To represent the function in the table, we need to determine a mathematical relationship between $x$ and $y$. Observing the values:

• When $x = 0$, $y = 24$
• When $x = 1$, $y = 36$
• When $x = 2$, $y = 54$
• When $x = 3$, $y = 81$

This pattern suggests a geometric progression in $y$. To confirm, calculate the ratios of successive $y$-values:

$\frac{36}{24} = 1.5, \quad \frac{54}{36} = 1.5, \quad \frac{81}{54} = 1.5$

Since the ratio is constant ($1.5$), the function can be represented as:

$y = 24 \cdot (1.5)^x$

Final equation:

$y = 24 \cdot (1.5)^x$

Verification:

1. For $x = 0$, $y = 24 \cdot (1.5)^0 = 24$ ✔
2. For $x = 1$, $y = 24 \cdot (1.5)^1 = 36$ ✔
3. For $x = 2$, $y = 24 \cdot (1.5)^2 = 54$ ✔
4. For $x = 3$, $y = 24 \cdot (1.5)^3 = 81$ ✔

The equation matches the table data.

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Do you want a deeper explanation or graphical representation of the function?

5 Related Questions:

1. How can you determine if a pattern follows a geometric progression?
2. What is the significance of the base in an exponential function?
3. How can you derive the formula for any geometric sequence?
4. What happens if the ratio in a geometric progression is less than 1?
5. How would you adjust the equation if the initial $y$-value changes?

Tip:

When working with tables, check if the differences (linear growth) or ratios (exponential growth) between successive values are consistent to determine the function type.