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Bot GPT-4o First, let's summarize the given data in the table:
\[
\begin{array}{c|c}
x & y \\
\hline
0 & 0 \\
1 & -2 \\
2 & 0 \\
3 & 6 \\
4 & 16 \\
\end{array}
\]
To determine which type of function best models the data, we can either graph the data and observe its shape or test the given equations.
### Step-by-Step Analysis:
#### 1. Quadratic Functions
Quadratic functions have the general form \( y = ax^2 + bx + c \).
Let's test the two given quadratic functions:
##### For \( y = 2x^2 + 4x - 3 \):
1. For \( x = 0 \):
\[
y = 2(0)^2 + 4(0) - 3 = -3 \quad (\text{does not match } y = 0)
\]
Since the function \( y = 2x^2 + 4x - 3 \) did not produce the correct \( y \) value for \( x = 0 \), we can discard this option.
##### For \( y = 2x^2 - 4x \):
1. For \( x = 0 \):
\[
y = 2(0)^2 - 4(0) = 0 \quad (\text{matches } y = 0)
\]
2. For \( x = 1 \):
\[
y = 2(1)^2 - 4(1) = 2 - 4 = -2 \quad (\text{matches } y = -2)
\]
3. For \( x = 2 \):
\[
y = 2(2)^2 - 4(2) = 8 - 8 = 0 \quad (\text{matches } y = 0)
\]
4. For \( x = 3 \):
\[
y = 2(3)^2 - 4(3) = 18 - 12 = 6 \quad (\text{matches } y = 6)
\]
5. For \( x = 4 \):
\[
y = 2(4)^2 - 4(4) = 32 - 16 = 16 \quad (\text{matches } y = 16)
\]
All values match the given data points.
#### 2. Exponential and Linear Functions
We also need to consider and rule out the other options:
##### For \( y = 2 \cdot 4^x \):
1. For \( x = 1 \):
\[
y = 2 \cdot 4^1 = 8 \quad (\text{not } -2)
\]
Clearly, this is not a match.
##### For \( y = -2x \):
1. For \( x = 1 \):
\[
y = -2(1) = -2 \quad (\text{matches})
\]
2. For \( x = 2 \):
\[
y = -2(2) = -4 \quad (\text{not } 0)
\]
The linear function \( y = -2x \) does not match all data points.
### Conclusion
The quadratic function \( y = 2x^2 - 4x \) best matches the data. Therefore, the correct function is:
\[ \boxed{y = 2x^2 - 4x} \]