Solve The Equation $-(6)^{x-1}+5=\left(\frac{2}{3}\right)^{2-x}$ By Graphing. Round The Solution To The Nearest Tenth.

by ADMIN 121 views

Introduction

In this article, we will explore how to solve the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$ by graphing. This method involves plotting the two sides of the equation as separate functions and finding the point(s) of intersection, which represent the solution(s) to the equation. We will also discuss the steps involved in graphing exponential functions and how to use graphing tools to find approximate solutions to the nearest tenth. Solving equations graphically provides a visual understanding of the solutions and can be particularly useful for equations that are difficult to solve algebraically.

Understanding the Equation

Before we dive into graphing, let's understand the equation $-(6){x-1}+5=\left(\frac{2}{3}\right)2-x}$. We have two expressions $-(6)^{x-1+5$ on the left side and $\left(\frac{2}{3}\right)^{2-x}$ on the right side. Each side represents a function of x. The left side involves an exponential function with a base of 6, while the right side involves an exponential function with a base of 2/3. Our goal is to find the value(s) of x where these two functions are equal.

To begin, let's rewrite the equation to clearly define the two functions we will be graphing. We can define the left side as $f(x) = -(6)^{x-1} + 5$ and the right side as $g(x) = \left(\frac{2}{3}\right)^{2-x}$. By graphing these two functions, the points where the graphs intersect will give us the solutions to the original equation. The x-coordinates of these intersection points are the values of x that satisfy the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$.

Let's analyze the behavior of each function. The function $f(x) = -(6)^{x-1} + 5$ is an exponential function that is reflected over the x-axis and shifted vertically upward by 5 units. As x increases, the term $(6)^{x-1}$ increases exponentially, but the negative sign in front reflects the graph, causing it to decrease rapidly. The addition of 5 shifts the entire graph upward. On the other hand, the function $g(x) = \left(\frac{2}{3}\right)^{2-x}$ is an exponential function with a base less than 1, which means it is a decreasing function. The exponent 2-x further affects the graph, causing it to increase as x decreases. Understanding these behaviors will help us predict the general shape of the graphs and the possible points of intersection.

Graphing the Functions

Now, let's graph the two functions $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$. You can use a graphing calculator, online graphing tool like Desmos or Wolfram Alpha, or graph paper to plot these functions. We'll discuss the general process and key points to consider for each method.

Using a Graphing Calculator or Online Tool

Using a graphing calculator or online tool like Desmos is the most efficient way to graph these functions. These tools allow you to input the equations directly and view the graphs. Here are the steps:

  1. Input the functions: Enter $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$ into the graphing calculator or online tool.
  2. Adjust the viewing window: You may need to adjust the viewing window to see the intersection points clearly. Start with a standard window (e.g., -10 to 10 for both x and y) and then zoom in or out as needed. Look for the area where the two graphs intersect.
  3. Identify the intersection points: Use the tool's features to find the coordinates of the intersection points. Most graphing calculators and online tools have a function to calculate the intersection points directly.

Graphing by Hand

If you are graphing by hand, you'll need to plot several points for each function to get a good representation of the graph. Here are the steps:

  1. Create a table of values: Choose a range of x-values and calculate the corresponding y-values for both functions. For example, you could choose x-values like -2, -1, 0, 1, 2, and 3.
  2. Plot the points: Plot the points from your table on a coordinate plane.
  3. Connect the points: Draw smooth curves through the plotted points to represent the graphs of the functions. Remember the general shape of exponential functions: $f(x) = -(6)^{x-1} + 5$ will be a decreasing curve, and $g(x) = \left(\frac{2}{3}\right)^{2-x}$ will be an increasing curve.
  4. Identify the intersection points: Look for the points where the two curves intersect. These points represent the solutions to the equation.

Finding the Solutions

Once you have graphed the functions, the next step is to find the x-coordinates of the intersection points. These x-values are the solutions to the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$.

Using a Graphing Calculator or Online Tool

Graphing calculators and online tools have built-in functions to find intersection points. Typically, you can use the "intersect" function in the calculator's menu or the corresponding feature in the online tool. This function will calculate the coordinates of the intersection points for you.

To use this feature:

  1. Access the "intersect" function: Look for the "intersect" option in the calculator's menu (usually under "CALC" or "2nd TRACE") or in the online tool's features.
  2. Select the curves: The calculator or tool will prompt you to select the two curves you want to find the intersection of (i.e., $f(x)$ and $g(x)$).
  3. Provide a guess: You may need to provide a guess for the intersection point. This helps the calculator narrow down the search. Move the cursor close to the intersection point you are interested in.
  4. Calculate the intersection: The calculator or tool will calculate the coordinates of the intersection point. The x-coordinate is the solution to the equation.

Estimating from the Graph

If you graphed the functions by hand, you'll need to estimate the x-coordinates of the intersection points from the graph. This method is less precise but can give you a good approximation.

  1. Identify the intersection points: Locate the points where the two curves intersect on your graph.
  2. Estimate the x-coordinates: Draw a vertical line from each intersection point to the x-axis. Estimate the x-value where the line intersects the x-axis. This value is an approximate solution to the equation.

Rounding to the Nearest Tenth

The question asks us to round the solutions to the nearest tenth. This means we need to round the x-coordinates of the intersection points to one decimal place.

  1. Identify the digit in the tenths place: This is the first digit after the decimal point.
  2. Look at the next digit (the hundredths place): If this digit is 5 or greater, round up the digit in the tenths place. If it is less than 5, leave the digit in the tenths place as it is.

For example, if the x-coordinate of an intersection point is 1.27, we would round it to 1.3. If the x-coordinate is 1.23, we would round it to 1.2.

Example and Solution

Let's apply the steps we've discussed to solve the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$ by graphing.

  1. Graph the functions: Graph $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$ using a graphing calculator or online tool.
  2. Identify the intersection points: Observe the graphs and locate the points of intersection. You should see one intersection point.
  3. Find the x-coordinate: Use the "intersect" function on your calculator or tool to find the x-coordinate of the intersection point. The approximate value is x ≈ 1.6.
  4. Round to the nearest tenth: The x-coordinate rounded to the nearest tenth is 1.6.

Therefore, the solution to the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$ rounded to the nearest tenth is x ≈ 1.6.

Conclusion

In this article, we have demonstrated how to solve the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$ by graphing. This method involves graphing the two sides of the equation as separate functions and finding the x-coordinates of the intersection points. We discussed the steps involved in graphing exponential functions, using graphing tools, and estimating solutions from a graph. We also covered how to round the solutions to the nearest tenth.

Graphing is a powerful tool for solving equations, especially those that are difficult to solve algebraically. It provides a visual representation of the solutions and can help you understand the behavior of the functions involved. By following the steps outlined in this article, you can confidently solve equations graphically and find approximate solutions to a high degree of accuracy. Remember to always double-check your solutions and consider the limitations of graphical methods, such as the potential for estimation errors.

Therefore, the solution to the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$ rounded to the nearest tenth is approximately x ≈ 1.6.

Find Approximate Solutions to Equations Graphically

Solving equations graphically is a powerful technique, especially for equations that are difficult to solve using algebraic methods. The graphical method involves plotting the functions on both sides of the equation and identifying the points of intersection. The x-coordinates of these points represent the solutions to the equation. This approach not only provides a visual representation of the solutions but also helps in understanding the behavior of the functions involved.

One of the primary advantages of solving equations graphically is its applicability to a wide range of functions, including exponential, logarithmic, trigonometric, and polynomial functions. While algebraic methods might be cumbersome or impossible to apply to certain equations, graphical methods can often provide approximate solutions with ease. Furthermore, graphical solutions offer insights into the number of solutions an equation might have, which can be challenging to determine algebraically.

Steps for Solving Equations Graphically

To effectively solve an equation graphically, follow these steps:

  1. Rewrite the equation: If necessary, rewrite the equation so that one side is equal to the other. For example, if you have the equation $f(x) = g(x)$, you already have the functions separated.
  2. Define the functions: Let $y_1 = f(x)$ and $y_2 = g(x)$. This step is crucial for graphing each side of the equation as a separate function.
  3. Graph the functions: Use a graphing calculator, an online graphing tool (such as Desmos or Wolfram Alpha), or graph paper to plot the functions $y_1$ and $y_2$. When using a graphing calculator or online tool, input the equations and adjust the viewing window to ensure the intersection points are visible.
  4. Identify the intersection points: Locate the points where the graphs of $y_1$ and $y_2$ intersect. These points represent the values of x for which $f(x) = g(x)$.
  5. Determine the x-coordinates: Find the x-coordinates of the intersection points. These x-values are the solutions to the original equation. If using a graphing calculator or tool, the "intersect" function can help determine these coordinates precisely.
  6. Approximate the solutions: If the solutions are not exact, estimate the x-coordinates from the graph. For more accurate approximations, use the numerical methods available in graphing calculators or software.

Graphing Tools and Their Advantages

Various tools are available for graphing functions, each with its advantages:

  • Graphing Calculators: Devices like those from Texas Instruments (TI-84, TI-Nspire) and Casio offer robust graphing capabilities. They can handle complex functions, find intersection points, and provide numerical approximations. Graphing calculators are particularly useful in educational settings and for exams where access to computers might be restricted.
  • Online Graphing Tools: Websites such as Desmos and Wolfram Alpha provide user-friendly interfaces for graphing functions. Desmos is known for its intuitive design and dynamic graphing capabilities, making it excellent for exploring mathematical concepts. Wolfram Alpha offers powerful computational abilities, including solving equations and providing detailed analysis of functions.
  • Graph Paper: While less precise, graph paper can be used to sketch graphs manually. This method helps in understanding the fundamental principles of graphing and is useful for simple functions.

Example: Solving an Equation Graphically

Consider the equation $x^2 - 4 = x + 2$. To solve this graphically:

  1. Rewrite: The equation is already in a suitable form.
  2. Define functions: Let $y_1 = x^2 - 4$ and $y_2 = x + 2$.
  3. Graph: Plot the functions $y_1$ and $y_2$ using a graphing calculator or an online tool. The graph of $y_1$ is a parabola, and the graph of $y_2$ is a straight line.
  4. Identify intersections: Observe the points where the parabola and the line intersect. There are two intersection points.
  5. Determine x-coordinates: Use the "intersect" function or estimate from the graph. The intersection points are at $x = -2$ and $x = 3$.
  6. Solutions: The solutions to the equation are $x = -2$ and $x = 3$.

Limitations and Considerations

While graphical methods are versatile, they have limitations:

  • Approximations: Graphical solutions are often approximations, especially when estimating from hand-drawn graphs. Numerical methods and graphing tools can provide more precise results.
  • Hidden Solutions: If the viewing window is not appropriately set, some intersection points (and thus solutions) might be missed. It is essential to explore different ranges of x and y values to ensure all solutions are captured.
  • Complex Solutions: For equations with complex solutions, graphical methods are not directly applicable as they primarily deal with real-number solutions.

Conclusion

Solving equations graphically is a valuable technique that complements algebraic methods. It provides a visual understanding of solutions and is particularly useful for equations that are difficult to solve analytically. By following the steps outlined and utilizing appropriate graphing tools, you can effectively find approximate solutions to a wide range of equations. Always consider the limitations of graphical methods and use them in conjunction with other techniques for a comprehensive approach to problem-solving.

Repair Input Keyword

Solve the equation $-(6){x-1}+5=\left(\frac{2}{3}\right){2-x}$ by graphing. Round the solution to the nearest tenth.

SEO Title

Solving $-(6){x-1}+5=(\frac{2}{3}){2-x}$ by Graphing: A Step-by-Step Guide