IDNLearn.com provides a reliable platform for finding accurate and timely answers. Find the solutions you need quickly and accurately with help from our knowledgeable community.
Sagot :
To graph the function [tex]\(f(x) = 2\left(\frac{1}{2}\right)^{x+2} + 8\)[/tex], let's follow a step-by-step approach to determine its key features and graph it accordingly.
### Step 1: Identify and plot the horizontal asymptote
The function [tex]\(f(x)\)[/tex] has a horizontal asymptote due to the term involving the exponent. As [tex]\(x\)[/tex] approaches positive or negative infinity, the term [tex]\(\left(\frac{1}{2}\right)^{x+2}\)[/tex] approaches 0. Therefore, the horizontal asymptote is determined by the constant term [tex]\(+8\)[/tex].
- Horizontal Asymptote: [tex]\(y = 8\)[/tex]
### Step 2: Calculate and plot points on the function
We choose specific [tex]\(x\)[/tex] values to find points with integer coordinates. It is ideal to pick [tex]\(x\)[/tex] values that result in simpler calculations.
#### Point 1: [tex]\(x = 0\)[/tex]
[tex]\[ f(0) = 2\left(\frac{1}{2}\right)^{0+2} + 8 \][/tex]
[tex]\[ f(0) = 2\left(\frac{1}{2}\right)^2 + 8 \][/tex]
[tex]\[ f(0) = 2 \cdot \frac{1}{4} + 8 \][/tex]
[tex]\[ f(0) = \frac{1}{2} + 8 \][/tex]
[tex]\[ f(0) = 8.5 \][/tex]
The point is [tex]\((0, 8.5)\)[/tex].
#### Point 2: [tex]\(x = 1\)[/tex]
[tex]\[ f(1) = 2\left(\frac{1}{2}\right)^{1+2} + 8 \][/tex]
[tex]\[ f(1) = 2\left(\frac{1}{2}\right)^3 + 8 \][/tex]
[tex]\[ f(1) = 2 \cdot \frac{1}{8} + 8 \][/tex]
[tex]\[ f(1) = \frac{1}{4} + 8 \][/tex]
[tex]\[ f(1) = 8.25 \][/tex]
The point is [tex]\((1, 8.25)\)[/tex].
### Step 3: Plotting the function
1. Plot the horizontal asymptote [tex]\(y = 8\)[/tex]: Draw a dashed line parallel to the x-axis at [tex]\(y = 8\)[/tex].
2. Plot the points [tex]\((0, 8.5)\)[/tex] and [tex]\((1, 8.25)\)[/tex]: Mark these points on the graph.
3. Draw the curve for the function [tex]\(f(x)\)[/tex]: Sketch the curve that passes through the plotted points and approaches the asymptote as [tex]\(x\)[/tex] moves towards positive or negative infinity. The shape should show exponential decay.
【Example Graph】:
y
|
10 + (0, 8.5)
|
9 + (1, 8.25)
8 --+ -------------------------------------------------------- x
|
| Asymptote: [tex]\(y = 8\)[/tex]
|
|
|
This graph efficiently represents the behavior of [tex]\(f(x) = 2\left(\frac{1}{2}\right)^{x+2} + 8\)[/tex], indicating the horizontal asymptote and two critical points with integer coordinates.
### Step 1: Identify and plot the horizontal asymptote
The function [tex]\(f(x)\)[/tex] has a horizontal asymptote due to the term involving the exponent. As [tex]\(x\)[/tex] approaches positive or negative infinity, the term [tex]\(\left(\frac{1}{2}\right)^{x+2}\)[/tex] approaches 0. Therefore, the horizontal asymptote is determined by the constant term [tex]\(+8\)[/tex].
- Horizontal Asymptote: [tex]\(y = 8\)[/tex]
### Step 2: Calculate and plot points on the function
We choose specific [tex]\(x\)[/tex] values to find points with integer coordinates. It is ideal to pick [tex]\(x\)[/tex] values that result in simpler calculations.
#### Point 1: [tex]\(x = 0\)[/tex]
[tex]\[ f(0) = 2\left(\frac{1}{2}\right)^{0+2} + 8 \][/tex]
[tex]\[ f(0) = 2\left(\frac{1}{2}\right)^2 + 8 \][/tex]
[tex]\[ f(0) = 2 \cdot \frac{1}{4} + 8 \][/tex]
[tex]\[ f(0) = \frac{1}{2} + 8 \][/tex]
[tex]\[ f(0) = 8.5 \][/tex]
The point is [tex]\((0, 8.5)\)[/tex].
#### Point 2: [tex]\(x = 1\)[/tex]
[tex]\[ f(1) = 2\left(\frac{1}{2}\right)^{1+2} + 8 \][/tex]
[tex]\[ f(1) = 2\left(\frac{1}{2}\right)^3 + 8 \][/tex]
[tex]\[ f(1) = 2 \cdot \frac{1}{8} + 8 \][/tex]
[tex]\[ f(1) = \frac{1}{4} + 8 \][/tex]
[tex]\[ f(1) = 8.25 \][/tex]
The point is [tex]\((1, 8.25)\)[/tex].
### Step 3: Plotting the function
1. Plot the horizontal asymptote [tex]\(y = 8\)[/tex]: Draw a dashed line parallel to the x-axis at [tex]\(y = 8\)[/tex].
2. Plot the points [tex]\((0, 8.5)\)[/tex] and [tex]\((1, 8.25)\)[/tex]: Mark these points on the graph.
3. Draw the curve for the function [tex]\(f(x)\)[/tex]: Sketch the curve that passes through the plotted points and approaches the asymptote as [tex]\(x\)[/tex] moves towards positive or negative infinity. The shape should show exponential decay.
【Example Graph】:
y
|
10 + (0, 8.5)
|
9 + (1, 8.25)
8 --+ -------------------------------------------------------- x
|
| Asymptote: [tex]\(y = 8\)[/tex]
|
|
|
This graph efficiently represents the behavior of [tex]\(f(x) = 2\left(\frac{1}{2}\right)^{x+2} + 8\)[/tex], indicating the horizontal asymptote and two critical points with integer coordinates.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.