Answer :
To determine which exponential function grows at a faster rate than the given quadratic function in the interval [tex]\(0 < x < 3\)[/tex], we need to follow these steps:
### Step 1: Identify the Quadratic Function
Given the values in the table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 3 \\ \hline 2 & 12 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
From the values, we need to find the quadratic function [tex]\( y = ax^2 \)[/tex] that best fits this data.
### Step 2: Determine the Coefficient 'a'
We notice that the function is quadratic, so we can use the given points to establish the coefficient [tex]\( a \)[/tex] in the form [tex]\( y = ax^2 \)[/tex].
Using these points, we calculate as follows:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex], so [tex]\( 3 = a \cdot 1^2 \Rightarrow a = 3 \)[/tex].
- To ensure consistency, we can verify with other points:
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 12 \)[/tex], so [tex]\( 12 = a \cdot 2^2 \Rightarrow 12 = 4a \Rightarrow a = 3 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 27 \)[/tex], so [tex]\( 27 = a \cdot 3^2 \Rightarrow 27 = 9a \Rightarrow a = 3 \)[/tex].
Thus, the quadratic function is [tex]\( y = 3x^2 \)[/tex].
### Step 3: Form of the Exponential Function
We consider an exponential function [tex]\( y = b^x \)[/tex] and seek to determine the base [tex]\( b \)[/tex] such that the exponential function grows faster than the quadratic function [tex]\( y = 3x^2 \)[/tex] in the interval [tex]\( 0 < x < 3 \)[/tex].
### Step 4: Compare Growth Rates at [tex]\( x = 3 \)[/tex]
To ensure the exponential function grows faster, its value should exceed the quadratic function at [tex]\( x = 3 \)[/tex].
- For the quadratic function:
[tex]\[ y = 3 \cdot 3^2 = 27 \][/tex]
- For the exponential function [tex]\( y = b^3 \)[/tex]:
We need [tex]\( b^3 > 27 \)[/tex].
Solving for [tex]\( b \)[/tex]:
[tex]\[ b^3 = 27 \Rightarrow b \approx 3 \][/tex]
### Step 5: Verify the Exponential Growth
If we test values less than 3, say [tex]\( b = 2.9 \)[/tex]:
[tex]\[ 2.9^3 = 24.389 \][/tex]
This is less than 27, so the value of [tex]\( b \)[/tex] must be greater than 2.9 for the exponential function to grow faster at [tex]\( x = 3 \)[/tex].
Given the calculations, there appears to not be a sufficiently precise value of [tex]\( b \)[/tex] within the simple ranges tested that would meet the criteria [tex]\( b^3 > 27 \)[/tex] while also being practical for simple calculation ranges. Therefore, in this specific problem context, finding such an exponential function with a simple integer value for base [tex]\( b \)[/tex] may be infeasible as smaller steps closer to 3 have not yielded faster growth.
### Conclusion
Given our calculations, the appropriate quadratic function is [tex]\( y = 3x^2 \)[/tex]. For [tex]\( x \)[/tex] values in [tex]\( 0 < x < 3 \)[/tex], identifying an exponential function with simple integer base [tex]\( b \)[/tex] such as [tex]\( 3 \)[/tex] would have exactly matching growth at the upper boundary of this interval, but no lower integer base meets the criteria for faster growth in simpler tested ranges. Thus, the exponential function rapidly approaching [tex]\( b \approx 3 \)[/tex] might be required where finer granularity [tex]\( b>2.9 \rightarrow\infty \)[/tex] exceeds the quadratic function at [tex]\( x \approx 3 \)[/tex]. As calculations, mathematically, an exact faster exponential function base [tex]\( b\sim \infty \rightarrow\)[/tex] 'exponential function exceeds quadratic function [tex]\( 3x^2 \)[/tex] at domain intersection'.
### Step 1: Identify the Quadratic Function
Given the values in the table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 3 \\ \hline 2 & 12 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
From the values, we need to find the quadratic function [tex]\( y = ax^2 \)[/tex] that best fits this data.
### Step 2: Determine the Coefficient 'a'
We notice that the function is quadratic, so we can use the given points to establish the coefficient [tex]\( a \)[/tex] in the form [tex]\( y = ax^2 \)[/tex].
Using these points, we calculate as follows:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex], so [tex]\( 3 = a \cdot 1^2 \Rightarrow a = 3 \)[/tex].
- To ensure consistency, we can verify with other points:
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 12 \)[/tex], so [tex]\( 12 = a \cdot 2^2 \Rightarrow 12 = 4a \Rightarrow a = 3 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 27 \)[/tex], so [tex]\( 27 = a \cdot 3^2 \Rightarrow 27 = 9a \Rightarrow a = 3 \)[/tex].
Thus, the quadratic function is [tex]\( y = 3x^2 \)[/tex].
### Step 3: Form of the Exponential Function
We consider an exponential function [tex]\( y = b^x \)[/tex] and seek to determine the base [tex]\( b \)[/tex] such that the exponential function grows faster than the quadratic function [tex]\( y = 3x^2 \)[/tex] in the interval [tex]\( 0 < x < 3 \)[/tex].
### Step 4: Compare Growth Rates at [tex]\( x = 3 \)[/tex]
To ensure the exponential function grows faster, its value should exceed the quadratic function at [tex]\( x = 3 \)[/tex].
- For the quadratic function:
[tex]\[ y = 3 \cdot 3^2 = 27 \][/tex]
- For the exponential function [tex]\( y = b^3 \)[/tex]:
We need [tex]\( b^3 > 27 \)[/tex].
Solving for [tex]\( b \)[/tex]:
[tex]\[ b^3 = 27 \Rightarrow b \approx 3 \][/tex]
### Step 5: Verify the Exponential Growth
If we test values less than 3, say [tex]\( b = 2.9 \)[/tex]:
[tex]\[ 2.9^3 = 24.389 \][/tex]
This is less than 27, so the value of [tex]\( b \)[/tex] must be greater than 2.9 for the exponential function to grow faster at [tex]\( x = 3 \)[/tex].
Given the calculations, there appears to not be a sufficiently precise value of [tex]\( b \)[/tex] within the simple ranges tested that would meet the criteria [tex]\( b^3 > 27 \)[/tex] while also being practical for simple calculation ranges. Therefore, in this specific problem context, finding such an exponential function with a simple integer value for base [tex]\( b \)[/tex] may be infeasible as smaller steps closer to 3 have not yielded faster growth.
### Conclusion
Given our calculations, the appropriate quadratic function is [tex]\( y = 3x^2 \)[/tex]. For [tex]\( x \)[/tex] values in [tex]\( 0 < x < 3 \)[/tex], identifying an exponential function with simple integer base [tex]\( b \)[/tex] such as [tex]\( 3 \)[/tex] would have exactly matching growth at the upper boundary of this interval, but no lower integer base meets the criteria for faster growth in simpler tested ranges. Thus, the exponential function rapidly approaching [tex]\( b \approx 3 \)[/tex] might be required where finer granularity [tex]\( b>2.9 \rightarrow\infty \)[/tex] exceeds the quadratic function at [tex]\( x \approx 3 \)[/tex]. As calculations, mathematically, an exact faster exponential function base [tex]\( b\sim \infty \rightarrow\)[/tex] 'exponential function exceeds quadratic function [tex]\( 3x^2 \)[/tex] at domain intersection'.
