The graph in the image shows points plotted on a curve, and the task asks for the equation of the function represented by the graph. Based on the given points $(0,12)$, $(1,6)$, and $(2,3)$, it seems that the curve follows an exponential decay pattern.

To determine the equation, let's assume the function has the form: $y = ab^x$ where $a$ and $b$ are constants to be determined, and $x$ is the input variable.

We can use the coordinates of the points to form equations and solve for $a$ and $b$.

1. From the point $(0, 12)$, substituting into the equation: $12 = ab^0 \implies 12 = a \times 1 \implies a = 12$

2. Substituting $a = 12$ into the general form of the equation, we get: $y = 12b^x$

3. Now use the point $(1, 6)$ to find $b$: $6 = 12b^1 \implies b = \frac{6}{12} = 0.5$

4. So, the equation of the function becomes: $y = 12(0.5)^x$

To confirm, let's check this equation with the third point $(2, 3)$: $y = 12(0.5)^2 = 12 \times 0.25 = 3$ which matches the given point $(2, 3)$.

Thus, the equation of the function is: $y = 12(0.5)^x$

Would you like more details or clarifications on this?

Here are 5 related questions:

1. How can we identify if a graph represents exponential growth or decay?
2. What is the general form of an exponential function?
3. How do we calculate the rate of decay from an exponential function?
4. What role does the base $b$ play in determining the behavior of the function?
5. Can exponential functions model real-world phenomena, like population decline?

Tip: When identifying exponential functions from graphs, pay attention to whether the function is decreasing or increasing, which helps determine if it's decay or growth.