My English lesson

My English lesson Graph symmetries and traslations DEFINITION A y = f(x) function is symmetric to the y-axis if f( x) = f(x) for each x of its domain. example O Verify that the function y = x2 1 is symmetric with respect to the y-axis. Said function has the R set as its domain. It is therefore possible to consider each element x R to algebraically verify that it satisfies the requested condition. Hence f( x) = ( x)2 1 = x2 1 = f(x) O Verify that the function y = x2 + x is not symmetric to the y-axis. In this case too, we observe that the function has R as its domain; we can thus consider each element x R and determine f( x): f( x) = ( x)2 + ( x) = x2 x f(x) We can achieve the same result by observing the corresponding graph for each function. y y 2 1 1 O 1 1 x O 1 1 x DEFINITION The graph of a y = f(x) function is symmetric to the origin O if f( x) = f(x) for each x of its domain. example O Verify that the function y = x3 + x is symmetric to the origin O. As in the previous examples, we observe that the function has R as its domain and therefore we can consider each element x R and determine f( x): 3 f( x) = ( x)3 + ( x) = x x = (x3 + x) = f(x) y 1 1 NOW IT S YOUR TURN N D Determine if the graphs of the following functions are symmetric to the y-axis, or the origin, or if they do not have any symmetries: a. y = x4 + 3x2 2 x b. y = _ cosx c. y = 3x3 5 52 O 1 x 1 We can verify the symmetry to the origin also by drawing a graph. A function, which is symmetric to the y-axis, is sometimes said to be an even function while a function which is symmetric to the origin is said to be an odd function.

Il Maraschini-Palma - volume 5
Il Maraschini-Palma - volume 5