Explore a diverse range of topics and get answers from knowledgeable individuals on IDNLearn.com. Our experts provide prompt and accurate answers to help you make informed decisions on any topic.
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.
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.
