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)=x³+4x²-8x-8; [-3.8, -3]

Verified step by step guidance

1. Understand that turning points of a polynomial function occur where its derivative equals zero, as these points correspond to local maxima or minima on the graph.
2. Find the first derivative of the function \( f(x) = x^3 + 4x^2 - 8x - 8 \). Use the power rule for differentiation: \( f'(x) = 3x^2 + 8x - 8 \).
3. Use the graphing calculator to plot the derivative function \( f'(x) = 3x^2 + 8x - 8 \) over the domain interval \( [-3.8, -3] \) to identify where \( f'(x) = 0 \) within this interval.
4. Find the x-values where the derivative equals zero (the roots of \( f'(x) \)) within the given domain. These x-values are the x-coordinates of the turning points.
5. Substitute each x-value found back into the original function \( f(x) \) to calculate the corresponding y-coordinates, giving the coordinates of the turning points to the nearest hundredth.

Key Concepts

Turning Points of a Polynomial Function

Turning points are points on the graph where the function changes direction from increasing to decreasing or vice versa. For polynomial functions, these correspond to local maxima or minima and occur where the derivative equals zero. Identifying turning points helps understand the shape and behavior of the graph.

Using a Graphing Calculator to Find Turning Points

A graphing calculator can plot the function and use built-in tools to locate turning points by finding where the slope (derivative) is zero. It allows zooming into a specific domain interval and provides numerical coordinates, which can be rounded to the nearest hundredth for precision.

Domain Restriction and Interval Analysis

Restricting the domain to a specific interval, such as [-3.8, -3], limits the search for turning points to that range. This focuses the analysis on relevant parts of the graph and ensures that only turning points within the given interval are considered, which is important for accurate and context-specific answers.