Question

Could you teach me how to solve this problem without using a graphing calculator? (My algebra teacher doesn’t allow graphing calculators on the test.)

Answer 1

The type of function we are going to work with here is an exponential function. These functions are given by equations like:

`f(x)=a\cdot b^x+c`

Where x is the variable and a, b and c fixed parameters. In this case c=0, a=1 and b=13/4. This number b, i.e. the one affected by the exponent is called the base so I'm going to refer to it with that word.

All exponential functions have similar graphs, they pass from being very close, almost pararel to the x-axis and then they begin to rapidly increase toward infinite y values. In the following picture you can see the graph of two exponential functions:

The red graph tends to zero for positive x values, this means that its equation has a negative sign in the exponent:

`\text{red(x)}=b^(-x)`

The blue graph on the other hand tends to zero for negative x values which means that there's no negative sign in the exponent. This is the case of our function r(x) so we can discard the upper left option.

Another thing to take into account is the offset of the function, that number that I labeled as "c" in the first equation. This offset translates the graph upwards or downwards depending on its sign.

For example, in this image the graphs decrease for negative x values which means their exponents don't have a negative sign but they all tend to different values which means their offsets are different. Blue's offset is 0 since it tends to 0 while gray's offset is 1 because it tends to 1. The same way you can deduce that the orange graph has an offset of -2. In the case of our problem, function r(x) doesn't have an offset which means that in one direction it has to tend to 0. This means we can discard the lower rigth option.

We have already discarded two options, if we discard one more we have the answer.

Now let's take r(x) and evaluate it in x=0:

`r(0)=((13)/(4))^0=1`

You don't need a calculator for this because any number raised to 0 lays 1. This means that at x=0 ,i.e. the y-intercept of the graph, the function has y=1.

Lower left graph intercepts the y-axis in 1 while the upper right does it closer to 0 so we can discard this last option.

In summary, we have only one possible option left, the one in the lower left. This is the correct option.