A polynomial function is a function of the form: All of these coefficients are real numbers n must be a positive integer Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 … The degree of the polynomial is the largest power on any x term in the polynomial.

Polynomial Functions The largest exponent within the polynomial determines the degree of the polynomial. Polynomial Function in General Form Degree Name of Function 1Linear 2Quadratic 3Cubic 4Quartic

x 0 Not a polynomial because of the square root since the power is NOT an integer Determine which of the following are polynomial functions. If the function is a polynomial, state its degree. A polynomial of degree 4. A polynomial of degree 0. We can write in an x 0 since this = 1. Not a polynomial because of the x in the denominator since the power is negative

Graphs of polynomials are smooth and continuous. No sharp corners or cusps No gaps or holes, can be drawn without lifting pencil from paper This IS the graph of a polynomial This IS NOT the graph of a polynomial

Smooth, Continuous Graphs Two important features of the graphs of polynomial functions are that they are smooth and continuous. By smooth, we mean that the graph contains only rounded curves with no sharp corners. By continuous, we mean that the graph has no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. These ideas are illustrated in the figure. Smooth rounded corner x x y y

Even polynomial functions: All exponents of the polynomial are even Cool feature: always symmetrical about the y-axis f(-x) = f(x) Odd polynomial functions: All exponents of the polynomial are odd Cool feature: always symmetrical about the origin f(-x) = - f(x)

Let’s look at the graph of where n is an even integer. and grows steeper on either side Notice each graph looks similar to x 2 but is wider and flatter near the origin between –1 and 1 The higher the power, the flatter and steeper

Let’s look at the graph of where n is an odd integer. and grows steeper on either side Notice each graph looks similar to x 3 but is wider and flatter near the origin between –1 and 1 The higher the power, the flatter and steeper

Let’s graph Looks like x 2 but wider near origin and steeper after 1 and -1 Reflects about the x-axis Translates up 2 So as long as the function is a transformation of x n, we can graph it, but what if it ’ s not? We ’ ll learn some techniques to help us determine what the graph looks like in the next slides.

LEFT RIGHT and HAND BEHAVIOR OF A GRAPH The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behavior of a graph.

Even Degree: Positive coefficient Negative Coefficient If the leading coefficient is positive, the graph rises to the left and to the right. If the leading coefficient is negative, the graph falls to the left and to the right. Rises right Rises left Falls left Falls right The Leading Coefficient Test

Even degree polynomials rise on both the left and right hand sides of the graph (like x 2 ) if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term. left hand behavior: rises right hand behavior: rises

Even degree polynomials fall on both the left and right hand sides of the graph (like - x 2 ) if the coefficient is negative. left hand behavior: falls right hand behavior: falls turning points in the middle

The Leading Coefficient Test For an odd degree: Positive coefficient negative coefficient If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises right Falls left Falls right Rises left

Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x 3 ) if the coefficient is positive. left hand behavior: falls right hand behavior: rises turning Points in the middle

Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x 3 ) if the coefficient is negative. left hand behavior: rises right hand behavior: falls turning points in the middle

Determining End Behavior Match each function with its graph. A. B. C. D.

Quartic Polynomials Look at the two graphs and discuss the questions given below. 1. How can you check to see if both graphs are functions? 3. What is the end behavior for each graph? 4. Which graph do you think has a positive leading coeffient? Why? 5. Which graph do you think has a negative leading coefficient? Why? 2. How many x-intercepts do graphs A & B have? Graph B Graph A

A polynomial of degree n can have at most n-1 turning points (so whatever the degree is, subtract 1 to get the most times the graph could turn). doesn’t mean it has that many turning points but that’s the most it can have Let’s determine left and right hand behaviour for the graph of the function: degree is 4 which is even and the coefficient is positive so the graph will look like x 2 looks off to the left and off to the right. The graph can have at most 3 turning points How do we determine what it looks like near the middle?

x and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x. (0,30) To find the x intercept we put 0 in for y. Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that ’ s okay)

We found the x intercept by putting 0 in for f(x) or y (they are the same thing remember). So we call the x intercepts the zeros of the polynomial since it is where it = 0. These are also called the roots of the polynomial. Can you find the zeros of the polynomial? There are repeated factors. (x-1) is to the 3 rd power so it is repeated 3 times. If we set this equal to zero and solve we get 1. We then say that 1 is a zero of multiplicity 3 (since it showed up as a factor 3 times). What are the other zeros and their multiplicities? -2 is a zero of multiplicity 2 3 is a zero of multiplicity 1

Multiplicity and x-Intercepts If r is a zero of even multiplicity, 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 a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

So knowing the zeros of a polynomial we can plot them on the graph. If we know the multiplicity of the zero, it tells us whether the graph crosses the x axis at this point (odd multiplicities CROSS) or whether it just touches the axis and turns and heads back the other way (even multiplicities TOUCH). Let’s try to graph: You don’t need to multiply this out but figure out what the highest power on an x would be if multiplied out. In this case it would be an x 3. Notice the negative out in front. What would the left and right hand behavior be? What would the y intercept be? (0, 4) Find the zeros and their multiplicity 1 of mult. 1 (so crosses axis at 1) -2 of mult. 2 (so touches at 2)

Text Example Solution Because the degree is odd (n  3) and the leading coefficient, 1, is positive, the graph falls to the left and rises to the right, as shown in the figure. Use the Leading Coefficient Test to determine the end behavior of the graph of Graph the quadratic function f (x)  x 3  3x 2  x  3. Falls left y Rises right x

Step 1 Determine end behavior. Because the degree is even (n  4) and the leading coefficient, 1, is positive, the graph rises to the left and the right: Solution Graph: f (x)  x 4  2x 2 . y x Rises left Rises right Text Example

 x 2  1)  x 2  1)  Factor. (x  1) 2 (x  1) 2  0 Express the factoring in more compact notation. (x  1) 2  or (x  1) 2  0 Set each factor equal to zero. x   1 x  1 Solve for x. (x  1)(x  1)(x  1)(x  1)  0 Factor completely. Step 2 Find the x-intercepts (zeros of the function) by setting f (x)   x 4  2x 2  0 Solution Graph: f (x)  x 4  2x 2 . Text Example cont.

Step 2 We see that -1 and 1 are both repeated zeros with multiplicity 2. Because of the even multiplicity, the graph touches the x-axis at  1 and 1 and turns around. Furthermore, the graph tends to flatten out at these zeros with multiplicity greater than one: Solution Graph: f (x)  x 4  2x 2 . Rises left Rises right x y 1 Text Example cont.

f  0 4   1   Step 3 Find the y-intercept. Replace x with 0 in f (x)   x  4x  1. There is a y-intercept at 1, so the graph passes through (0, 1). Solution Graph: f (x)  x 4  2x 2 . 1 Rises left Rises right x y 1 Text Example cont.

Solution Graph: f (x)  x 4  2x 2 . y x Text Example cont.

Graphing a Polynomial Function f (x)  a n x n  a n-1 x n-1  a n-2 x n-2    a 1 x  a 0 (a n  0) 1.Use the Leading Coefficient Test to determine the graph's end behavior. 2.Find x-intercepts by setting f (x)  0 and solving the resulting polynomial equation. If there is an x- intercept at r as a result of (x  r) k in the complete factorization of f (x), then: a. If k is even, the graph touches the x-axis at r and turns around. b. If k is odd, the graph crosses the x-axis at r. c. If k > 1, the graph flattens out at (r, 0). 3. Find the y-intercept by setting x equal to 0 and computing f (0).

G RAPHING P OLYNOMIAL F UNCTIONS END BEHAVIOR FOR POLYNOMIAL FUNCTIONS C ONCEPT S UMMARY > 0even f (x)+  f (x) +  > 0odd f (x)–  f (x) +  < 0even f (x)–  f (x) –  < 0odd f (x)+  f (x) –  a n n x –  x + 

Example Find the x-intercepts and multiplicity of f(x) = 2(x+2) 2 (x-3) Solution: x=-2 is a zero of multiplicity 2 or even x=3 is a zero of multiplicity 1 or odd

 x 4  4x 3  4x 2  We now have a polynomial equation. Solution We find the zeros of f by setting f (x) equal to 0. x 2 (x 2  4x  4)  0 Factor out x 2. x 2 (x  2) 2  0 Factor completely. x 2  or (x  2) 2  0 Set each factor equal to zero. x  0 x  2 Solve for x. x 4  4x 3  4x 2  0 Multiply both sides by  1. (optional step) Find all zeros of f (x)   x 4  4x 3  4x 2. Text Example

x f (x) –3 –7 –2 3 – – Graphing Polynomial Functions Graph f (x) = x 3 + x 2 – 4 x – 1. S OLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. The degree is odd and the leading coefficient is positive, so f (x) – as x – and f (x) + as x +.

x f (x) –3 –21 –2 0 – –16 3 –105 The degree is even and the leading coefficient is negative, so f (x) – as x – and f (x) – as x +. Graphing Polynomial Functions Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x. S OLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.