Cubic Function Graph

4 Key excerpts on "Cubic Function Graph"

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  eBook - ePub

  Mathematics for Business Analysis

  • Paul Turner, Justine Wood(Authors)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  x . The general form of such functions can be written as

  (2.14)

  where the input variable x is a real number. A function of the form (2.14) is referred to as an nth order polynomial function because n is the highest power of x included.

  We have already seen that linear functions produce a straight-line relationship in the Cartesian plane. If we introduce higher-order powers into the relationship, then the shape of the output function will change. For example, a quadratic function takes the form . When drawn in the Cartesian plane, this produces a curved relationship, the slope of which will change around some critical point. Consider, for example, the case shown in Figure 2.13 , where we have , , and . This produces the curve shown in the diagram, in which the slope is negative for values of x less than −1/2, and positive for values of x greater than −1/2. We also see that it cuts the x axis at two points, where and . The introduction of cubic terms into a polynomial function will produce even more general shapes. For example, if we graph the function , as shown in Figure 2.14 . We can observe that it has two turning points and cuts the x -axis in three places.

  FIGURE 2.13  A quadratic function in the Cartesian plane.

  FIGURE 2.14  A cubic function in the Cartesian plane.

  Turning points are defined as points at which the slope of the function changes sign, and the roots , or zeros , of the function are defined as points at which . If the roots are real, then the condition means that the function cuts the x -axis at such points. As the order of the function increases, the potential number of turning points and real roots increase. However, this is not necessarily the case. For example, the fourth-order polynomial function , has only a single turning point and does not cross the x -axis for any real value of x . The order of the polynomial puts an upper limit on both these features. For example, we can say that a cubic function has at most two turning points, and that it cuts the x -axis at most three times. In general, the maximum number of turning points is one less than the order of the polynomial (n -1), and the maximum number of real roots is equal to the order (n

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  eBook - ePub

  Introductory Regression Analysis

  with Computer Application for Business and Economics

  • Allen Webster(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  A polynomial is an expression of finite length constructed from variables (also known as indeterminants) and constants using only the operations of addition, subtraction, and multiplication. Exponents must be positive whole integers.

  Now do you know why Figures 9.1E and 9.1F are not polynomials or quadratics? Because polynomials must have positive integers for exponents. Figure 9.1E can be written as Y = 500 − 10X −2 . The exponent is not positive. The same is true for Figure 9.1F .

  Polynomial relationships are extremely common in many business and economic relationships. The quadratic is a useful way to explore parabolic ∪-shaped average cost curves, while the total revenue curves and total product curves are graphed as ∩-shaped parabolas. The cubic function is illustrative of total cost curves and the logit models we studied in Chapter 6 .

  However, it is suggested that polynomials of an order greater than two be avoided if possible. The introduction of another RHS variable such as X 3 leads to a loss of a degree of freedom.

  More serious is the fact that the use of an additional explanatory variable increases the likelihood of infecting the model with multicollinearity. Nevertheless, while in some cases it may be necessary to use cubics, it should be done with caution. The next section examines the different polynomial models.

  Figure 9.2A is a polynomial but not a quadratic. Figure 9.2B is neither. Figure 9.2C is neither. Figure 9.2D is both because it can also be written as Y = 17X − 0.25X 2 . Can you answer why in all four cases?

  A linear function will have a constant slope throughout its entire length. This means that the change in Y given a one-unit change in X will be the same at all values of X . The slope of a non-linear function will vary over its length. The change in Y given a one-unit change in X will differ for different values of X

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  eBook - ePub

  Higher Engineering Mathematics

  • John Bird(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  When a mathematical equation is known, co-ordinates may be calculated for a limited range of values, and the equation may be represented pictorially as a graph, within this range of calculated values. Sometimes it is useful to show all the characteristic features of an equation, and in this case a sketch depicting the equation can be drawn, in which all the important features are shown, but the accurate plotting of points is less important. This technique is called ‘curve sketching’ and can involve the use of differential calculus, with, for example, calculations involving turning points.

  If, say, y depends on, say, x , then y is said to be a function of x and the relationship is expressed as y = f (x ); x is called the independent variable and y is the dependent variable.

  In engineering and science, corresponding values are obtained as a result of tests or experiments. Here is abrief resume of standard curves, some of which have been met earlier in this text. (i) Straight line

  The general equation of a straight line is y = mx + c , where m is the gradient

  (

  i .e .  

  d y

  d x

  )

  and c is the y -axis intercept.

  Two examples are shown in Fig. 20.1

  Figure 20.1

  (ii) Quadratic graphs

  The general equation of a quadratic graph is y = ax 2 + bx + c , and its shape is that of a parabola. The simplest example of a quadratic graph, y = x 2 , is shown in Fig. 20.2 .

  Figure 20.2

  (iii) Cubic equations

  The general equation of a cubic graph is y = ax 3 + bx 2 + cx + d .

  The simplest example of a cubic graph, y = x 3 , is shown in Fig. 20.3 .

  Figure 20.3

  (iv) Trigonometric functions (see Chapter 15 , page 162)

  Graphs of y = sin θ , y = cos θ and y = tan θ are shown in Fig. 20.4 .

  Figure 20.4

  (v) Circle (see Chapter 14 , page 155)

  The simplest equation of a circle is x 2 + y 2 = r 2 , with centre at the origin and radius r , as shown in Fig. 20.5 .

  Figure 20.5

  More generally, the equation of a circle, centre (a, b ), radius r , is given by:

  (

  x − a

  )

  2

  +

  (

  y − b

  )

  2

  =

  r 2

  Figure 20.6 shows a circle

  (

  x − 2

  )

  2

  +

  (

  y − 3

  )

  2

  = 4

  Figure 20.6

  (vi) Ellipse The equation of an ellipse is

  x 2

  a 2

  +

  y 2

  b 2

  = 1

  and the general shape is as shown in Fig. 20.7

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  Mathematics for Scientific and Technical Students

  • H. Davies, H.G. Davies, G.A. Hicks(Authors)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  y .

  2. The stress S (MN /m 2 ) in a plate varies with the distance x (cm) from the centre and is given by S = – x 2 + 9x – 14. Draw the graph for the range x = 1 to x = 8 cm. Find (a) the greatest positive stress in the plate and the position at which it occurs, (b) the position(s) at which there is no stress in the plate.

  9.11 Cubic graphs

  Cubic graphs have equations which involve variables with the power of 3 such as

  y = ax 3 + bx 2 + cx + d  where a , b , c , d are constants.

  For a cubic equation b , c , or d may be equal to zero.

  Fig. 9.27

  The simplest cubic equations are y = x 3 and y = –x 3 . The graphs of these functions are shown in Fig. 9.27 .

  Graphs of the more general equations are shown in Fig 9.28 . These graphs have local maximum and minimum points or turning points. The value of the coefficient a determines the quadrant from which the graph commences on the left-hand side.

  Fig. 9.28a

  Fig. 9.28b

  Fig. 9.28c

  The cubic equation ax 3 + bx 2 + cx + d = 0 has

  1. 3 real roots at A , B , C in Fig. 9.28(a) , that is, where the graph cuts the x -axis, where y = 0;

  2. 1 real root only at A in Fig. 9.28(b) ;

  3. 1 real root at A and two equal real roots at B in Fig. 9.28(c) .

  9.12 Exponential graphs

  An equation such as y = 4x , where the variable x is an index, is called an exponential equation. 4x is called a function of x ; 4 is called the base number. In Chapter 1

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