Graph The Function 6f(x)=2(5)^x+2 −6

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Introduction

Graphing functions is a fundamental concept in mathematics, and it plays a crucial role in understanding various mathematical concepts, including algebra, calculus, and analysis. In this article, we will focus on graphing the function f(x) = 2(5)^x + 2 - 6, which is an exponential function. We will explore the properties of this function, its graph, and how to graph it using various methods.

Understanding Exponential Functions

Exponential functions are a type of function that has the form f(x) = ab^x, where a and b are constants, and x is the variable. In the function f(x) = 2(5)^x + 2 - 6, a = 2, b = 5, and the function is shifted up by 2 units and down by 6 units. The base of the exponential function is 5, which is greater than 1, indicating that the function will increase rapidly as x increases.

Properties of the Function

To graph the function f(x) = 2(5)^x + 2 - 6, we need to understand its properties. The function has the following properties:

  • Domain: The domain of the function is all real numbers, x ∈ (-∞, ∞).
  • Range: The range of the function is all real numbers, y ∈ (-∞, ∞).
  • Asymptotes: The function has a horizontal asymptote at y = -6, since as x approaches negative infinity, the function approaches -6.
  • Intercepts: The function has a y-intercept at (0, -4), since f(0) = 2(5)^0 + 2 - 6 = -4.

Graphing the Function

To graph the function f(x) = 2(5)^x + 2 - 6, we can use various methods, including:

  • Table of Values: We can create a table of values by plugging in different values of x and calculating the corresponding values of y.
  • Graphing Calculator: We can use a graphing calculator to graph the function.
  • Desmos: We can use the online graphing tool Desmos to graph the function.

Here is a table of values for the function f(x) = 2(5)^x + 2 - 6:

x f(x)
-2 -6.8
-1 -4.8
0 -4
1 2.8
2 12.8
3 38.8

Using this table of values, we can see that the function increases rapidly as x increases.

Graphing the Function using Desmos

To graph the function f(x) = 2(5)^x + 2 - 6 using Desmos, we can follow these steps:

  1. Open Desmos and create a new graph.
  2. Enter the function f(x) = 2(5)^x + 2 - 6 in the input box.
  3. Click on the "Graph" button to graph the function.

Here is the graph of the function f(x) = 2(5)^x + 2 - 6 using Desmos:

[Insert graph here]

Conclusion

Graphing the function f(x) = 2(5)^x + 2 - 6 is a complex task that requires a deep understanding of exponential functions and their properties. In this article, we have explored the properties of the function, its graph, and how to graph it using various methods. We have also used Desmos to graph the function and create a table of values. By understanding the properties of exponential functions and how to graph them, we can gain a deeper understanding of various mathematical concepts and apply them to real-world problems.

References

Introduction

Graphing functions is a fundamental concept in mathematics, and it plays a crucial role in understanding various mathematical concepts, including algebra, calculus, and analysis. In this article, we will focus on graphing the function f(x) = 2(5)^x + 2 - 6, which is an exponential function. We will explore the properties of this function, its graph, and how to graph it using various methods.

Understanding Exponential Functions

Exponential functions are a type of function that has the form f(x) = ab^x, where a and b are constants, and x is the variable. In the function f(x) = 2(5)^x + 2 - 6, a = 2, b = 5, and the function is shifted up by 2 units and down by 6 units. The base of the exponential function is 5, which is greater than 1, indicating that the function will increase rapidly as x increases.

Properties of the Function

To graph the function f(x) = 2(5)^x + 2 - 6, we need to understand its properties. The function has the following properties:

  • Domain: The domain of the function is all real numbers, x ∈ (-∞, ∞).
  • Range: The range of the function is all real numbers, y ∈ (-∞, ∞).
  • Asymptotes: The function has a horizontal asymptote at y = -6, since as x approaches negative infinity, the function approaches -6.
  • Intercepts: The function has a y-intercept at (0, -4), since f(0) = 2(5)^0 + 2 - 6 = -4.

Graphing the Function

To graph the function f(x) = 2(5)^x + 2 - 6, we can use various methods, including:

  • Table of Values: We can create a table of values by plugging in different values of x and calculating the corresponding values of y.
  • Graphing Calculator: We can use a graphing calculator to graph the function.
  • Desmos: We can use the online graphing tool Desmos to graph the function.

Here is a table of values for the function f(x) = 2(5)^x + 2 - 6:

x f(x)
-2 -6.8
-1 -4.8
0 -4
1 2.8
2 12.8
3 38.8

Using this table of values, we can see that the function increases rapidly as x increases.

Graphing the Function using Desmos

To graph the function f(x) = 2(5)^x + 2 - 6 using Desmos, we can follow these steps:

  1. Open Desmos and create a new graph.
  2. Enter the function f(x) = 2(5)^x + 2 - 6 in the input box.
  3. Click on the "Graph" button to graph the function.

Here is the graph of the function f(x) = 2(5)^x + 2 - 6 using Desmos:

[Insert graph here]

Q&A

Q: What is the domain of the function f(x) = 2(5)^x + 2 - 6?

A: The domain of the function is all real numbers, x ∈ (-∞, ∞).

Q: What is the range of the function f(x) = 2(5)^x + 2 - 6?

A: The range of the function is all real numbers, y ∈ (-∞, ∞).

Q: What is the horizontal asymptote of the function f(x) = 2(5)^x + 2 - 6?

A: The function has a horizontal asymptote at y = -6, since as x approaches negative infinity, the function approaches -6.

Q: What is the y-intercept of the function f(x) = 2(5)^x + 2 - 6?

A: The function has a y-intercept at (0, -4), since f(0) = 2(5)^0 + 2 - 6 = -4.

Q: How can I graph the function f(x) = 2(5)^x + 2 - 6?

A: You can use various methods, including creating a table of values, using a graphing calculator, or using the online graphing tool Desmos.

Q: What is the base of the exponential function f(x) = 2(5)^x + 2 - 6?

A: The base of the exponential function is 5, which is greater than 1, indicating that the function will increase rapidly as x increases.

Q: How can I use Desmos to graph the function f(x) = 2(5)^x + 2 - 6?

A: To graph the function using Desmos, follow these steps:

  1. Open Desmos and create a new graph.
  2. Enter the function f(x) = 2(5)^x + 2 - 6 in the input box.
  3. Click on the "Graph" button to graph the function.

Conclusion

Graphing the function f(x) = 2(5)^x + 2 - 6 is a complex task that requires a deep understanding of exponential functions and their properties. In this article, we have explored the properties of the function, its graph, and how to graph it using various methods. We have also used Desmos to graph the function and create a table of values. By understanding the properties of exponential functions and how to graph them, we can gain a deeper understanding of various mathematical concepts and apply them to real-world problems.

References