The Natural Base e

In order to match the function with its graph, we will first determine the type of the given function, then plot the graph. Lastly, we will compare the outcome with graphs given in the exercise.

Type of Function

To determine whether the given function represents exponential growth or exponential decay, let's recall two properties of the natural base exponential function, $y = a e^{r x} .$

$y = a e^{r x}$
$a > 0$ and $r > 0$	$a > 0$ and $r < 0$
Exponential growth function	Exponential decay function

Let's now consider the given function. $y = 0.75 e^{x} \Leftrightarrow y = 0.75 e^{1 x}$ Since both $a = 0.75$ and $r = 1$ are greater than zero, the function is an exponential growth function.

Graph

Next, let's make a table of values to graph the function.

$x$	$0.75 e^{x}$	$= 0.75 e^{x}$
$- 1$	$0.75 e^{- 1}$	$\approx 0.28$
$0$	$0.75 e^{0}$	$0.75$
$1$	$0.75 e^{1}$	$\approx 2.04$

Finally, we will plot the points and connect them with a smooth curve.

Conclusion

Since $y = 0.75 e^{x}$ is an exponential growth function and the points $(0, 0.75)$ and $(1, 2.04)$ belong to its graph, we can conclude that it is represented by the graph in choice C.