Question

Using a chart of values, graph the function f(x)=3^x

Answer 1

Hey there!

For this problem, I would assume that you must make your own chart of values. You can do this by simply plugging in numbers like 1, 2, 3, etc... Since your graph won't be linear, you might want to do some negative numbers, as well. 

Also because your graph isn't linear, you might want to figure out or determine on your own just how far your curve will go. You can then figure out how many times you need to plug in an additional numbers for your function to then graph. 

I've attached a digital graph of this function, perhaps you can find it useful. 

Hope this helped you out! :-)

Answer 2

Let's approach this by understanding how an exponential works, to begin.

Any exponential in the form: $f(x) = 3^{x}$ will cut the y-axis once and once only. This is because the exponential graph is a monotonic increasing graph for all real values of x.

We can prove this by taking the derivative. When we take the derivative of a function, we are finding the slope or gradient of the function at an arbitrary point on the graph.

$\text{Differentiating the function:} f'(x) = \frac{d(3^{x})}{dx}$
$f(x) = e^{ln(3^{x})}$
$f(x) = e^{xln3}$
$f'(x) = ln(3) \cdot e^{xln3}$

Thus, we can conclude for any value of x, the gradient will always be increasing. Now, by plotting some critical values, we can find that the graph hits the y-axis once and once only at the value y = 1.

At this point, we've sort of have an idea as to what the graph will do, but the most critical points are the endpoints, when x approaches infinity and negative infinity. In order to understand what they do at those points, instead of plotting those points, let's use a limit.

$\lim_{x \to \pm \infty}(3^{x})$

Limits are a crucial tool in understanding what a graph does as the terms get arbitrarily huge and infinitesimally small. We can see that as x approaches infinity, we approach some really big number faster than a  quadratic, linear line, and any function other than factorials.

However, let's approach the negative side. If we were to plug in negative infinity into the function, we get:
$f(-\infty) = 3^{-\infty}$
$f(-\infty) = \frac{1}{3^{\infty}}$

Now, we know that for a really big number in the denominator, the function will shrink smaller and smaller to a value, 0. Because the numerator is positive, we can see that the graph will NEVER hit below 0. So, we can conclude that the graph has a limit at x ≠ 0.

Since we know the endpoints, the gradient at any value, and the y-intercept, we can graph the function. Refer to the other answer for the correct graph of 3ˣ.