The graph shown in the image appears to be an exponential decay function, based on the curve and the provided points: $(0, 162), (1, 54), (2, 18), (3, 6), (4, 2)$.

An exponential function typically takes the form:

$y = ab^x$

Where:

• $a$ is the initial value (the y-intercept, or the value when $x = 0$),
• $b$ is the base of the exponential function, which represents the rate of growth or decay.

Step 1: Determine $a$

From the graph, the point $(0, 162)$ tells us that when $x = 0$, $y = 162$. This means that $a = 162$.

Thus, the equation becomes: $y = 162 \cdot b^x$

Step 2: Determine $b$

Next, we use another point to solve for $b$. Let's use the point $(1, 54)$.

Substitute $x = 1$ and $y = 54$ into the equation: $54 = 162 \cdot b^1$

Solve for $b$: $b = \frac{54}{162} = \frac{1}{3}$

Final Equation:

Now that we know $a = 162$ and $b = \frac{1}{3}$, the equation representing the graph is:

$y = 162 \left( \frac{1}{3} \right)^x$

This is the equation that represents the given graph.

Do you want more details or explanations on any part of the solution?

Here are five follow-up questions to deepen your understanding:

1. What happens to the graph when the value of $b$ is greater than 1?
2. How do you interpret the rate of decay in this exponential function?
3. Can you verify the equation by plugging in other points from the graph?
4. How would the graph change if $a$ were negative?
5. How do you solve for $b$ if you're given other types of points?

Tip: Exponential functions grow or decay at a constant rate relative to their current value, which makes them ideal for modeling populations, radioactive decay, and interest growth.