Functions+Notes+_updated_.pdf

Graphically b is the y intercept of the line y mx b

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Graphically. b is the y -intercept of the line y = mx + b . Examples 2.8. The function f ( x ) = 2 x +1 has slope m = 2 and y -intercept b = 1, while the line y = - 1 . 5 x + 3 . 5 has slope m = - 1 . 5 and y -intercept b = 3 . 5. Computing the Slope of a Line We can always determine the slope of a line if we are given two (or more) points of the line. Assume that ( x 1 , y 1 ) and ( x 2 , y 2 ) are two points of a line. Then its slope m can be computed as follows: m = Δ y Δ x = y 2 - y 1 x 2 - x 1
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2. Functions 15 Computing the y -intercept Once we know the slope m of a line, and also the coordinates of a point ( x 1 , y 1 ), the y -intercept of such a line is b = y 1 - mx 1 Examples 2.9. Find equations for the following straight lines: a. Through the points (1 , 2) and (3 , - 1) . b. Through (2 , - 2) and parallel to the line 3 x + 4 y = 5 . c. Horizontal through ( - 9 , 5) . d. Vertical through ( - 9 , 5) . Solution: In any case, we need to calculate the slope m and the y - intercept b . a. Because we are given two points on the line, we can use the slope formula: m = y 2 - y 1 x 2 - x 1 = - 1 - 2 3 - 1 = - 3 2 We now have the slope of the line, and so b = y 1 - mx 1 = 2 - - 3 2 = 7 2 Thus, the equation of the line is y = - 3 2 x + 7 2 b. Keeping in mind that parallel lines have the same slope, we can conclude that m = - 3 4 , since the line 3 x +4 y = 5 can be written as y = - 3 4 x + 5 4 . We now use the formula for the y -intercept: b = y 1 - mx 1 = - 2 - - 3 4 (2) = - 1 2 Thus, the equation of the line is y = - 3 4 x - 1 2
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16 J. S´ anchez-Ortega c. The slope is m = 0 because the line is horizontal. Thus, it has the form y = b . Since we have been given a point, namely ( - 9 , 5), we can conclude that the equation of the line is y = 5. d. Since the line is vertical, its equation is of the form x = c for some real number c . Notice that the slope is undefined in this case. We have that all the points in the line are of the form ( c, y ) for y any real number. Using that ( - 9 , 5) is a point of our line, we can conclude that their equation is x = - 9. See also Example 1, page 81 in the textbook.
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2. Functions 17 Below we could see how different slopes look like. Notice that the larger the absolute value of the slope, the steeper is the line.
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18 J. S´ anchez-Ortega 2.5 Quadratic Functions Linear functions are very useful but unfortunately in real life, the relationship between two quantities is often best modeled by a curved line rather a straight line. The simplest function with a graph that is not a straight line is a quadratic function . A quadratic function of the variable x is a function that can be written in the form f ( x ) = ax 2 + bx + c , where a , b and c are fixed real numbers with a 6 = 0. The graph of a quadratic function is a parabola A summary of some features of parabolas follows. It will help us to sketch the graph of any quadratic function.
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2. Functions 19 We can also write the x -intercepts as x = - b 2 a ± b 2 - 4 ac 2 a Notice that they are located symmetrically on either side of the vertical line x = - b/ (2 a ).
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