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. ƒ(x)=x^3+4x^2-8x-8; [-3.8, -3]

Verified step by step guidance

Step 1: Enter the polynomial function \( f(x) = x^3 + 4x^2 - 8x - 8 \) into the graphing calculator.

Step 2: Set the viewing window of the graphing calculator to the domain interval \([-3.8, -3]\).

Step 3: Use the graphing calculator's feature to find the turning points (local maxima and minima) within the specified interval.

Step 4: Identify the coordinates of the turning points by analyzing the graph within the interval.

Step 5: Round the coordinates of the turning points to the nearest hundredth as required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Turning Points

Turning points are points on a graph where the function changes direction from increasing to decreasing or vice versa. For polynomial functions, these points occur where the first derivative of the function equals zero. Identifying turning points is crucial for understanding the shape and behavior of the graph.

Derivatives

The derivative of a function measures the rate at which the function's value changes at any given point. For polynomial functions, the first derivative can be calculated using power rules. Finding the derivative is essential for determining the turning points, as these occur where the derivative is zero.

Graphing Calculators

Graphing calculators are powerful tools that can plot functions, calculate derivatives, and find specific points on graphs. They can be used to visualize the behavior of polynomial functions and to accurately determine coordinates of turning points within a specified domain. Familiarity with using a graphing calculator is important for efficiently solving problems in calculus and algebra.