Functions and Their Graphs

Practice 1:

Practice 2: $g(t) = t^2 – 5t$.
$g(1) = 1^2 – 5(1) = –4$.  $g(–2) = (–2)^2 – 5(–2) = 14$.
$g(w + 3) = (w + 3)^2 –5(w + 3)  =  w^2 + 6w + 9 – 5w – 15  =  w^2 + w – 6$.
$g(x + h) = (x + h)  – 5(x + h)  =  x^2 + 2xh + h^2  – 5x – 5h$.
$g(x + h) – g(x)  = ( x^2 + 2xh + h^2  – 5x – 5h ) – ( x^2 – 5x )  =  2xh + h^2 – 5h$.
 
$\dfrac{g(x + h) – g(x)}{h}    =  \dfrac{2xh + h^2 – 5h}{h}     =   2x + h – 5$.
 
Practice 3: (a)  Friend (b)  Friend  (c) At  $t = 40$.  Before that your friend is walking faster and increasing the distance between    you.  Then you start to walk faster than your friend and start to catch up.  (d) Friend.  You are walking faster than your friend at  $t = 50$, but you still have    not caught up.
 
Practice 4: (a) The x–coordinate is increasing. (b)  The  x–increment  $∆x$  is  decreasing.  (c) The slope of the line through  $P$  and  $Q$  is  decreasing.
 
Practice 5: See Fig. 31.