# Question 092e0

Jul 26, 2017

Start with a simpler function and adjust the scales as indicated below.

#### Explanation:

Start with the simpler function $g \left(v\right) = \sqrt{v}$
Hopefully you will know the graph for this function, but if not the following table should give you some points that you can use to sketch it:
color(white)("XXX"){:(ul(v),color(white)("xx"),ul(g(v))),(0,,0),(1,,1),(2,,4),(3,,9),(4,,16):}
The graph should look like:

If we replace $v$ with $\left(x + 1\right)$ then the only effect this has on our graph is to shift the $g$-axis.
color(white)("XXX"){:(ul(x),color(white)("xx"),ul(v),color(white)("xx"),ul(g(v)),color(white)("xx"),ul(g(x+1))),(-1,,0,,0,,0),(0,,1,,1,,1),(1,,2,,4,,4),(2,,3,,9,,9),(3,,4,,16,,16):}
which would give a modified graphic form:

(Note that the actual curve has not changed.)

Now if we modify the vertical axis, replacing $g \left(x + 1\right)$ with $f \left(x\right)$ by changing the scale so that each value on the vertical axis is replaced by $3$ times its original value (i.e. $f \left(x\right) = 3 \times g \left(x + 1\right) = 3 \sqrt{x = 1}$)
color(white)("XXX"){:(ul(x),color(white)("xx"),ul(v),color(white)("xx"),ul(g(v)),color(white)("xx"),ul(g(x+1)),color(white)("xx"),ul(f(x)=3g(x))),(-1,,0,,0,,0,,0),(0,,1,,1,,1,,3),(1,,2,,4,,4,,12),(2,,3,,9,,9,,27),(3,,4,,16,,16,,48):}#
and our final graph looks like: