19) what is the effect on the graph of the parent function f(x) = x when f(x) is replaced with f(x) + 3? a) shifts the line 3 units up b) shifts the line 3 units down c) increases the slope of the line by a factor of 3 d) decreases the slope of the line by a factor of 3

Describe the possible echelon forms of the standard matrix for a linear transformation t where t: set of real numbers is onto.

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .

What is the least common multiple for 6 and 8? what is the least common multiple for 4 and 12 ? what is the least common multiple for 11 and 12? what is the least common multiple for 3 and 6?