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^4-7x^3+13x^2+6x-28; [-1, 0]

Verified step by step guidance

Step 1: Understand that turning points of a polynomial function are points where the function changes direction, which occur at local maxima or minima.

Step 2: To find turning points, first find the derivative of the function \( f(x) = x^4 - 7x^3 + 13x^2 + 6x - 28 \). The derivative \( f'(x) \) will help identify critical points.

Step 3: Calculate the derivative: \( f'(x) = 4x^3 - 21x^2 + 26x + 6 \).

Step 4: Set the derivative equal to zero to find critical points: \( 4x^3 - 21x^2 + 26x + 6 = 0 \). Solve this equation for \( x \) within the interval \([-1, 0]\).

Step 5: Use a graphing calculator to plot \( f(x) \) and \( f'(x) \) to visually identify the turning points within the interval \([-1, 0]\). The x-coordinates where \( f'(x) = 0 \) are the turning points. Evaluate \( f(x) \) at these x-coordinates to find the corresponding y-coordinates.

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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 equals zero. Identifying turning points is crucial for understanding the shape and behavior of the graph within a specified interval.

Graphing Calculator Usage

A graphing calculator is a powerful tool that allows users to visualize functions and perform complex calculations. It can be used to find derivatives, evaluate functions at specific points, and graph polynomial equations. Familiarity with the calculator's features is essential for efficiently finding turning points and interpreting the results.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. The degree of the polynomial determines its general shape and the number of turning points it can have. Understanding the characteristics of polynomial functions is vital for analyzing their graphs and behavior over specified intervals.