Polynomial Functions and Their Graphs. Definition of a Polynomial Function: numbers, with a n 0. The function defined by

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1 Polynomial Functions and Their Graphs Definition of a Polynomial Function: Let n be a nonnegative number and let a n, a n 1, a 2, a 1, a 0 be real numbers, with a n 0. The function defined by f(x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 is called a polynomial function of degree n. The number a n, the coefficient of the variable to the highest power, is called the leading coefficient. The maximum number of turning points is given by (n-1) Polynomial functions are continuous, there are no holes or breaks, it is smooth, and there are no sharp curves. The domain of all polynomial functions is the set of all real numbers. The following are examples of polynomial functions The following are not polynomial functions 1 Page

2 Leading Term Behavior (Leading Coefficient Test) If a n x n is the leading term of a polynomial function, then the behavior of the graph as x or x can be described as follows: If n is even, and a n > 0 If n is even, and a n < 0 If n is odd, and a n > 0 If n is odd, and a n < 0 Given the graphs, give the sign of the leading term, and the degree. 2 Page

3 Examples Give the end behavior and degree of the following polynomials a) f(x) = 4x 2 + 9x x 2 + 4x + 1 b) f(x) = 4x 3 + 3x 2 x 3 c) f(x) = 4x 3 (x 1) 2 (x + 5) 3 Page

4 Solving Equations by Factoring Finding Zeros If f is a polynomial function, then the values of x for which f(x) equals zero are called the zeros of f. These values of x are the roots, or solutions of the polynomial equation f(x) = 0. Each real root of the polynomial equation appears as an x-intercept of the graph of the polynomial function. To get the zeros of a polynomial function. Step 1: Get zero on one side of equation Step 2: Factor completely Step 3: Set each factor equal to zero and solve to get the answers. (if ab=0, then a=0 or b=0) Example Find the zeros of the following polynomial functions a) y 2 6y = 27 b) 2x 2 3x = 35 c) x 3 2x 2 9x = 18 4 Page

5 To find the zeros of a polonomail equation by calculator: Step1: Graph the function - Hit the Y= button near the top of your calculator - Enter your function - Hit the GRAPH button near the top of your calculator. Step2: Adjust the view of the graph so that the x-intercepts are visible in the screen. - Hit the ZOOM button near the top of your calculator to make these adjustments. For Zeros Step3: Select Zero - When graph is showing hit 2 nd (orange button) Calc (option listed above TRACE ) button. - Then select zero - Now the bottom of the screen says left bound? (move cursor to a point left of the vertex, and hit enter) - Now the bottom of the screen says right bound? (move cursor to a point right of the vertex, and hit enter) - Now the bottom of the screen says guess? (move cursor to the vertex or very close to it, and hit enter) - The calculator will display the x-intercept at the bottom of the screen. - Repeat this process for each x-intercept. 5 Page

6 Multiplicity of zeros- If r is a zero of even multiplicity (exponent of the zero), then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether the multiplicity of a zero is even or odd, graphs tend to flatten out near zeros with multiplicity greater than one. Example f(x) = 1 (x + 1)(2x 2 3)2 Find the zeros and their multiplicity. And graph the polynomial function. 6 Page

7 Intermediate Value Theorem For any polynomial function P(x) with real coefficients, suppose that for a b, P(a) and P(b) are opposite signs. Then the function has a real zero between a and b. Example Determine if the polynomial function f(x) = x 3 + x 2 6x has a real zero between -4 and -2. Example Determine if the polynomial function f(x) = x 3 + x 2 6x has a real zero between -1 and 3. 7 Page

8 Graphing a polynomial function by hand. If P(x) is a polynomial function of degree n, the graph of the function has. - At most n real zeros, and thus at most n x-intercepts. - At most (n-1) turning points. Steps for Graphing Step 1: Use leading term test to determine the end behavior. Step 2: Find the zeros of the function; draw a line diagram. Step 3: Draw a chart and pick test values in your intervals (intervals determined by your x-int.) Step 4: Determine you y-intercept (i.e. solve for P(0)) Step 5: Choose any additional points that may be useful. Example Graph the polynomial function P(x) = 2x 4 + 3x 3 Step1: Step 2: 8 Page

9 Step3: Step4: 9 Page

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n- 1,, a 2, a 1, a 0, be real numbers with a n 0. The function defined by f (x) a