Graph This Function: ${ Y = \frac{5}{6}(3)^x }$Plot Two Points To Graph The Function.
Introduction
Graphing functions is an essential skill in mathematics, particularly in algebra and calculus. It allows us to visualize the behavior of a function and understand its properties. In this article, we will graph the function y = (5/6)(3)^x and explore its characteristics.
Understanding the Function
The given function is y = (5/6)(3)^x. This is an exponential function, where the base is 3 and the coefficient is 5/6. The function can be rewritten as y = (5/6) * 3^x.
Properties of Exponential Functions
Exponential functions have several properties that are essential to understand when graphing them. Some of these properties include:
- Domain: The domain of an exponential function is all real numbers.
- Range: The range of an exponential function is all positive real numbers.
- End behavior: As x approaches negative infinity, y approaches 0. As x approaches positive infinity, y approaches infinity.
- Intercepts: The y-intercept is the point where x = 0. The x-intercept is the point where y = 0.
Graphing the Function
To graph the function y = (5/6)(3)^x, we need to find two points that lie on the graph. We can do this by substituting different values of x into the function and finding the corresponding values of y.
Finding Two Points
Let's find two points on the graph by substituting x = 0 and x = 1 into the function.
- For x = 0: y = (5/6)(3)^0 = (5/6)(1) = 5/6
- For x = 1: y = (5/6)(3)^1 = (5/6)(3) = 5/2
Plotting the Points
Now that we have found two points on the graph, we can plot them on a coordinate plane.
| x | y |
|---|---|
| 0 | 5/6 |
| 1 | 5/2 |
Graphing the Function
Using the two points we found, we can graph the function y = (5/6)(3)^x. The graph will be a curve that passes through the two points.
Characteristics of the Graph
The graph of the function y = (5/6)(3)^x has several characteristics that are worth noting.
- Asymptote: The graph has a horizontal asymptote at y = 0.
- Intercepts: The y-intercept is (0, 5/6) and the x-intercept is (1, 0).
- End behavior: As x approaches negative infinity, y approaches 0. As x approaches positive infinity, y approaches infinity.
Conclusion
Graphing the function y = (5/6)(3)^x allows us to visualize its behavior and understand its properties. We found two points on the graph by substituting different values of x into the function and plotted them on a coordinate plane. The graph has several characteristics, including a horizontal asymptote, intercepts, and end behavior.
Applications of Graphing Functions
Graphing functions has several applications in mathematics and other fields. Some of these applications include:
- Modeling real-world phenomena: Graphing functions can be used to model real-world phenomena, such as population growth, chemical reactions, and economic systems.
- Solving equations: Graphing functions can be used to solve equations, such as quadratic equations and systems of equations.
- Optimization: Graphing functions can be used to optimize functions, such as finding the maximum or minimum value of a function.
Final Thoughts
Introduction
In our previous article, we graphed the function y = (5/6)(3)^x and explored its characteristics. In this article, we will answer some common questions about graphing this function.
Q: What is the domain of the function y = (5/6)(3)^x?
A: The domain of the function y = (5/6)(3)^x is all real numbers. This means that the function is defined for all values of x.
Q: What is the range of the function y = (5/6)(3)^x?
A: The range of the function y = (5/6)(3)^x is all positive real numbers. This means that the function will always produce a positive value for y.
Q: What is the y-intercept of the function y = (5/6)(3)^x?
A: The y-intercept of the function y = (5/6)(3)^x is (0, 5/6). This means that when x = 0, y = 5/6.
Q: What is the x-intercept of the function y = (5/6)(3)^x?
A: The x-intercept of the function y = (5/6)(3)^x is (1, 0). This means that when y = 0, x = 1.
Q: How do I graph the function y = (5/6)(3)^x?
A: To graph the function y = (5/6)(3)^x, you can use a graphing calculator or a computer program. You can also plot two points on the graph and use them to draw the curve.
Q: What are some common mistakes to avoid when graphing the function y = (5/6)(3)^x?
A: Some common mistakes to avoid when graphing the function y = (5/6)(3)^x include:
- Not using a graphing calculator or computer program: Graphing the function by hand can be difficult and may lead to errors.
- Not plotting enough points: Plotting only two points may not give you a complete picture of the graph.
- Not using a coordinate plane: Failing to use a coordinate plane can make it difficult to accurately plot the points.
Q: How can I use the graph of the function y = (5/6)(3)^x to solve problems?
A: The graph of the function y = (5/6)(3)^x can be used to solve problems in a variety of ways. For example:
- Finding the value of y for a given value of x: You can use the graph to find the value of y for a given value of x.
- Finding the value of x for a given value of y: You can use the graph to find the value of x for a given value of y.
- Solving equations: You can use the graph to solve equations involving the function y = (5/6)(3)^x.
Q: What are some real-world applications of the function y = (5/6)(3)^x?
A: The function y = (5/6)(3)^x has several real-world applications, including:
- Modeling population growth: The function can be used to model population growth in a variety of situations.
- Modeling chemical reactions: The function can be used to model chemical reactions and predict the outcome of different reactions.
- Modeling economic systems: The function can be used to model economic systems and predict the outcome of different economic scenarios.
Conclusion
Graphing the function y = (5/6)(3)^x is an essential skill in mathematics that allows us to visualize the behavior of a function and understand its properties. By answering some common questions about graphing this function, we can gain a deeper understanding of its characteristics and how to use it to solve problems.