Quantitative Techniques (Finance) 03 Polynomial Functions Felix Chan October 006 Introduction This topic discusses the properties and the applications of polynomial functions, specifically, linear and quadratic functions. These functions form the basic building blocks of calculus, and therefore it is crucial to obtain a thorough understanding of their properties. Polynomial A k th -order polynomial function is a function f : R Y such that f(x) = a 0 + a x + a x +... + a k x k, () or alternatively k f(x) = a i x i. () where the domain, Y, is a subset of R depending on k. The following subsections will discuss the cases for k = (linear function) and k = (quadratic function) in details. i=0
3 y= x+ 5-4 - 0 3 4 5 - y= x Figure : Linear Functions with Different Intercepts. Linear Function Definition. Let f : R R, then f is a linear function if f(x) = a 0 + a x with a 0 R and a R. The constant a and a 0 are often referred to as the slope (gradient or rate of change) and the intercept of the linear function, respectively. Example. Let y = f(x) = x + 4 then f(x) is a linear function with a = and a 0 = 4. Graphically, a linear function represents a straight line on the Cartesian plane, where the location and steepness of the straight line are controlled by the values of a 0 and a. To understand this, consider two linear functions, y = x + and y = x. Notice that in both cases, the slope, a is set to. Notice in figure, both linear function has the same slope but they have a different intercepts. The term intercept is referring to the value of y when x = 0, that is, the value of y when the line cuts through the y-axis. Obviously, when the line cuts through the y-axis, the corresponding x-value would be 0.
y = a 0 + a x x = 0 y = a 0 Therefore, the constant, a 0, is the intercept. Now that we know a 0 represents the value of y when x = 0, the next natural question would be, what is the value of x when y = 0? That is, where is the point on the x-axis the linear function goes through? The question can be solved by considering the following arguments: y = a 0 + a x Set y = 0 0 = a 0 + a x x = a 0 a Therefore, the linear function cuts through the x-axis at x = a 0 /a. The value of x when y = 0 is often referred as the x intercept. Example. Consider the linear function y = x + 4. In this case, a = and a 0 = 4 then the x intercept is x = (4)/ =. The slope, a, represents the sensitivity of y with respect to the changes in x. More formally, a represents the changes in the value of y with respect to a unit change in x. To see this, consider the following arguments: y = a 0 + a x y + δ = a 0 + a (x + ) 3
where x + represents a unit increase in x, where δ represents the impact on the value of y with respect to the unit change in x. So multiply a through the bracket on the right hand side gives y + δ = a 0 + a x + a y + δ = a 0 + a x + a }{{} y y + δ = y + a δ = a. Therefore, the value of y will change by a amount when x increases by unit. Example 3. Consider the linear equation y = 3x 5. In this case, a = 3 and a 0 = 5. Therefore, the linear function will cut through the y-axis at y = 5 and it will cut through the x-axis at x = 5/3. Moreover, y will increase by 3 units when there is a unit increase in x. Example 4. Consider the linear equation y = 3x +. In this case, a = 3 and a 0 =. Therefore, the linear function will cut through the y-axis at y = and it will cut through the x-axis at x = /3. Moreover, a unit increase in x will lead to a 3 units decrease in y. Notice the sign of a determines the type of relationship between x and y. Specifically, if a > 0 then x and y are said to be positively or directly related. That is, the values of x and y will change in the same directions. If a < 0 then x and y are said to be negatively or inversely related. That is, the values of x and y will change in the opposite direction. This concept can be easily understood by examine the plots of the two linear functions in the previous examples as shown in Figure. As shown in Figure, for y = 3x 5, the values of y increases as x increases where for y = 3x + the values of y decreases as x increases. In the case of linear function, the two typical questions are:. Given a linear function, how to locate the straight line on the Cartesian plane? 4
y= 3x+ y=3x 5-0 3 4 5 6 - -4 Figure : Two Linear Functions with different sign in a x-intercept - 0 3 4 5 6 - y= x 4 y-intercept -4-5 Figure 3: Drawing the Linear Function y = x 4. Given two points on the Cartesian plane, how to derive the linear function that would pass through the two points? The solution to the first question is straightforward, as all we need to do is to join up two points that belong to the straight line. The easiest way to do this is to work out the x-intercept and the y-intercept, as demonstrated in the following example: Example 5. Consider y = x 4, for x = 0, y = 4 so the y-intercept is 4 and for y = 0, x = 4, so the x-intercept is 4, so the function represents a straight line that passes through (0, 4) and (4, 0) on the Cartesian plane. For the second problem, we need to solve a 0 and a given two points on the straight line. Recall 5
that the slope in the linear function, a, represents how much y changes per unit change in x, so given two points (x, y ) and (x, y ), the changes in y per unit change in x is simply a = y y x x. (3) So the slope, a can be easily obtained by using the formula above. In order to solve for a 0, we know that the function must pass through both (x, y ) and (x, y ), so the function must satisfy y = a 0 + a x and therefore a 0 = y a x. (4) Notice the calculation of a 0 involves only (x, y ) in the derivation above. However the straight line must pass through both points, so it is also possible to calculate a 0 using (x, y ). In fact, using the similar arguments as above, it is possible to show that a 0 = y a x = y a x. (5) Example 6. Derive the linear function that passes through (, ) and (4, 9). In this case, (x, y ) = (, ) and (x, y ) = (4, 9). So the slope can be calculated as a = y y x x = 9 4 = 8 = 4 6
3 y= x 5-4 - 0 3 4 5 y= x -. Figure 4: Plots of y = x and y = x Moreover, the y-intercept can be calculated as a 0 = y a x = 4() = 7. Therefore, the equation of the straight line that passes through (, ) and (4, 9) is y = 7 + 4x.. Quadratic Function Definition. A polynomial of order, that is k =, is also known as the quadratic equation. It is a function, f : R Y, such that f(x) = a 0 + a x + a x. (6) where Y R depending on a 0, a and a. Figure 4 contains the plot of y = x and y = x. Notice the plots of these functions are mirror reflection to each other through the x-axis. 7
4 No Intercept 3 One Intercept -5-4 - 0 3 4 - Two Intercepts Figure 5: Three Scenarios for the Quadratic Equations In the case of quadratic functions, a 0 represents the y-intercept where the x-intercept can be found by solving the following equation 0 = a 0 + a x + a x (7) which can be solved by using the following quadratic equation: x = a ± a 4a a 0 a. (8) Notice the number of x-intercept depends on d = a 4a a 0, namely, if d > 0 then the quadratic equation has x-intercepts, if d = 0 then the quadratic equation has x-intercept and if d < 0 then the quadratic equation has no x-intercept. The graphical illustration of these scenarios can be demonstrated in Figure 5. Example 7. Consider y = x + x. In this case, a =, a = and a 0 =. Obviously, when x = 0, y =, so the y-intercept is and for the x-intercepts, the solution to x+x = 0 8
is x = a ± a 4a a 0 a = ( ) ± ( ) 4()() () =, and therefore the x-intercept is. Notice in the example above, d = a 4a a 0 = 0 and hence there is only one x-intercept. Example 8. Consider y = x x 3. In this case, a =, a = and a 0 = 3. Again, when x = 0, y = 3 so the y-intercept is. For the x-intercepts, the solutions to x x 3 = 0 are x = a ± a 4a a 0 a = ( ) ± ( ) 4()( 3) () = 3 or In the example above, there are two x-intercepts, namely y = 0 when x = 3 or x =. Notice in this case, d = a 4a a 0 = 4 > 0 and hence there are two intercepts. Example 9. Consider y = x +. In this case, a =, a = 0 and a 0 =. When x = 0, y = so the y-intercept is. For the x-intercepts, the solution to x + = 0 is x = = 4()() () 4 Notice in order to solve the equation above, we must find the square root of 4 which does not belong to R. In such case, there is no x-intercept. Another important properties of the quadratic function is the axis of symmetry. Note that 9
- 0 3 4 5 6 7 8 - x = a a 4a 0 a a x = + a 4a 0a a a x = a a Figure 6: An Illustration on the Quadratic Equation and the Axis of Symmetry every quadratic function has either a maximum point or a minimum point. In both cases, the x value of the optimal point is in the middle of the two x-intercepts. Moreover, the shape of the quadratic function is symmetric around the optimal point and thus the axis of symmetry can be calculated as x = a a. (9) The quadratic equation and the axis of symmetry formula can be best demonstrated in Figure 6..3 Cubic Function Definition 3. A 3 rd order polynomial is also known as a cubic function, which is a function f : R Y such that f(x) = a 0 + a x + a x + a 3 x 3, (0) where Y R and a i R i = 0,..., 3. The exact range depends on the values of the coefficients. Figure 7 contains the plot of a standard cubic function, y = x 3. Similar to the other two cases a 0 represents the y-intercept and the number of x-intercepts ranged from to 3 in the case of cubic function, as demonstrated in Figure 8. Although explicit formula to calculate the x-intercepts for a cubic function is available, it would 0
3 y= x 3 5-4 - 0 3 4 5 - Figure 7: Plot of the standard Cubic Function, y = x 3 not be necessary for us to know them in this course and hence we will omit the introduction. In the tutorial, you will explore the different representation of quadratic and cubic functions.
3 intercept 5-4 - 0 3 4 5-5.5 0-7.5-5.5 0.5 5 7.5 0.5 intercepts -5 5.5 0-7.5-5.5 0.5 5 7.5 0.5 3 intercepts -5 Figure 8: Cubic Functions with Different Number of Intercepts