Graph The Function $f(x) = -3(3)^{x-3}$ On The Axes Below. You Must Plot The Asymptote And Any Two Points With Integer Coordinates.Asymptote:

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Understanding the Function

The given function is f(x)=−3(3)x−3f(x) = -3(3)^{x-3}. This is an exponential function with a base of 3 and a coefficient of -3. The function has a horizontal asymptote, which is a horizontal line that the function approaches as x goes to positive or negative infinity.

Graphing the Asymptote

To graph the asymptote, we need to find the horizontal line that the function approaches as x goes to positive or negative infinity. Since the function is an exponential function, the asymptote will be a horizontal line at y = 0.

Graphing the Function

To graph the function, we need to plot two points with integer coordinates. We can choose any two values of x and calculate the corresponding values of y.

Let's choose x = 0 and x = 1.

For x = 0, we have:

f(0)=−3(3)0−3=−3(3)−3=−3⋅127=−19f(0) = -3(3)^{0-3} = -3(3)^{-3} = -3 \cdot \frac{1}{27} = -\frac{1}{9}

For x = 1, we have:

f(1)=−3(3)1−3=−3(3)−2=−3⋅19=−13f(1) = -3(3)^{1-3} = -3(3)^{-2} = -3 \cdot \frac{1}{9} = -\frac{1}{3}

So, the two points with integer coordinates are (0, -1/9) and (1, -1/3).

Plotting the Graph

To plot the graph, we can use a graphing calculator or a computer program. We can plot the asymptote at y = 0 and the two points with integer coordinates. We can then use the graphing tool to draw the graph of the function.

Discussion

The graph of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} is a decreasing exponential function. The function approaches the horizontal asymptote at y = 0 as x goes to positive or negative infinity. The graph of the function is a smooth curve that approaches the asymptote as x increases or decreases.

Properties of the Function

The function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} has several properties that are important to understand.

  • Domain: The domain of the function is all real numbers, x ∈ (-∞, ∞).
  • Range: The range of the function is all real numbers, y ∈ (-∞, 0).
  • Horizontal Asymptote: The horizontal asymptote of the function is y = 0.
  • Vertical Asymptote: The function has no vertical asymptote.
  • End Behavior: The function approaches the horizontal asymptote as x goes to positive or negative infinity.

Conclusion

In conclusion, the graph of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} is a decreasing exponential function that approaches the horizontal asymptote at y = 0 as x goes to positive or negative infinity. The function has several important properties, including a domain of all real numbers, a range of all real numbers, and a horizontal asymptote at y = 0.

Graph

Here is the graph of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}:

### Graph of the Function

#### Asymptote

*   Horizontal Asymptote: y = 0

#### Points with Integer Coordinates

*   (0, -1/9)
*   (1, -1/3)

#### Graph

The graph of the function is a smooth curve that approaches the horizontal asymptote as x increases or decreases.

References

Frequently Asked Questions

Q: What is the domain of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

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

Q: What is the range of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

A: The range of the function is all real numbers, y ∈ (-∞, 0).

Q: What is the horizontal asymptote of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

A: The horizontal asymptote of the function is y = 0.

Q: Does the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} have a vertical asymptote?

A: No, the function does not have a vertical asymptote.

Q: What is the end behavior of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

A: The function approaches the horizontal asymptote as x goes to positive or negative infinity.

Q: How do I graph the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

A: To graph the function, you can use a graphing calculator or a computer program. You can plot the asymptote at y = 0 and the two points with integer coordinates. You can then use the graphing tool to draw the graph of the function.

Q: What are some important properties of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

A: Some important properties of the function include:

  • Domain: all real numbers, x ∈ (-∞, ∞)
  • Range: all real numbers, y ∈ (-∞, 0)
  • Horizontal Asymptote: y = 0
  • Vertical Asymptote: none
  • End Behavior: approaches horizontal asymptote as x goes to positive or negative infinity

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} in real-world applications?

A: Yes, the function can be used in real-world applications such as modeling population growth or decay, chemical reactions, and financial investments.

Q: How do I find the value of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} at a specific value of x?

A: To find the value of the function at a specific value of x, you can plug the value of x into the function and simplify.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is constant?

A: No, the function is an exponential function, which means that the rate of change is not constant.

Q: How do I determine the type of function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} is?

A: The function is an exponential function because it has the form f(x)=abxf(x) = ab^x, where a and b are constants.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is increasing or decreasing?

A: Yes, the function can be used to model a situation where the rate of change is increasing or decreasing.

Q: How do I find the derivative of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

A: To find the derivative of the function, you can use the chain rule and the fact that the derivative of bxb^x is bxln⁡bb^x \ln b.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is constant and the function is increasing or decreasing?

A: No, the function is an exponential function, which means that the rate of change is not constant.

Q: How do I find the integral of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3}?

A: To find the integral of the function, you can use the fact that the integral of bxb^x is bxln⁡b\frac{b^x}{\ln b}.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing?

A: Yes, the function can be used to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing.

Q: How do I determine the type of asymptote the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} has?

A: The function has a horizontal asymptote at y = 0.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is constant and the function is increasing or decreasing?

A: No, the function is an exponential function, which means that the rate of change is not constant.

Q: How do I find the value of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} at a specific value of x using a graphing calculator?

A: To find the value of the function at a specific value of x using a graphing calculator, you can enter the function and the value of x into the calculator and press the "calculate" button.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing?

A: Yes, the function can be used to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing.

Q: How do I determine the type of function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} is using a graphing calculator?

A: To determine the type of function using a graphing calculator, you can enter the function into the calculator and observe the graph. If the graph is a smooth curve that approaches the horizontal asymptote as x increases or decreases, then the function is an exponential function.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is constant and the function is increasing or decreasing?

A: No, the function is an exponential function, which means that the rate of change is not constant.

Q: How do I find the value of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} at a specific value of x using a computer program?

A: To find the value of the function at a specific value of x using a computer program, you can enter the function and the value of x into the program and run the program.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing?

A: Yes, the function can be used to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing.

Q: How do I determine the type of function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} is using a computer program?

A: To determine the type of function using a computer program, you can enter the function into the program and observe the graph. If the graph is a smooth curve that approaches the horizontal asymptote as x increases or decreases, then the function is an exponential function.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is constant and the function is increasing or decreasing?

A: No, the function is an exponential function, which means that the rate of change is not constant.

Q: How do I find the derivative of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} using a computer program?

A: To find the derivative of the function using a computer program, you can enter the function into the program and use the program's built-in derivative function.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing?

A: Yes, the function can be used to model a situation where the rate of change is increasing or decreasing and the function is increasing or decreasing.

Q: How do I find the integral of the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} using a computer program?

A: To find the integral of the function using a computer program, you can enter the function into the program and use the program's built-in integral function.

Q: Can I use the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} to model a situation where the rate of change is constant and the function is increasing or decreasing?

A: No, the function is an exponential function, which means that the rate of change is not constant.

Q: How do I determine the type of asymptote the function f(x)=−3(3)x−3f(x) = -3(3)^{x-3} has using a computer program?

A: To determine the type of asymptote using a computer program, you can enter the function into the