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Chapter 3: Problem 88

Describe a strategy for graphing a polynomial function. In your description, mention intercepts, the polynomial's degree, and turning points.

Short Answer

Expert verified
To graph a polynomial: 1) Find the x- and y- intercepts; 2) Determine the degree and leading coefficient of the polynomial to understand its general shape; 3) Find the turning points by setting the derivative of the function equal to zero and solve for x.

Step by step solution

01

Identify the X and Y Intercepts

Intercepts are the points where the graph crosses the x-axis and y-axis. The x-intercepts are found by setting the polynomial equal to zero and solving for x. The y-intercept is found by setting \(x = 0\). Compute these values.
02

Determine the Degree and Leading Coefficient

The degree of the polynomial is the highest power of x in the polynomial equation, which determines the general shape of the graph. The sign of the leading coefficient determines whether the graph opens up or down. If the degree is odd, and the leading coefficient is positive, the end behavior will be in opposite directions (from down to up). If it's negative, the end will behave from up to down.
03

Find the Turning Points

The turning points of a polynomial function occur when the graph changes direction from moving upward to moving downward, or vice versa. A polynomial of degree n will have at most n-1 turning points. To find the exact x-values of turning points, take the derivative of the polynomial, set it equal to zero, and solve for x.

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