Type The Correct Answer In Each Box. Round Your Answers To One Decimal Place.Use The Function G ( X ) = 4 ( 0.6 ) X G(x)=4(0.6)^x G ( X ) = 4 ( 0.6 ) X To Complete The Table And Find The Y Y Y -intercept. \[ \begin{tabular}{|c|c|} \hline X$ & G ( X ) G(x) G ( X ) \ \hline -10 &

Mar 13, 2025 by ADMIN 279 views

**Exploring Exponential Functions: Completing the Table and Finding the Y-Intercept**

Introduction

Exponential functions are a fundamental concept in mathematics, describing how a quantity changes over time or space. In this article, we will explore the function $g(x)=4(0.6)^x$ and use it to complete a table of values. We will also find the $y$ -intercept of the function, which is a crucial aspect of understanding its behavior.

The Function $g(x)=4(0.6)^x$

The given function is an exponential function with a base of 0.6 and a coefficient of 4. The function can be written in the form $g(x)=ab^x$ , where $a$ is the coefficient and $b$ is the base. In this case, $a=4$ and $b=0.6$ .

Completing the Table

To complete the table, we need to find the values of $g(x)$ for different values of $x$ . We will use the function $g(x)=4(0.6)^x$ to calculate the values.

$x$	$g(x)$
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10

Calculating the Values

To calculate the values of $g(x)$ , we will use the function $g(x)=4(0.6)^x$ . We will substitute the values of $x$ into the function and calculate the corresponding values of $g(x)$ .

$x$	$g(x)$
-10	4(0.6)^(-10) = 4(0.6)^(-10) ≈ 0.00006103515625
-9	4(0.6)^(-9) ≈ 0.00006103515625
-8	4(0.6)^(-8) ≈ 0.00006103515625
-7	4(0.6)^(-7) ≈ 0.00006103515625
-6	4(0.6)^(-6) ≈ 0.00006103515625
-5	4(0.6)^(-5) ≈ 0.00006103515625
-4	4(0.6)^(-4) ≈ 0.00006103515625
-3	4(0.6)^(-3) ≈ 0.00006103515625
-2	4(0.6)^(-2) ≈ 0.00006103515625
-1	4(0.6)^(-1) ≈ 0.66666666666667
0	4(0.6)^0 = 4
1	4(0.6)^1 ≈ 2.4
2	4(0.6)^2 ≈ 1.44
3	4(0.6)^3 ≈ 0.864
4	4(0.6)^4 ≈ 0.5184
5	4(0.6)^5 ≈ 0.31088
6	4(0.6)^6 ≈ 0.186624
7	4(0.6)^7 ≈ 0.1113728
8	4(0.6)^8 ≈ 0.06682368
9	4(0.6)^9 ≈ 0.040049056
10	4(0.6)^10 ≈ 0.0240298352

Finding the Y-Intercept

The $y$ -intercept of a function is the point where the function intersects the $y$ -axis. In other words, it is the value of $y$ when $x=0$ . To find the $y$ -intercept of the function $g(x)=4(0.6)^x$ , we can substitute $x=0$ into the function.

$g(0) = 4(0.6)^0 = 4$

Therefore, the $y$ -intercept of the function $g(x)=4(0.6)^x$ is 4.

Conclusion

Q: What is an exponential function?

A: An exponential function is a mathematical function that describes a relationship between two quantities, where one quantity is a constant power of the other. In the function $g(x)=4(0.6)^x$ , the base is 0.6 and the coefficient is 4.

Q: What is the base of an exponential function?

A: The base of an exponential function is the constant that is raised to the power of the input variable. In the function $g(x)=4(0.6)^x$ , the base is 0.6.

Q: What is the coefficient of an exponential function?

A: The coefficient of an exponential function is the constant that multiplies the base raised to the power of the input variable. In the function $g(x)=4(0.6)^x$ , the coefficient is 4.

Q: How do you calculate the values of an exponential function?

A: To calculate the values of an exponential function, you can use the formula $g(x)=ab^x$ , where $a$ is the coefficient and $b$ is the base. You can substitute the values of $x$ into the formula and calculate the corresponding values of $g(x)$ .

Q: What is the y-intercept of an exponential function?

A: The $y$ -intercept of an exponential function is the point where the function intersects the $y$ -axis. In other words, it is the value of $y$ when $x=0$ . To find the $y$ -intercept of the function $g(x)=4(0.6)^x$ , you can substitute $x=0$ into the function.

Q: How do you complete a table of values for an exponential function?

A: To complete a table of values for an exponential function, you can use the formula $g(x)=ab^x$ , where $a$ is the coefficient and $b$ is the base. You can substitute the values of $x$ into the formula and calculate the corresponding values of $g(x)$ .

Q: What is the significance of the y-intercept in an exponential function?

A: The $y$ -intercept of an exponential function is significant because it represents the starting point of the function. It is the value of $y$ when $x=0$ , and it can be used to determine the behavior of the function.

Q: Can you give an example of an exponential function?

A: Yes, an example of an exponential function is $g(x)=2(1.5)^x$ . This function has a base of 1.5 and a coefficient of 2.

Q: How do you graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the points on a coordinate plane.

Q: What is the difference between an exponential function and a linear function?

A: An exponential function is a function that describes a relationship between two quantities, where one quantity is a constant power of the other. A linear function, on the other hand, is a function that describes a straight line. Exponential functions and linear functions have different properties and behaviors.

Q: Can you give an example of a linear function?

A: Yes, an example of a linear function is $f(x)=2x+3$ . This function has a slope of 2 and a y-intercept of 3.

Q: How do you determine the type of function (exponential or linear) that best fits a set of data?

A: To determine the type of function that best fits a set of data, you can use a graphing calculator or a computer program. You can also use a table of values to plot the points on a coordinate plane and determine the type of function that best fits the data.