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Chapter 9: Problem 45

Many nonlinear systems cannot be solved algebraically, so graphical analysis is the only way to determine the solutions of such systems. Use a graphing calculator to solve each nonlinear system. Give \(x\) - and \(y\) -coordinates to the nearest hundredth. $$\begin{aligned} &y=\log (x+5)\\\ &y=x^{2} \end{aligned}$$

Short Answer

Expert verified
The points of intersection are approximately (-3.68, 13.54) and (0.54, 0.29).

Step by step solution

01

Graph each equation

On a graphing calculator, input and plot the equations separately. The first equation is \( y = \log(x+5) \) and the second equation is \( y = x^2 \).
02

Identify the points of intersection

Look for the points where the graphs of the two equations intersect. These points are the solutions of the nonlinear system.
03

Determine the coordinates

Use the graphing calculator's feature to trace and find the exact coordinates of the points of intersection. Make sure to round each coordinate to the nearest hundredth.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Using a Graphing Calculator
To solve nonlinear systems graphically, a graphing calculator can be a powerful tool. The first step is to enter each equation into the calculator. For this exercise, input the equations \( y = \log(x+5) \) and \( y = x^2 \).* Here's how to do it:
  • Turn on your graphing calculator.
  • Access the graphing mode, usually by pressing a button labeled 'Y=' or similar.
  • Input \( y = \log(x+5) \) in the first available entry line.
  • Input \( y = x^2 \) in the next line.
Once entered, adjust the viewing window so you can see the curves clearly. This ensures that the intersection points, which are the system's solutions, will be visible.
Finding Intersection Points
After graphing the equations, the next step is to identify where the graphs intersect. These intersection points represent the solutions of the system and occur where both equations have the same \(x\) and \(y\) values.
To find these points:
  • Use the 'Calculate' or 'Trace' feature on your graphing calculator.
  • Navigate to the intersection points manually by following the graph with your cursor.
  • Confirm the intersection points using the calculator鈥檚 built-in feature, often labeled 'Intersect'.
Be sure to zoom in appropriately to get a precise view, ensuring that the coordinates you identify are as accurate as possible.
Rounding Coordinates
Once you identify the intersection points, it's important to round the coordinates to the nearest hundredth. This improves readability and precision in your final solution.
To round the coordinates:
  • Look at the third digit after the decimal point.
  • If it is 5 or greater, round the second digit up by one.
  • If it is less than 5, keep the second digit as it is.
For example, if the x-coordinate is 1.234, it rounds to 1.23. If it鈥檚 1.236, it rounds to 1.24.
By following these steps, you'll derive accurately rounded intersection points that represent the solution to the nonlinear system.

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