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:
```
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```
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.
