# How do you graph g(x) = -(3/2)^x?

Oct 12, 2015

Actual graph plot shown. Explanation following it!

#### Explanation:

It is a good thing if you are able to interpret the expression/equation in such a way that you build up a mental picture of what the numbers are actually doing. The structure of an equation is modelling behaviour in some way. First the graph:

Clarifying a point: $g \left(x\right)$ is a short hand way of writing that some form of mathematical operation is applied to the variable of $x$.
IN this case the process is given the name of "g".
Hence $g \text{ of } x$.

To be able to graph this we have to include the consequence of that operation which is a value. This is the dependant variable and when we "equate" this to $g \left(x\right)$ we then have a "consequence" and a "cause" that we can plot on a graph. This in effect is $y = g \left(x\right) = - {\left(\frac{3}{2}\right)}^{x}$.

Look at the overall structure of the equation. It is $- \text{something}$. That is, it is always negative. So y is always negative so the plot never occurs above y=0.

The absolute value of the number that the operation is carried out on is a is "bigger" than 1 ( absolute value means that it has been converted to positive). So if multiplied by itself repeatedly it gets bigger and bigger.

But this number will always be negative so its value as $x$ increases becomes more and more less than 0.

Note that less than 0 does not necessarily mean smaller.

By the way; when $0 < x < 1$ then $0 > y < \text{absolute} \left(- \frac{3}{2}\right)$
or another way of saying it: $0 > y > \left(- \frac{3}{2}\right)$

when $x = 0$ then ${\left(\frac{3}{2}\right)}^{0} = 1$ but we have $- 1 \times {\left(\frac{3}{2}\right)}^{0} = - 1$

When $x < 0$ then we have $- \frac{1}{\frac{3}{2}} ^ x$
in which case as ${\left(\frac{3}{2}\right)}^{x}$ gets bigger and bigger, whilst still negative. Remember that -25 is bigger than -1. It is just negatively bigger.