Basic Maths Section [E] Function and Graph

SECTION [E] : FUNCTION AND GRAPH FUNCTION Physics involves study of natural phenomena and describes them in terms of several physical quantities. A mathematical formulation of interdependence of these physical quantities is necessary for a concise and precise description of the phenomena. These mathematical formulae are expressed in form of equations and known as function. Thus, a function describing a physical process expresses an unknown physical quantity in terms of one or more known physical quantities. We call the unknown physical quantity as dependent variable and the known physical quantities as independent variables. For the sake of simplicity, we consider a function that involves a dependant variable y and only one independent variable x. It is denoted y=f(x) and is read as y equals to f of x. Here f(x) is the value of y for a given x. Following are some examples of functions. y=2x+1, y=2x2+3x+1, y=sinx, y=n (2x+1) Knowledge of the dependant variable for different values of the independent variable, and how it changes when the independent variable varies in an interval is collectively known as behavior of the function. Ex. In the given figure, each box represents a function machine. A function machine illustrates what it does with the input. Input (x) Double the input and add three Square root Output (z) of the input Which of the following statements are correct? (A) y=2x+3 Sol. (B) y=2(x+3) (C) z  2 x  (D) z  2  x  (C) GRAPH OF A FUNCTION Graph is diagrammatic representation of a function and allows us to visualize it. To plot a graph the dependant variable (here y) is usually taken on the ordinate and the independent variable (here x) on the abscissa. Graph being an alternative way to represent a function does not require elaborate calculations and explicitly shows behavior of the function in a concerned interval. Ex. Consider a body moving with constant speed of 2 m/s in a straight line. When you start your stopwatch, you observe the body 1 m away from a fixed point on the line. Suggest suitable physical quantities, write a function and draw its graph describing motion of the body. Sol. Distance x of the body from the given fixed point and time t measured by the stopwatch are the suitable variables. If we consider the fixed point as the origin, distance x is known as the position coordinate of the body. In the following figure it is shown that the body is on point A at the instant t = 0 and after a time t it reaches another point B covering a distance, which equals to product of speed and time interval. Thus, distance s covered by the body in time t is given by the following equation. s = 2t O x=0 s=2t A x0=1 t=0 B x=1+2s t Position coordinate x(m) BASIC MATHS & VECTOR Time(s) With the help of the above figure, position coordinate x of the body at any time t is given by the following equation, which is the required function describing motion of the body. x=2t+1 Graph of this equation is also shown in the adjoining figure. GRAPHS OF SOME COMMONLY USED FUNCTIONS Linear, parabolic, trigonometric and exponential functions are the most common in use. (i) Straight line Equation and its Graph When the dependant variable y varies linearly with the independent variable x, the relationship between them is represented by a linear equation of the type given below. The equation is also shown in graph by an arbitrary line. y= mx + c Here m & c are known as slope of the line and intercept on the y-axis, respectively. Slope Slope of a line is a quantitative measure to express the inclination of the line. It is expressed by ratio of change in ordinate to change in abscissa. Slope of a line m y2  y1 y  = slope of tangent x2  x1 x When the x and the y axes are scaled identically, slope equals to tangent of the angle, which line makes with the positive x-axis. m=tan Sometimes the slope is also called gradient and expressed by the term “y in ’” where  is the length along the line l   x  2   y  . 2 It is positive if y increases with increase in x, negative if y decreases with increase in x, zero if y remains unchanged with change in x and infinite if y changes but x remains unchanged. For these cases the line is inclined up, inclined down, parallel to x-axis and parallel to y-axis respectively as shown in the adjoining figure by lines A, B, C and D respectively. y B D A C x -x -y INTERCEPT It equals to the value of ordinate y, where the line cuts the y-axis. It may be positive, negative or zero for lines crossing the positive y-axis, negative y-axis and passing through the origin respectively. Ex. Write equations for the straight lines shown in the following graphs. y y (A) x x (B) y (C) y x (D) x BASIC MATHS & VECTOR y y (E) (F) x Sol. (A) y  32 x  3 ; Ex. A parallelogram ABCD is shown in figure. (B) y  32 x  2 ; x (C) y=5; (D) y=–x+5 y B (0,1) (2,0) x C (-2,0) A D (0,-1) Column I Sol. Column II (A) Equation of side AB (P) 2y + x =2 (B) Equation of side BC (Q) 2y – x = 2 (C) Equation of side CD (R) 2y + x = –2 (D) Equation of side DA (S) 2y –x =–2 (T) y + 2x =2 (A) Q (B) P (C) S (D) R 1 x 1 2 For side AB : m  1 0 1  , c  1    0  2 2 For side BC : m  20  2, c  1  y  2 x  1 0 1 For side CD : m  1 0   1 1  , c  1  y  x  1 2 20 2 For side DA : m  1 1  0 1   , c  1  y   x  1 2 0   2  2 y (E) y  23 x ; (F) x=4 Ex. A variable y increases from y1 = 2 to y2 = 8 linearly with another variable x in the interval x1 = 0 to x2 = 10. Express y as function of x and draw its graph. Sol. Linear variation is represented by a linear equation of the form y=mx+c. To represent the function on graph we have to join two points whose coordinates are (x1, y1) and (x2, y2) i.e. (0, 2) and (10, 8). Slope of the line is m  y2  y1 8  2 3   x2  x1 10  0 5 y x From the graph, intercept is c=2. Now the required equation is y  Ex. 3 x2 5 The graph shows a linear relation between variable y and x. Consider two quantities p and q defined by the equations . y y p x q yb x b 0 0 a x As x changes from zero to a, which of the following statements are correct according to the graph? Sol. (A) Quantity p increases and q decrease. (B) Quantity p decrease and q increases. (C) Quantity p decreases and q remain constant. (D) Quantity p increases and q remain constant. (C) q is slope of the given line, which is a constant for a straight line. p is slope of the line join origin and point on the line, which is decrease as x increases. Ex. Frequency f of a simple pendulum depends on its length  and acceleration g due to gravity according to the following equation f Ans. 1 g 2 l Graph between which of the following quantities is a straight line? (A) f on the ordinate and l on the abscissa (B) f on the ordinate and  on the abscissa 2 (C) f on the ordinate and l on the abscissa (D) f2 on the ordinate and 1/ on the abscissa (D) BASIC MATHS & VECTOR (ii) Parabola equation and its graph y A function of the form y=ax2+bx+c is known as parabola. The simplest parabola has the form y=ax2. Its graph is shown in the following figure. -x GRAPH OF SOME TRIGONOMETRIC FUNCTIONS Among all the trigonometric functions, sinusoidal function, which includes sine and cosine both is most common in use. Sine Function y = a sin x x y a x -a /2 Here, a is known as the amplitude and equals to the maximum magnitude of y. In the adjoining figure graph of a sine function is shown, which has amplitude a units. Cosine Function y = a cos x y a x -a /2 Here, a is known as the amplitude and equals to the maximum magnitude of y. In the adjoining figure graph of a cosine function is shown, which has amplitude a units. (iii) Exponential function and its graph y Behavior of several physical phenomena is described by exponential a function to the base e. Here e is known as Euler’s Number. e = 2.718218 y=ae—x Most commonly used exponential function has the form y=ae–x. In the adjoining figure graph of this function is shown. x Ex. In the given figure is shown a variable y varying exponentially on another variable x. Study the graph carefully. y Which of the following equations best suits the shown graph? –x –x 3 (A) y = 3 – e (B) y = 1 – 4e (C) y = 1 – 3e–x (D) y = 3 – 4e–x Sol. (D) Shift the curve (–4e–x) in positive y-direction by 3 units. (iv) Circle and Ellipse 2 -1 x 2 y2 Ellipse : 2  2  1 a b Circle : x + y = a 2 x 2 y y b a (0,0) a x (0,0) a x