can ne1 plz explain me in brief how can we draw the graph of
h(x) = f(x) + g(x)
i.e., the sum of two functions......
e.g., y = x+sin x
y = x2+x3
y= x + 1/x
is dere any direct approach for such graphs or we need to find maxima, minima.... point where x-coordinate nd y-co-ordinate is o... plz help me
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3 Answers
62one approach that i would suggest..
if f(x) =0 then the sum will be equal to g(x) and vice versa..
check the differentiability fo the function and its continuity... and find its value whereever it is not differentiable..
also find the value of the sum (using limit!) at these points...
also find the zeros...
23frm sanchit's cb
Step 1) Check the domain of function (In set of real nos. only)
Step 2) Find the range of function
Step 3) Find the X intercept(roots) and Y intercepts
Step 4) Check whether function is even or odd.
Step 5) Find the period of function
Step 6) Check continuity and differentiability of function.
Step 7) Find the limit of function at the points where it is not continuous.
Step 8) Find asymptotes (if any)
By this time,we have a rough idea of how graph looks...but for complete graphing (i.e perfection) we have to use differential calculus now.
Step 9) Find derivative of function.Use this to get the intervals in which function increases,decreases and remains constant
Step10) Find maxima, minima, inflexion points through an appropriate derivative.
Now your graph is ready....You must be thinking that doing these long 10 steps would take a lot of time but believe me if u have done practice u wont take more than 30 seconds to do all this work.
Sometimes we are in a situation when transformation of graph is needed.....Here are some simple transformations:
1) The graph of the function y= f (x + a) is obtained from the graph of the function y = f(x) by translating the latter graph along the x-axis by IaI scale units in the direction opposite to the sign of a .
2) The graph y=f(x)+b is obtained from the graph of the function y= f (x) by translating the latter graph along the y-axis by IbI scale units in the direction opposite to the sign of b
3) The graph of the function y =f(kx) (k > 0) is obtained from the graph of the function y=f(x) by "compressing" the latter graph against the y-axis in the horizontal direction k times at k > 1 and by "stretching" it in the horizontal direction from the y-axis I/k times at k < 1.
4) The graph of the function y = kf (x) (k > 0) is obtained from the graph of function y = f (x) by "stretching" it in the horizontal direction k times at k > 1 and "compressing" it against the x-axis (i. e. vertically) I/k times at k < 1 (see Fig. 21).
5)The graph of the function y = - f(x) is symmetrical to that of the function y = f(x) about the x-axis, while the graph of the function y = f(-x) is symmetrical to that of the function y = f(x) about the y-axis.
6) The graph of the function y = f(IxI) is obtained from the graph of the function y = f(x) in the following way:
For x >= 0 the graph of the function y = f (x) is retained, then this retained part of the graph is reflected symmetrically about the y-axis, thus determining the graph of the function for x =< 0
7)The graph of the function y = If(x)I is constructed from the graph y = f (x) in the following way:
The portion of the graph of the function y = f(x) lying above the x-axis remains unchanged, its other portion located below the x-axis being transformed symmetrically about the x-axis
8)The graphs of the more complicated functions y = |f(k|x|+a) + b| are drawn from the graph of y = f(x) by applying consecutively the above transformations .
