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To identify the graph of the equation [tex]\( y = 20 \left(\frac{1}{4}\right)^x \)[/tex], let's break down the components of this exponential function and understand its behavior step-by-step.
### 1. Understanding the Exponential Function
The equation [tex]\( y = 20 \left(\frac{1}{4}\right)^x \)[/tex] is in the form of [tex]\( y = a \cdot b^x \)[/tex]:
- [tex]\( a = 20 \)[/tex]
- [tex]\( b = \frac{1}{4} \)[/tex]
Here, [tex]\( a \)[/tex] is a constant multiplier and [tex]\( b \)[/tex] is the base of the exponent.
### 2. The Base [tex]\(\frac{1}{4}\)[/tex]
Since [tex]\( b = \frac{1}{4} \)[/tex], which is a fraction less than 1:
- The function [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] is a decreasing function.
- As [tex]\( x \)[/tex] increases, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] approaches 0.
### 3. The Multiplier 20
The multiplier [tex]\( a = 20 \)[/tex] scales the function vertically:
- It stretches the graph upwards by a factor of 20.
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 20 \cdot 1 = 20 \)[/tex].
### 4. Behavior and Key Points
Let's evaluate the function at some specific points to understand its shape:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 20 \left(\frac{1}{4}\right)^0 = 20 \cdot 1 = 20 \][/tex]
The graph passes through the point [tex]\((0, 20)\)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 20 \left(\frac{1}{4}\right)^1 = 20 \cdot \frac{1}{4} = 5 \][/tex]
The graph passes through the point [tex]\((1, 5)\)[/tex].
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 20 \left(\frac{1}{4}\right)^{-1} = 20 \cdot 4 = 80 \][/tex]
The graph passes through the point [tex]\((-1, 80)\)[/tex].
- As [tex]\( x \)[/tex] approaches infinity ([tex]\(+\infty\)[/tex]):
[tex]\[ \left(\frac{1}{4}\right)^x \to 0 \quad \text{and hence} \quad y \to 0 \][/tex]
The graph asymptotically approaches the x-axis but never touches it.
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\(-\infty\)[/tex]):
[tex]\[ \left(\frac{1}{4}\right)^x \to \infty \quad \text{and hence} \quad y \to \infty \][/tex]
The graph rises sharply to very large values.
### 5. Graphing the Function
Given these key points and behavior:
- The graph is a decreasing curve.
- It starts from [tex]\( y = \infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex].
- It intercepts at [tex]\( y = 20 \)[/tex] when [tex]\( x = 0 \)[/tex].
- It approaches [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], but never touches the x-axis.
### 6. Sketching the Graph
You can visualize the graph with the following properties:
- Passes through [tex]\((-1, 80)\)[/tex], [tex]\((0, 20)\)[/tex], and [tex]\((1, 5)\)[/tex].
- Asymptotically approaches the x-axis.
The graph looks like this:
Always remember, for [tex]\( x \)[/tex] approaching negative values, the value of y grows rapidly, forming an exponential decay curve. For positive values of [tex]\( x \)[/tex], the function value diminishes towards zero.
By understanding these steps, you can accurately plot the exponential function [tex]\( y = 20 \left(\frac{1}{4}\right)^x \)[/tex] and predict its graphical behavior.
### 1. Understanding the Exponential Function
The equation [tex]\( y = 20 \left(\frac{1}{4}\right)^x \)[/tex] is in the form of [tex]\( y = a \cdot b^x \)[/tex]:
- [tex]\( a = 20 \)[/tex]
- [tex]\( b = \frac{1}{4} \)[/tex]
Here, [tex]\( a \)[/tex] is a constant multiplier and [tex]\( b \)[/tex] is the base of the exponent.
### 2. The Base [tex]\(\frac{1}{4}\)[/tex]
Since [tex]\( b = \frac{1}{4} \)[/tex], which is a fraction less than 1:
- The function [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] is a decreasing function.
- As [tex]\( x \)[/tex] increases, [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] approaches 0.
### 3. The Multiplier 20
The multiplier [tex]\( a = 20 \)[/tex] scales the function vertically:
- It stretches the graph upwards by a factor of 20.
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 20 \cdot 1 = 20 \)[/tex].
### 4. Behavior and Key Points
Let's evaluate the function at some specific points to understand its shape:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 20 \left(\frac{1}{4}\right)^0 = 20 \cdot 1 = 20 \][/tex]
The graph passes through the point [tex]\((0, 20)\)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 20 \left(\frac{1}{4}\right)^1 = 20 \cdot \frac{1}{4} = 5 \][/tex]
The graph passes through the point [tex]\((1, 5)\)[/tex].
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 20 \left(\frac{1}{4}\right)^{-1} = 20 \cdot 4 = 80 \][/tex]
The graph passes through the point [tex]\((-1, 80)\)[/tex].
- As [tex]\( x \)[/tex] approaches infinity ([tex]\(+\infty\)[/tex]):
[tex]\[ \left(\frac{1}{4}\right)^x \to 0 \quad \text{and hence} \quad y \to 0 \][/tex]
The graph asymptotically approaches the x-axis but never touches it.
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\(-\infty\)[/tex]):
[tex]\[ \left(\frac{1}{4}\right)^x \to \infty \quad \text{and hence} \quad y \to \infty \][/tex]
The graph rises sharply to very large values.
### 5. Graphing the Function
Given these key points and behavior:
- The graph is a decreasing curve.
- It starts from [tex]\( y = \infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex].
- It intercepts at [tex]\( y = 20 \)[/tex] when [tex]\( x = 0 \)[/tex].
- It approaches [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], but never touches the x-axis.
### 6. Sketching the Graph
You can visualize the graph with the following properties:
- Passes through [tex]\((-1, 80)\)[/tex], [tex]\((0, 20)\)[/tex], and [tex]\((1, 5)\)[/tex].
- Asymptotically approaches the x-axis.
The graph looks like this:
Always remember, for [tex]\( x \)[/tex] approaching negative values, the value of y grows rapidly, forming an exponential decay curve. For positive values of [tex]\( x \)[/tex], the function value diminishes towards zero.
By understanding these steps, you can accurately plot the exponential function [tex]\( y = 20 \left(\frac{1}{4}\right)^x \)[/tex] and predict its graphical behavior.
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