1. Bellwork
Reflection:
Go through your notebook and list what you learned your First quarter.
List what I should do as a teacher to make next quarter more successful for you.

2. Graphing Polynomial Functions
Chapter 4
Section1

3. Definition: polynomial(Take Note)
A polynomial is a monomial or a sum of monomials. A polynomial function is a function of the form
Where 𝑎 𝑛 ≠ 0, the exponents are all whole numbers, and the coefficients are all real numbers. For this function, 𝑎 𝑛 is the leading coefficient, n is the degree, and a 0 is the constant term. A polynomial function is in standard form when its terms are written in descending order of exponents from left to right.

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

6. Determine which of the following are polynomial functions
Determine which of the following are polynomial functions. If the function is a polynomial, state its degree.
A polynomial of degree 4.
We can write in an x0 since this = 1.
x 0
A polynomial of degree 0.
Not a polynomial because of the square root since the power is NOT an integer
Not a polynomial because of the x in the denominator since the power is negative

7. Identifying Polynomial Functions
Decide whether each function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient

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

9. The end behavior of a function’s graph is the behavior of the graph as x approaches positive infinity (+∞) or negative infinity (−∞). For the graph of a polynomial function, the end behavior is determined by the function’s degree and the sign of its leading coefficient.(Take Note)

10. Bellwork

11. (Take Note) 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 x2 but is wider and flatter near the origin between –1 and 1
The higher the power, the flatter and steeper

12. (Take Note) Let’s look at the graph of where n is an odd integer.
Notice each graph looks similar to x3 but is wider and flatter near the origin between –1 and 1
and grows steeper on either side
The higher the power, the flatter and steeper

13. Let’s graph Translates up 2 Reflects about the x-axis
Looks like x2 but wider near origin and steeper after 1 and -1
So as long as the function is a transformation of xn, 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.

14. Bellwork
Describing End Behavior

15. HAND BEHAVIOUR OF A GRAPH
RIGHT
LEFT
HAND BEHAVIOUR 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 behaviour of a graph.

16. Even degree polynomials rise on both the left and right hand sides of the graph (like x2) 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 behaviour: rises
right hand behaviour: rises

17. Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the coefficient is negative.
turning points in the middle
left hand behaviour: falls
right hand behaviour: falls

18. Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the coefficient is positive.
turning Points in the middle
right hand behaviour: rises
left hand behaviour: falls

19. Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the coefficient is negative.
turning points in the middle
left hand behaviour: rises
right hand behaviour: falls

20. How do we determine what it looks like near the middle?
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 x2 looks off to the left and off to the right.
How do we determine what it looks like near the middle?
The graph can have at most 3 turning points

21. To find the x intercept we put 0 in for y.
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.
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)
(0,30)

22. 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 3rd 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

23. What would the left and right hand behavior be?
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:
What would the left and right hand behavior be?
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 x3. Notice the negative out in front.
What would the y intercept be?
1 of mult. 1 (so crosses axis at 1)
-2 of mult. 2 (so touches at 2)
(0, 4)
Find the zeros and their multiplicity

24. Steps for Graphing a Polynomial
Determine left and right hand behaviour by looking at the highest power on x and the sign of that term.
Determine maximum number of turning points in graph by subtracting 1 from the degree.
Find and plot y intercept by putting 0 in for x
Find the zeros (x intercepts) by setting polynomial = 0 and solving.
Determine multiplicity of zeros.
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.

25. Find and plot y intercept by putting 0 in for x
Let’s graph:
Determine left and right hand behavior by looking at the highest power on x and the sign of that term.
Find and plot y intercept by putting 0 in for x
Determine maximum number of turns in graph by subtracting 1 from the degree.
Find the zeros (x intercepts) by setting polynomial = 0 and solving.
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.
Determine multiplicity of zeros.
0 multiplicity 2 (touches) 3 multiplicity 1 (crosses) -4 multiplicity 1 (crosses)
Zeros are: 0, 3, -4
Multiplying out, highest power would be x4
Degree is 4 so maximum number of turns is 3
Here is the actual graph. We did pretty good. If we’d wanted to be more accurate on how low to go before turning we could have plugged in an x value somewhere between the zeros and found the y value. We are not going to be picky about this though since there is a great method in calculus for finding these maxima and minima.

26. What is we thought backwards
What is we thought backwards? Given the zeros and the degree can you come up with a polynomial? Find a polynomial of degree 3 that has zeros –1, 2 and 3.
What would the function look like in factored form to have the zeros given above?
Multiply this out to get the polynomial. FOIL two of them and then multiply by the third one.

27. Sketching a Graph
Sketch a graph of the polynomial function f having these characteristics. • f is increasing when x < 0 and x > 4. • f is decreasing when 0 < x < 4. • f(x) > 0 when −2 < x < 3 and x > 5. • f(x) < 0 when x < −2 and 3 < x < 5. Use the graph to describe the degree and leading coefficient of f

28. Scetch a graph

29. Sketch a graph of the polynomial function f having these characteristics. • f is decreasing when x < −1.5 and x > 2.5; f is increasing when −1.5 < x < 2.5. • f(x) > 0 when x < −3 and 1 < x < 4; f(x) < 0 when −3 < x < 1 and x > 4. Use the graph to describe the degree and leading coefficient of f.

30. In Exercises 33–36, describe the x-values for which (a) f is increasing or decreasing, (b) f(x) > 0, and (c) f(x) < 0.