Functions+Notes+_updated_.pdf

Example 210 sketch the graph of f x 2 x 2 x 6 by hand

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Example 2.10. Sketch the graph of f ( x ) = - 2 x 2 + x + 6 by hand. Soluci´on: Here a = - 2, b = 1, c = 6. Because a < 0 the parabola is concave down. The x -coordinate of the vertex is x = - b 2 a = - 1 2( - 2) = 1 4 To get the y -coordinate of the vertex, we compute f (1 / 4): f 1 4 = - 2 1 4 2 + 1 4 + 6 = - 2 16 + 1 4 + 6 = 49 8 Thus, the coordinates of the vertex are V = 1 4 , 49 8
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20 J. S´ anchez-Ortega Next, we compute the x -intercepts (if any). To do so, we solve the equation: - 2 x 2 + x + 6 = 0 x = - 1 ± p 1 2 - 4( - 2)6 2( - 2) = - 1 ± 49 - 4 = - 1 ± 7 - 4 , x 1 = 2 , x 2 = - 3 2 Thus P = (2 , 0), Q = ( - 3 2 , 0) are in the graph of f . Now, we can sketch the graph of f : -1.5 0.25 2 6.125 x y f ( x ) = - 2 x 2 + x + 6 V P Q See also Example 2, pages 622 and 623 in the textbook.
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2. Functions 21 2.6 Exponential Functions As we will see the quadratic functions can be used to model many nonlinear situations. However, exponential functions give better models in some appli- cations like, for example, the growth or depreciation of financial investments. We start this section by recalling the laws o exponents. An exponential function has the form f ( x ) = Ab x , where A and b are constants with A 6 = 0 and b positive and not equal to 1 . We call b the base of the exponential function. Notice that the domain of f ( x ) is R ; the y -intercept is A (obtained by setting x = 0);
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22 J. S´ anchez-Ortega What can we say about b ? Let us see an example! Consider the exponen- tial function f ( x ) = 3(2 x ). Here A = 3 and b = 2. From the following table we can see that the value of y is multiplied by b = 2 for every increase of 1 in x . If we decrease x by 1, the y coordinate gets divided by b = 2. On the graph if we move one unit to the right from any point on the curve, the y coordinate doubles. Thus, the curve becomes dramatically steeper as the value of x increases. This phenomenon is called exponential growth , which is illustrated by an exponential function when b > 1. If b < 1 then the corresponding exponential function models the opposite phenomenon, called exponential decay . In general:
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2. Functions 23 See also Quick Examples, and Examples 1 and 2 in pages 634, 635 (Textbook). The Number e The number e is the limiting of the quantities ( 1 + 1 m ) m as m gets larger and larger, and has the value 2 . 71828182845904523536 . . . The number e is one of the most important in Mathematics, it is irrational. The number e can be used to express any exponential function as follows: f ( x ) = Ae rx , where A and r are constants. If r is positive, then f models exponential growth; if r is negative, f models exponential decay. Notice that f ( x ) = Ae rx = A ( e r ) x = Ab x , b = e x . 2.7 Logarithmic Functions Logarithms are used to model real-world phenomena in numerous fields, in- cluding Finance. From the equation 2 3 = 8 we can see that the power to which we need to raise 2 in order to get 8 is 3. We abbreviate the phrase “the power to which we need to raise 2 in order to get 8” as log 2 8. Thus, another way of writing the equation 2 3 = 8 is log 2 8 = 3 .
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