How can I decipher which graph belongs to which equation?

Question

Aside from plotting points, how else can I tell which graph is which?

Answer 1

We know that $\ln (2) < \ln (7)$, if $x<0$, then

$$x\ln (2) > x \ln (7)$$

$$2^x>7^x$$

Similar for the second case, work with $\ln (4)$ and $\ln (3)$ to compare the graph.

Answer 2

Consider the 1st graph.

For x<0, 2^x > 7^x

So now you need to determine which curve is having higher values for the same values of x

The curve taking higher values for same values of x will always be above the curve taking lower values for same values of x.

So, the red curve is 2^x and the blue curve is 7^x

Consider the 2nd graph

i) x > 0

3^-x > 4^-x

So, 3 ^-x will be higher in the region x<0

ii) x < 0

4^-x > 3^-x

So, 4^-x will be higher in the region x<0

By observing the graph it can be concluded that the red curve is 4^-x and the blue curve is 3^-x