Solved on Jan 23, 2024

Find the decay equation from the options: $y=6\left(\frac{1}{3}\right)^{x}$ , $y=\frac{1}{2}(3)^{x}$ , $y=3^{x}-4$ , $y=2(4)^{x}$ .

STEP 1

Assumptions
1. The equation represents a decay process.
2. A decay process is characterized by a decrease in the value of the function as the independent variable increases.
3. The base of the exponential function in a decay process is between 0 and 1.

STEP 2

Analyze option a) to determine if it represents a decay process.
$y=6\left(\frac{1}{3}\right)^{x}$

STEP 3

Identify the base of the exponential function in option a).
The base is $\frac{1}{3}$ , which is between 0 and 1.

STEP 4

Conclude whether option a) represents a decay process.
Since the base $\frac{1}{3}$ is between 0 and 1, option a) represents a decay process.

STEP 5

Analyze option b) to determine if it represents a decay process.
$y=\frac{1}{2}(3)^{x}$

STEP 6

Identify the base of the exponential function in option b).
The base is $3$ , which is greater than 1.

STEP 7

Conclude whether option b) represents a decay process.
Since the base $3$ is greater than 1, option b) does not represent a decay process.

STEP 8

Analyze option c) to determine if it represents a decay process.
$y=3^{x}-4$

STEP 9

Identify the base of the exponential function in option c).
The base is $3$ , which is greater than 1.

STEP 10

Conclude whether option c) represents a decay process.
Since the base $3$ is greater than 1, option c) does not represent a decay process.

STEP 11

Analyze option d) to determine if it represents a decay process.
$y=2(4)^{x}$

STEP 12

Identify the base of the exponential function in option d).
The base is $4$ , which is greater than 1.

STEP 13

Conclude whether option d) represents a decay process.
Since the base $4$ is greater than 1, option d) does not represent a decay process.

STEP 14

Determine the correct option that represents a decay process.
Based on the analysis, the correct option that represents a decay process is a) $y=6\left(\frac{1}{3}\right)^{x}$ .
The equation that represents a decay is: a) $y=6\left(\frac{1}{3}\right)^{x}$ .

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Find the decay equation from the options: $y=6\left(\frac{1}{3}\right)^{x}$ , $y=\frac{1}{2}(3)^{x}$ , $y=3^{x}-4$ , $y=2(4)^{x}$ .

STEP 1

Assumptions
1. The equation represents a decay process.
2. A decay process is characterized by a decrease in the value of the function as the independent variable increases.
3. The base of the exponential function in a decay process is between 0 and 1.

STEP 2

Analyze option a) to determine if it represents a decay process.
$y=6\left(\frac{1}{3}\right)^{x}$

STEP 3

Identify the base of the exponential function in option a).
The base is $\frac{1}{3}$ , which is between 0 and 1.

STEP 4

Conclude whether option a) represents a decay process.
Since the base $\frac{1}{3}$ is between 0 and 1, option a) represents a decay process.

STEP 5

Analyze option b) to determine if it represents a decay process.
$y=\frac{1}{2}(3)^{x}$

STEP 6

Identify the base of the exponential function in option b).
The base is $3$ , which is greater than 1.

STEP 7

Conclude whether option b) represents a decay process.
Since the base $3$ is greater than 1, option b) does not represent a decay process.

STEP 8

Analyze option c) to determine if it represents a decay process.
$y=3^{x}-4$

STEP 9

Identify the base of the exponential function in option c).
The base is $3$ , which is greater than 1.

STEP 10

Conclude whether option c) represents a decay process.
Since the base $3$ is greater than 1, option c) does not represent a decay process.

STEP 11

Analyze option d) to determine if it represents a decay process.
$y=2(4)^{x}$

STEP 12

Identify the base of the exponential function in option d).
The base is $4$ , which is greater than 1.

STEP 13

Conclude whether option d) represents a decay process.
Since the base $4$ is greater than 1, option d) does not represent a decay process.

STEP 14

Determine the correct option that represents a decay process.
Based on the analysis, the correct option that represents a decay process is a) $y=6\left(\frac{1}{3}\right)^{x}$ .
The equation that represents a decay is: a) $y=6\left(\frac{1}{3}\right)^{x}$ .

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Find the decay equation from the options: y=6(13)xy=6\left(\frac{1}{3}\right)^{x}y=6(31​)x, y=12(3)xy=\frac{1}{2}(3)^{x}y=21​(3)x, y=3x−4y=3^{x}-4y=3x−4, y=2(4)xy=2(4)^{x}y=2(4)x.

STEP 1

Assumptions1. The equation represents a decay process.2. A decay process is characterized by a decrease in the value of the function as the independent variable increases.3. The base of the exponential function in a decay process is between 0 and 1.

STEP 2

Analyze option a) to determine if it represents a decay process.y=6(13)xy=6\left(\frac{1}{3}\right)^{x}y=6(31​)x

STEP 3

Identify the base of the exponential function in option a).The base is 13\frac{1}{3}31​, which is between 0 and 1.

STEP 4

Conclude whether option a) represents a decay process.Since the base 13\frac{1}{3}31​ is between 0 and 1, option a) represents a decay process.

STEP 5

Analyze option b) to determine if it represents a decay process.y=12(3)xy=\frac{1}{2}(3)^{x}y=21​(3)x

STEP 6

Identify the base of the exponential function in option b).The base is 333, which is greater than 1.

STEP 7

Conclude whether option b) represents a decay process.Since the base 333 is greater than 1, option b) does not represent a decay process.

STEP 8

Analyze option c) to determine if it represents a decay process.y=3x−4y=3^{x}-4y=3x−4

STEP 9

Identify the base of the exponential function in option c).The base is 333, which is greater than 1.

STEP 10

Conclude whether option c) represents a decay process.Since the base 333 is greater than 1, option c) does not represent a decay process.

STEP 11

Analyze option d) to determine if it represents a decay process.y=2(4)xy=2(4)^{x}y=2(4)x

STEP 12

Identify the base of the exponential function in option d).The base is 444, which is greater than 1.

STEP 13

Conclude whether option d) represents a decay process.Since the base 444 is greater than 1, option d) does not represent a decay process.

STEP 14

Determine the correct option that represents a decay process.Based on the analysis, the correct option that represents a decay process is a) y=6(13)xy=6\left(\frac{1}{3}\right)^{x}y=6(31​)x.The equation that represents a decay is: a) y=6(13)xy=6\left(\frac{1}{3}\right)^{x}y=6(31​)x.

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Find the decay equation from the options: $y=6\left(\frac{1}{3}\right)^{x}$ , $y=\frac{1}{2}(3)^{x}$ , $y=3^{x}-4$ , $y=2(4)^{x}$ .

Assumptions
1. The equation represents a decay process.
2. A decay process is characterized by a decrease in the value of the function as the independent variable increases.
3. The base of the exponential function in a decay process is between 0 and 1.

Analyze option a) to determine if it represents a decay process.
$y=6\left(\frac{1}{3}\right)^{x}$

Identify the base of the exponential function in option a).
The base is $\frac{1}{3}$ , which is between 0 and 1.

Conclude whether option a) represents a decay process.
Since the base $\frac{1}{3}$ is between 0 and 1, option a) represents a decay process.

Analyze option b) to determine if it represents a decay process.
$y=\frac{1}{2}(3)^{x}$

Identify the base of the exponential function in option b).
The base is $3$ , which is greater than 1.

Conclude whether option b) represents a decay process.
Since the base $3$ is greater than 1, option b) does not represent a decay process.

Analyze option c) to determine if it represents a decay process.
$y=3^{x}-4$

Identify the base of the exponential function in option c).
The base is $3$ , which is greater than 1.

Conclude whether option c) represents a decay process.
Since the base $3$ is greater than 1, option c) does not represent a decay process.

Analyze option d) to determine if it represents a decay process.
$y=2(4)^{x}$

Identify the base of the exponential function in option d).
The base is $4$ , which is greater than 1.

Conclude whether option d) represents a decay process.
Since the base $4$ is greater than 1, option d) does not represent a decay process.

Determine the correct option that represents a decay process.
Based on the analysis, the correct option that represents a decay process is a) $y=6\left(\frac{1}{3}\right)^{x}$ .
The equation that represents a decay is: a) $y=6\left(\frac{1}{3}\right)^{x}$ .