Question

Write an exponential equation?

Answer 1

Answer:

$y = 3^{-x-2}$

Step-by-step explanation:

Since the graph is exponential, the default form of formula is going to be:

y = aˣ;

First question is, what information do we have?

I can see 3 distinct points: (2, 1), (1, 3) and (0, 9);

Normal exponential curve have a positive gradient at every point, increasing as you proceed along the x-axis to ∞;

The opposite is true of the graph in the image which tells us it is reflected in the y-axis as compared to a default exponential graph;

Any exponential function will have the point (0, 1);

Considering we have (2, 1), there seems to be a translation of 2 units rightwards;

Then considering the pattern of the y-values of the points identified above, which is 1, 3 and 9;

3⁰ = 1, 3¹ = 3 and 3² = 9;

This suggests a = 3;

We then combine this like so:

So, if we start with f(x) = 3ˣ

Then we need to apply the transformations identified above in the order of stretching, then reflections and finally translations;

Applying the reflection first;

$g(x) = f(-x) = 3^{-x}$

Then, the translation of 2 units rightwards like so;

$g(x-2) = 3^{-(x-2)}$

There is no apparent stretching, there could be, but, we don't know so, best thing to do is to test what we have thus far and check if it matches the graph in the image;

So, we end up with:

$y = 3^{-x-2}$

Now, to test this;

Let x = 0, then:

$y = 3^{-(0)+2} \\\\ y = 3^{2} \\\\ y = 9$

This matches (0, 9);

Similarly, when x = 1, then y = 3;

And x = 2, then y = 1;

This matches the graph, so this seems to be the equation of the graph.