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Sagot :
To analyze the exponential function [tex]\( f(x) = -3(3)^x \)[/tex], let's go through the details step-by-step.
### Step 1: Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]
- The function is in the form [tex]\( f(x) = a \cdot b^x \)[/tex].
- Here, [tex]\( a = -3 \)[/tex] and [tex]\( b = 3 \)[/tex].
### Step 2: Calculate the [tex]\( y \)[/tex]-intercept
- The [tex]\( y \)[/tex]-intercept is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = -3 \cdot 3^0 = -3 \cdot 1 = -3 \][/tex]
- Thus, the [tex]\( y \)[/tex]-intercept is [tex]\( -3 \)[/tex].
### Step 3: Evaluate the function for given [tex]\( x \)[/tex]-values
Let's calculate [tex]\( f(x) \)[/tex] for [tex]\( x = -2, -1, 0, 1, \)[/tex] and [tex]\( 2 \)[/tex]:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -3 \cdot 3^{-2} = -3 \cdot \frac{1}{3^2} = -3 \cdot \frac{1}{9} = -\frac{1}{3} \approx -0.333 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -3 \cdot 3^{-1} = -3 \cdot \frac{1}{3} = -1 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -3 \cdot 3^0 = -3 \cdot 1 = -3 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -3 \cdot 3^1 = -3 \cdot 3 = -9 \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -3 \cdot 3^2 = -3 \cdot 9 = -27 \][/tex]
So the table filled in looks as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & -0.333 & -1 & -3 & -9 & -27 \\ \hline \end{array} \][/tex]
### Step 4: Analyze the end behavior
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- This is because as [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] grows exponentially, making [tex]\( -3(3^x) \)[/tex] become increasingly negative.
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow 0 \)[/tex].
- This is because as [tex]\( x \)[/tex] decreases, [tex]\( 3^x \)[/tex] approaches zero, making [tex]\( -3(3^x) \)[/tex] approach zero (from the negative side).
### Summary
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( -3 \)[/tex]
- End Behavior:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex]
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow 0 \)[/tex]
### Step 1: Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]
- The function is in the form [tex]\( f(x) = a \cdot b^x \)[/tex].
- Here, [tex]\( a = -3 \)[/tex] and [tex]\( b = 3 \)[/tex].
### Step 2: Calculate the [tex]\( y \)[/tex]-intercept
- The [tex]\( y \)[/tex]-intercept is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = -3 \cdot 3^0 = -3 \cdot 1 = -3 \][/tex]
- Thus, the [tex]\( y \)[/tex]-intercept is [tex]\( -3 \)[/tex].
### Step 3: Evaluate the function for given [tex]\( x \)[/tex]-values
Let's calculate [tex]\( f(x) \)[/tex] for [tex]\( x = -2, -1, 0, 1, \)[/tex] and [tex]\( 2 \)[/tex]:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -3 \cdot 3^{-2} = -3 \cdot \frac{1}{3^2} = -3 \cdot \frac{1}{9} = -\frac{1}{3} \approx -0.333 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -3 \cdot 3^{-1} = -3 \cdot \frac{1}{3} = -1 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -3 \cdot 3^0 = -3 \cdot 1 = -3 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -3 \cdot 3^1 = -3 \cdot 3 = -9 \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -3 \cdot 3^2 = -3 \cdot 9 = -27 \][/tex]
So the table filled in looks as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & -0.333 & -1 & -3 & -9 & -27 \\ \hline \end{array} \][/tex]
### Step 4: Analyze the end behavior
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- This is because as [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] grows exponentially, making [tex]\( -3(3^x) \)[/tex] become increasingly negative.
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow 0 \)[/tex].
- This is because as [tex]\( x \)[/tex] decreases, [tex]\( 3^x \)[/tex] approaches zero, making [tex]\( -3(3^x) \)[/tex] approach zero (from the negative side).
### Summary
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( -3 \)[/tex]
- End Behavior:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex]
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow 0 \)[/tex]
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