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Chapter 3: Problem 88

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. $$f(x)=x^{3}-x+3 ;[-1,0]$$

Short Answer

Expert verified
The coordinates of the turning points are approximately \((-0.58, 3.42)\).

Step by step solution

01

Input the Function

Enter the function \( f(x) = x^3 - x + 3 \) into the graphing calculator.
02

Set the Domain

Set the domain interval to \([-1, 0]\) on the graphing calculator to focus on the required region for finding the turning points.
03

Graph the Function

Graph the function \( f(x) = x^3 - x + 3 \) in the specified domain interval. Observe the graph and identify the regions where the turning points appear.
04

Use the Calculator's Turning Point Feature

Utilize the graphing calculator's feature to find the turning points (local maxima and minima) within the given interval.
05

Record the Coordinates

Record the coordinates of the turning points as provided by the calculator. Ensure to round the values to the nearest hundredth.

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Key Concepts

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

Graphing Polynomial Functions
Polynomial functions are mathematical expressions involving powers of a variable, typically written in the form of a sum of terms with coefficients and variables raised to various powers. To graph a polynomial function using a graphing calculator, you need to:
  • Enter the polynomial equation. In this case, the function is given as \( f(x) = x^3 - x + 3 \).
  • Set the appropriate viewing window to ensure the entire domain of interest is visible. For the interval \[ -1, 0 \], you may adjust the x-axis and y-axis ranges accordingly.
  • Graph the function. Most graphing calculators will automatically display the function after inputting the expression and setting the domain.
By viewing the graph, you can visualize where the function rises and falls, leading to the identification of key points such as turning points.
Local Maxima and Minima
In the context of polynomial functions, turning points are known as local maxima and minima.
  • A local maximum is a point where the function value is higher than any nearby points. It appears as the peak in the specified interval.
  • A local minimum, on the other hand, is a point where the function value is lower than any nearby points. It appears as a trough.
To find these points using a graphing calculator:
  • Enter the function into the calculator.
  • Set the domain interval you are interested in, as we set it to \[ -1, 0 \].
  • Use the calculator鈥檚 built-in features to determine the turning points within the interval. This may often be found under tools like 'calculate' or 'analyze'.
Once located, record the turning points. For the given example, you'd round the coordinates of these points to the nearest hundredth, ensuring precision in your answers.
Domain Interval
The domain of a function is the set of all possible input values (x-values) for which the function is defined. When graphing polynomial functions, it's important to set the correct domain interval to focus on the area of interest. In this exercise, the domain interval \[ -1, 0 \] means that only x-values between -1 and 0 will be considered. This is crucial for:
  • Reducing distractions from irrelevant parts of the graph.
  • Concentrating the analysis on the section where turning points need to be identified.
Set the domain by adjusting the viewing window on your graphing calculator. This way, when you plot the function, the graph displayed will only show the behavior of the function in the interval you specified. This precise focus helps in accurately locating and calculating the turning points within the desired range.

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