Graph the polynomial in the given viewing rectangle. Find...

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places.
$$
y=x^{4}-18 x^{2}+32, \quad[-5,5] \text { by }[-100,100]
$$

Key Concepts

Graphing on a Viewing Rectangle

When graphing a function within a specified viewing rectangle, one must consider both the domain and range limits imposed by the rectangular boundaries. This ensures that key features such as intercepts, local extrema, and turning points are clearly visible and accurately represented within the context of the chosen coordinate system.

Local Extrema

Local extrema are points on a graph where a function reaches a local maximum (a peak) or a local minimum (a trough), meaning that the function's value is higher or lower than the values at nearby points. The identification of local extrema usually involves finding the critical points using differentiation and then determining the nature of each point with additional tests, like the second derivative test or a sign analysis.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. For smooth polynomial functions, setting the derivative equal to zero provides the 'critical numbers' where the function may have a local minimum or maximum. Identifying these points is a fundamental step in locating local extrema.

Polynomial Functions

Polynomial functions are algebraic expressions involving variables raised to whole number exponents and coefficients. They have smooth and continuous graphs, and their degree determines the end behavior and the maximum number of turning points. Understanding their structure is essential for analyzing and graphing them effectively.

Differentiation

Differentiation is the process of computing the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. For polynomial functions, differentiation is performed term-by-term, and it is a key tool in finding critical points where the slope equals zero.

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