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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
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