https://file.aiquickdraw.com/mathbot/file/17304702892089sbg2mp5.jpg

The table you provided shows values of \( x \) and \( y \):

\[
\begin{array}{c|ccccc}
x & 0 & 2 & 4 & 6 \\
\hline
y & 1 & 3 & 9 & ? \\
\end{array}
\]

### Step 1: Determine the Pattern
Looking at the values in the \( y \)-row (1, 3, 9), it seems that \( y \) is following an exponential pattern rather than a linear one. Observing the relationship between consecutive \( y \)-values:

- From \( y = 1 \) to \( y = 3 \), the value triples (approximately),
- From \( y = 3 \) to \( y = 9 \), the value also triples.

This suggests an exponential function of the form:
\[
y = a \cdot b^x
\]
where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the common ratio.

### Step 2: Find the Function
Since \( y(0) = 1 \), we know \( a = 1 \). The value triples each time, so \( b = 3 \). Therefore, the function can be written as:
\[
y = 1 \cdot 3^{x/2} = 3^{x/2}
\]

### Step 3: Find \( y \) When \( x = 6 \)
To find the missing \( y \)-value when \( x = 6 \):
\[
y = 3^{6/2} = 3^3 = 27
\]

### Conclusion
The missing \( y \)-value when \( x = 6 \) is **27**.

Would you like further details on exponential functions or have any questions?

---

**Here are 5 related questions to further explore exponential functions:**

1. How do we differentiate between linear and exponential functions based on patterns?
2. What is the general form of an exponential function, and how do we determine the base?
3. How can we identify if a given sequence is geometric or arithmetic?
4. How do we calculate growth rates in exponential functions?
5. What are real-world applications of exponential functions?

**Tip:** Exponential functions grow or decay at a constant rate per interval, making them useful for modeling rapid changes like population growth or radioactive decay.

Fill in the table with the missing y-value. y is either a linear function of x or an exponential function of x.

Learn how to identify and complete exponential function tables by finding missing values. This example demonstrates filling in missing y-values based on an exponential growth pattern. Ideal for high school students (Grades 9-11) studying exponential functions and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation