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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.
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