MAM
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.
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:
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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