it, • 2y – 4x = 8 • x + y = 4**Solution**• 2y – 4x = 8 y - 2x = 4 y = 2x + 4 • m =2, c = 4**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.**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.**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**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**Finding the slope of a line**• The slope m of the line passing through (x1, y1) and (x2, y2) is • Where**Example**Find the slope of the line passing through each pair of the following points, • (-2, 0) and (3, 1) • (3, 5) and (2, 1)**Solution**• a) • b)**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.**Example**• Find the slope of the line passing through the points (1,3) and (2,3). • Solution**The slope of a vertical line**• For a vertical line x is constant, thus the slope is not defined.**Example**• Find the slope of the line passing through the points (7,2) and (7,3). • Solution Undefined**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)**Example**• Find the equation of the line with a slope 3 and passes through the point (-4,5) • Solution**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**Example**• Find the equation of line passing through the two points (2,3), (-1, 1). • Solution**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.**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)**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**Example: Using a linear model to make future predictions**• In the last example, can you predict the sales per share in the year 2011?**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**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