Working with quadratic and exponential graphs. Jackie Nicholas

Transcription

1 Mathematics Learning Centre Working with quadratic and eponential graphs Jackie Nicholas c 2005 Universit of Sdne

2 Mathematics Learning Centre, Universit of Sdne 1 1 Working with graphs The quadratic, eponential and logarithmic functions frequentl occur in economics or statistics. We will look at each in turn with the aim of giving ou some understanding of these functions and their graphical representation. 1.1 Quadratic functions A quadratic function has an equation of the form = a 2 + b + c where a 0,b and c are constants. Here are two scenarios where ou ma come across a quadratic function. A firm s revenue, R, is given as a function of output, q, b the formula, R = 200q 2q 2. The flow-densit relationship of traffic flow on a highwa can be described b the epression: k 2. Here q is given as a quadratic function of k, where u f and are constant. The graph of a quadratic is alwas a parabola and looks like Figure 1 or Figure 2. or Fig. 1 Fig. 2 The coefficient of the squared term determines whether our graph is like Figure 1 or Figure 2. For the general polnomial = a 2 + b + c, ifa>0 we get a parabola like that in Figure 1, while if a<0, we get a parabola like that in Figure 2. So, R = 200q 2q 2 will look like the parabola in Figure 2, as the coefficient of the q 2 term is negative. Note that R has a maimum value. For a general quadratic with equation = a 2 +b+c, there is a maimum or a minimum value when = b 2a.

3 Mathematics Learning Centre, Universit of Sdne 2 For R = 200q 2q 2 will have a maimum at q = 200 2( 2) =50. Note also that the parabolas in Figures 1 and 2 are smmetric about the vertical line through the maimum or minimum. To graph our quadratic we now need information that orients it on the coordinate aes. One thing we can do is determine the coordinates of the maimum or minimum. For the general polnomial, = a 2 + b + c we know this maimum or minimum occurs when = b. To find the coordinate of this maimum or minimum we substitute this 2a value into = a 2 + b + c: = a( b 2a )2 + b( b ab2 )+c = 2a 4a b2 2 2a + c = b2 4a + c. This gives us the coordinates of the maimum or minimum. For our eample R = 200q 2q 2, there is a maimum at q = 50 so the maimum value of R is R = 200(50) 2(50) 2 = = We now need either one or both intercepts of the parabola with the coordinate aes. If = 0 this is eas to find for = a 2 + b + c, as when =0, = c. If we want the points, if an, where the parabola = a 2 + b + c crosses the ais, we set = 0 and solve a 2 + b + c = 0 b factorising if we can, or b using the quadratic formula if required. For our eample, R = 200q 2q 2, when q =0,R =0. If R = 0 then 200q 2q 2 =2q(100 q) = 0 so, q =0orq = 100. This tells us that the parabola crosses the ais at (0, 0) and (100, 0). We now have sufficient information to sketch the graph of R = 200q 2q R q

4 Mathematics Learning Centre, Universit of Sdne 3 Eample Sketch the graph of k 2 if u f and are both positive numbers. Solution k 2 will look like Figure 2 since u f and are both positive. Note that q has a maimum value. k 2 will have a maimum value when k = 2( u f ) = 2. k 2 has a maimum at k = 2 u f so the maimum value for q is q = u f ( 2 ) u f ( 2 )2 = 2u f u f 4 = u f 4. k 2, when k =0,q =0. If q = 0 then u f k u f k 2 = 0 so, u f k(1 k )=0. That is when k =0ork =. So, the parabola crosses the k ais at k = 0 and k =. We now have sufficient information to sketch the graph of k 2. (u f )/4 q 0 /2 kj k

5 Mathematics Learning Centre, Universit of Sdne 4 2 Eponential Functions There is a class of functions called the eponential functions that are important in economics and statistics. For eample, a Poisson probabilit distribution function is given b: P (X) = e λ λ X. X! In this eample, e λ is an eponential function, as its variable, λ, isinthepoweror eponent. 2.1 The functions =2 and =2 The easiest function of this tpe to graph is the function =2 and we grph this function in Figure Figure 1: Graph of the function f() =2 You should be aware of several important features of this graph. The function f() =2 is alwas positive (the graph of the function never cuts the - ais), although the value of the function gets ver close to zero for values of ver large negative (ie a long wa to the left along the -ais). For eample, when = 5 wehave 2 = The function 2 increases ver rapidl for large values of. From the rules of eponents, 2 +1 =2 2. In words, the value of 2 doubles if is increased b 1. The graph of =2 intercepts the -ais at = =1. You should epect this because Figure 2 displas the graph of the function f() =2. How is the graph of =2 related to the graph of =2?

6 Mathematics Learning Centre, Universit of Sdne Figure 2: Graph of the function =2 Well, if we set = 1 then 2 =2 1 = 1, which is the value which would have been 2 obtained b setting = 1 in the function =2. In the same wa we see that if we set = 7 in the function =2 then we obtain the same value as we would b setting = 7 in the function =2. Proceeding like this we see that the graph of the function =2 is the reflection in the -ais of the graph of =2. Compare Figure 1 with Figure 2. It follows that 2 =(2 1 ) =( 1 2 ). The function =2 is the same as the function =( 1 2 ), and so 2 (+1) =( 1 2 )+1 = 1 2 (1 2 ) =( 1 2 ) 2. In words, the value of the function =2 is decreased b a factor of 1 2 b 1. if is increased 2.2 The functions = e and = e There is a number called e which has a special importance in mathematics. Like the number π, the number e is an irrational number, which is equivalent to saing that it has a non-terminating, non-repeating decimal representation. In other words we can never write down eactl what e is. To 5 decimal places it is equal to , but this is just an approimation of the correct value. Unless ou reall need to write down an approimate value for e it is more convenient and accurate to leave the smbol e in epressions involving this number. For eample, it is preferable to write 2e rather than or

7 Mathematics Learning Centre, Universit of Sdne 6 In mathematics the functions e and e are particularl important. Because of this we have graphed them in Figure 3. You can see how similar these functions are to the eponential functions, 2 and 2. The function = e is often referred to as the eponential function, and is even given another special smbol, ep, so that ep() =e and ep( ) =e. = e - 20 = e Figure 3: Graphs of e and e. Eample The densit function of the eponential distribution is given b: = 1 λ e λ where 0. Sketch a graph of this function when λ =0.5. Solution When λ =0.5, we have = e 0.5 =2e 2. Therefore, when =0, = 2. The graph of the densit function is given in Figure Figure 4: Graphs of =2e 2.