Section 3.2: Introduction to Polynomial Functions

Determine zeros of functions

Examples 3, SPK 2, 3

f(x)→∞

"f(x) approaches infinity" (the y value becomes infinitely large in the positive direction)

f(x)→-∞

"f(x) approaches negative infinity" (the y value becomes infinitely "large" in the negative direction)

x→∞

"x approaches infinity" (x becomes infinitely large in the positive direction)

x→-∞

"x approaches negative infinity" ( x becomes infinitely "large" in the negative direction)

To graph a polynomial function defined by y = f(x),

1. Use the leading term to determine the end behavior of the graph. 2. Determine the y-intercept by evaluating f(0). 3. Determine the real zeros of f and their multiplicities (these are the x-intercepts of the graph of f). 4. Plot the x- and y-intercepts and sketch the end behavior. 5. Draw a sketch starting from the left-end behavior. Connect the x- and y-intercepts in the order that they appear from left to right. Remember: cross the x-axis if the zero has an odd multiplicity, touch but not cross if the zero has an even multiplicity. 6.If the test for symmetry is easy to apply, plot additional points. f is an even function (symmetric to the y-axis) if f(-x) = f(x). f is an odd function (symmetric to the origin) if f(-x) = -f(x). 7. Plot more points if a greater level of accuracy is desired. In particular, to estimate the location of turning points, find several points between two consecutive x-intercepts. Example 7, 8

Intermediate Value Theorem:

Let f be a polynomial function. For a < b, if f (a) and f (b) have opposite signs, then f has at least one zero on the interval [a, b]. Example 9, SKP5, 6, 7

Zero of Multiplicity

Number or times a factor appears in a polynomial. Example 4, SKP4

Turning points

The turning points of a polynomial function are the points where the function changes from increasing to decreasing or vice versa. The turning points correspond to points where the function has relative maxima and relative minima. -The graph of a polynomial function of degree n has at most n - 1 turning points. Example 6

If c is a zero of EVEN multiplicity, then the graph TOUCHES the x-axis at c.

True

If c is a zero of ODD multiplicity, then the graph CROSSES the x-axis at c.

True

End behavior of a function

general direction that the function follow as x approaches ∞or -∞?

Determine End Behavior of a Polynomial

use the following chart f(x)=An(X^n) Example 1,2 SKP 1