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)=x⁴-7x³+13x²+6x-28; [-1, 0]

Verified step by step guidance

1. Understand that turning points of a polynomial function occur where the derivative equals zero, as these points correspond to local maxima, minima, or points of inflection.
2. Find the first derivative of the function \( f(x) = x^4 - 7x^3 + 13x^2 + 6x - 28 \). Use the power rule for differentiation: \( f'(x) = 4x^3 - 21x^2 + 26x + 6 \).
3. Use a graphing calculator to graph the derivative \( f'(x) \) and identify the values of \( x \) in the domain interval \( [-1, 0] \) where \( f'(x) = 0 \). These \( x \)-values are the candidates for turning points.
4. For each \( x \)-value found, calculate the corresponding \( y \)-coordinate by substituting \( x \) back into the original function \( f(x) \). This gives the coordinates of the turning points.
5. Round the coordinates to the nearest hundredth as requested, and verify that these points lie within the given domain interval \( [-1, 0] \).

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 approximate turning points by graphing the function and using built-in tools like 'maximum' and 'minimum' to locate local extrema. It allows setting domain restrictions to focus on specific intervals, providing coordinates accurate to a desired decimal place, such as the nearest hundredth.

Domain Interval Restriction

Restricting the domain interval means considering the function only within a specified range of x-values, here [-1, 0]. This limits the search for turning points to that interval, ensuring that only relevant points are identified and reported, which is important for precise and context-specific analysis.