IDNLearn.com connects you with a community of experts ready to answer your questions. Ask your questions and receive comprehensive and trustworthy answers from our experienced community of professionals.
Sagot :
Alright, let's go through the problem step by step.
### Step 1: Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]
Given the exponential function [tex]\( f(x) = 3 \cdot 2^x \)[/tex], we can identify the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\(a\)[/tex] is the coefficient in front of the exponential term, which is [tex]\(3\)[/tex].
- [tex]\(b\)[/tex] is the base of the exponent, which is [tex]\(2\)[/tex].
### Step 2: Calculate [tex]\( f(x) \)[/tex] for given [tex]\( x \)[/tex] values
We are given a table with specific [tex]\( x \)[/tex] values: [tex]\(-2, -1, 0, 1, 2\)[/tex]. We will plug these [tex]\( x \)[/tex] values into the function [tex]\( f(x) = 3 \cdot 2^x \)[/tex] to find their corresponding [tex]\( f(x) \)[/tex] values.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 3 \cdot 2^{-2} = 3 \cdot \frac{1}{4} = 0.75 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 3 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = 1.5 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 \cdot 2^0 = 3 \cdot 1 = 3 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3 \cdot 2^1 = 3 \cdot 2 = 6 \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 3 \cdot 2^2 = 3 \cdot 4 = 12 \][/tex]
Thus, the completed table is:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & -2 & -1 & 0 & 1 & 2 \\ \hline $f ( x )$ & 0.75 & 1.5 & 3 & 6 & 12 \\ \hline \end{tabular} \][/tex]
### Step 3: Identify the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept is the value of the function when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( 3 \)[/tex].
### Step 4: Determine the end behavior
For the end behavior, we analyze what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex]:
- As [tex]\( x \to \infty \)[/tex], the term [tex]\( 2^x \)[/tex] grows exponentially, and thus [tex]\( f(x) \to \infty \)[/tex]. Therefore, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], the term [tex]\( 2^x \)[/tex] approaches [tex]\( 0 \)[/tex] because [tex]\( 2^x \)[/tex] is a fraction that gets smaller and smaller. So, [tex]\( f(x) \to 0 \)[/tex]. Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
### Summary
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 3 \)[/tex]
- End behavior:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex]
The graph of the exponential function [tex]\( f(x) = 3 \cdot 2^x \)[/tex] will show the values calculated in the table, with the [tex]\( y \)[/tex]-intercept at [tex]\( 3 \)[/tex], growing rapidly as [tex]\( x \)[/tex] increases, and approaching [tex]\( 0 \)[/tex] as [tex]\( x \)[/tex] decreases.
### Step 1: Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]
Given the exponential function [tex]\( f(x) = 3 \cdot 2^x \)[/tex], we can identify the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\(a\)[/tex] is the coefficient in front of the exponential term, which is [tex]\(3\)[/tex].
- [tex]\(b\)[/tex] is the base of the exponent, which is [tex]\(2\)[/tex].
### Step 2: Calculate [tex]\( f(x) \)[/tex] for given [tex]\( x \)[/tex] values
We are given a table with specific [tex]\( x \)[/tex] values: [tex]\(-2, -1, 0, 1, 2\)[/tex]. We will plug these [tex]\( x \)[/tex] values into the function [tex]\( f(x) = 3 \cdot 2^x \)[/tex] to find their corresponding [tex]\( f(x) \)[/tex] values.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 3 \cdot 2^{-2} = 3 \cdot \frac{1}{4} = 0.75 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 3 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = 1.5 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 \cdot 2^0 = 3 \cdot 1 = 3 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3 \cdot 2^1 = 3 \cdot 2 = 6 \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 3 \cdot 2^2 = 3 \cdot 4 = 12 \][/tex]
Thus, the completed table is:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & -2 & -1 & 0 & 1 & 2 \\ \hline $f ( x )$ & 0.75 & 1.5 & 3 & 6 & 12 \\ \hline \end{tabular} \][/tex]
### Step 3: Identify the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept is the value of the function when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( 3 \)[/tex].
### Step 4: Determine the end behavior
For the end behavior, we analyze what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex]:
- As [tex]\( x \to \infty \)[/tex], the term [tex]\( 2^x \)[/tex] grows exponentially, and thus [tex]\( f(x) \to \infty \)[/tex]. Therefore, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], the term [tex]\( 2^x \)[/tex] approaches [tex]\( 0 \)[/tex] because [tex]\( 2^x \)[/tex] is a fraction that gets smaller and smaller. So, [tex]\( f(x) \to 0 \)[/tex]. Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
### Summary
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 3 \)[/tex]
- End behavior:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex]
The graph of the exponential function [tex]\( f(x) = 3 \cdot 2^x \)[/tex] will show the values calculated in the table, with the [tex]\( y \)[/tex]-intercept at [tex]\( 3 \)[/tex], growing rapidly as [tex]\( x \)[/tex] increases, and approaching [tex]\( 0 \)[/tex] as [tex]\( x \)[/tex] decreases.
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.
