Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Sure! Let's graph the exponential function [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex].
We will first understand how the function behaves, then choose five points to plot, and finally draw the asymptote.
1. Behavior of the Function:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \left(\frac{1}{4}\right)^0 = 1 \][/tex]
- For [tex]\( x > 0 \)[/tex]:
As [tex]\( x \)[/tex] increases, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] gets smaller because we are raising 1/4 to a positive power.
- For [tex]\( x < 0 \)[/tex]:
As [tex]\( x \)[/tex] decreases, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] gets larger because raising a fraction to a negative power results in a large positive number.
2. Five Points to Plot:
- Point 1: For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \left(\frac{1}{4}\right)^{-2} = \left(\frac{4}{1}\right)^2 = 16 \][/tex]
- Point 2: For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \left(\frac{1}{4}\right)^{-1} = \frac{4}{1} = 4 \][/tex]
- Point 3: For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \left(\frac{1}{4}\right)^0 = 1 \][/tex]
- Point 4: For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \left(\frac{1}{4}\right)^1 = \frac{1}{4} = 0.25 \][/tex]
- Point 5: For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \left(\frac{1}{4}\right)^2 = \left(\frac{1}{4}\right) \cdot \left(\frac{1}{4}\right) = \frac{1}{16} = 0.0625 \][/tex]
3. Drawing the Asymptote:
The horizontal asymptote for the function [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex] is [tex]\( y = 0 \)[/tex]. This is because as [tex]\( x \)[/tex] becomes very large and positive, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] approaches 0 but never actually reaches it.
4. Plotting Points and Drawing the Graph:
Now, let's plot these five points on a coordinate plane and draw the curve.
- Point 1: (-2, 16)
- Point 2: (-1, 4)
- Point 3: (0, 1)
- Point 4: (1, 0.25)
- Point 5: (2, 0.0625)
These points help us understand the shape of the curve, which should decrease rapidly as x increases towards positive infinity.
The curve will pass through the points we calculated and get closer and closer to the asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] increases.
Here's a visual representation and script for the graphing steps:
1. Mark the 5 points on a graph.
2. Connect the points smoothly to show the exponential decay.
3. Draw the horizontal asymptote at [tex]\( y = 0 \)[/tex].
The resulting graph should look like this:
```
^
|
|
|
|
|
|
|
|
|
|* -----------------------------
|________________________________>
```
The points (2, 0.0625), (1, 0.25), (0, 1), (-1, 4), and (-2, 16) are marked, and the horizontal asymptote of [tex]\( y = 0 \)[/tex] is drawn as a dashed line.
We will first understand how the function behaves, then choose five points to plot, and finally draw the asymptote.
1. Behavior of the Function:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \left(\frac{1}{4}\right)^0 = 1 \][/tex]
- For [tex]\( x > 0 \)[/tex]:
As [tex]\( x \)[/tex] increases, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] gets smaller because we are raising 1/4 to a positive power.
- For [tex]\( x < 0 \)[/tex]:
As [tex]\( x \)[/tex] decreases, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] gets larger because raising a fraction to a negative power results in a large positive number.
2. Five Points to Plot:
- Point 1: For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \left(\frac{1}{4}\right)^{-2} = \left(\frac{4}{1}\right)^2 = 16 \][/tex]
- Point 2: For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \left(\frac{1}{4}\right)^{-1} = \frac{4}{1} = 4 \][/tex]
- Point 3: For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \left(\frac{1}{4}\right)^0 = 1 \][/tex]
- Point 4: For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \left(\frac{1}{4}\right)^1 = \frac{1}{4} = 0.25 \][/tex]
- Point 5: For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \left(\frac{1}{4}\right)^2 = \left(\frac{1}{4}\right) \cdot \left(\frac{1}{4}\right) = \frac{1}{16} = 0.0625 \][/tex]
3. Drawing the Asymptote:
The horizontal asymptote for the function [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex] is [tex]\( y = 0 \)[/tex]. This is because as [tex]\( x \)[/tex] becomes very large and positive, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] approaches 0 but never actually reaches it.
4. Plotting Points and Drawing the Graph:
Now, let's plot these five points on a coordinate plane and draw the curve.
- Point 1: (-2, 16)
- Point 2: (-1, 4)
- Point 3: (0, 1)
- Point 4: (1, 0.25)
- Point 5: (2, 0.0625)
These points help us understand the shape of the curve, which should decrease rapidly as x increases towards positive infinity.
The curve will pass through the points we calculated and get closer and closer to the asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] increases.
Here's a visual representation and script for the graphing steps:
1. Mark the 5 points on a graph.
2. Connect the points smoothly to show the exponential decay.
3. Draw the horizontal asymptote at [tex]\( y = 0 \)[/tex].
The resulting graph should look like this:
```
^
|
|
|
|
|
|
|
|
|
|* -----------------------------
|________________________________>
```
The points (2, 0.0625), (1, 0.25), (0, 1), (-1, 4), and (-2, 16) are marked, and the horizontal asymptote of [tex]\( y = 0 \)[/tex] is drawn as a dashed line.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
