Download x → +∞ means

Advanced Functions
Warm Up Day 8
Name____________
1. Divide using Synthetic Division.
P(x) = 5x3-18x+2 by x-1
2. Divide using long division.
x4+3x3-5x+3x2-1 by x-4
3. Use the remainder theorem to find
P(c).
4. Use synthetic division and Factor
Theorem to determine whether the
given binomial is a factor of P(x).
P(x) = 3x3+4x2-27x-36, x-4
P(x) = -2x3-2x2-x-20, c=10
Now can you apply what you know to completely factor and solve polynomials?
If we know a factor, we can use synthetic division to rewrite the polynomial
as a product of the binomial and the reduced polynomial. Then can you
factor more?
5. Factor f(x) = 3x3-4x2-28x-16 completely given that x + 2 is a factor.
If we know a zero, we can use synthetic division to rewrite the polynomial as
a product of the binomial and the reduced polynomial. Then can you find the
remaining solutions?
6. Given a polynomial function f and a known zero of f, find all the zeros.
f(x) = x3 - 2x 2- 21x - 18; -3
NOTES: 3.2 Graphing Polynomial Functions
A polynomial function is a function of the form:
f(x) = anxn + an−1xn−1 + . . . + a1x + a0
What are an , an−1 ,
...,
a1 , a0 ? _______________
What are n, n-1, n-2, … ? ________________
Note: all coefficients must be real numbers and all exponents of variables must be whole numbers!!
Example: 2x3  5x2  4x  7
Name the leading coefficient________, the constant_________, and the degree_________,
the linear term __________, the quadratic term __________.
Give an example of each type of polynomial:
Degree
Type
Constant
Standard Form
Linear
Quadratic
Cubic
Quartic
Nth degree polynomial
Some graphing fun. Sketch the general shape of each function:
f(x) = 2x
f(x) = x3
f(x) = x3 + 1
f(x) = −x3
f(x) = −x3 + 1
Now, repeat:
f(x) = −2x
What happened to the leading coefficient? What happened to the graph?
Sketch again:
f(x) = 2x2
f(x) = 2x2 + 3
f(x) = x4
f(x) = −2x2 + 3
f(x) = −x4
Now, repeat:
f(x) = −2x2
What happened to the leading coefficient? What happened to the graph?
To summarize end behavior…
x → +∞ means “as x approaches positive infinity (x gets large)”
x → −∞ means “as x approaches negative infinity (x gets small)”
an> 0
an< 0
an> 0
an< 0
Odd degree function
x → −∞
x → +∞
Even degree function
x → −∞
x → +∞
Turning Points:
 The graph of every polynomial with degree n
has at most n – 1 turning points.
For example, if the degree is 8, there can be no more than _____ turning points.
Alternatively, if there are 7 turning points, the lowest degree will be ____.
Vocabulary:
Absolute Maximum__________________________________________
Absolute Minimum__________________________________________
Relative Minimum___________________________________________
Relative Maximum___________________________________________
Example: Analyze the graphs. Give the (approximate) coordinates of any local maximums and
minimums. State the real zeros. Determine the lowest degree that the function could have.
a)
b)
As x +, f(x)_____
As x +, f(x)_____
As x -, f(x)_____
As x -, f(x)_____
Real Zeros: __________________
Real Zeros: __________________
Relative Maximum(s):___________
Relative Maximum(s):___________
Relative Minimum(s):____________
Relative Minimum(s):____________
Absolute Maximum:_____________
Absolute Maximum:_____________
Absolute Minimum:______________
Absolute Minimum:______________
Number of Turning Points:________
Number of Turning Points:________
Lowest Degree:_______
Lowest Degree:_______
Example: Sketch the graph of f ( x)  x( x  2)( x  1)( x  2) .
Label x- and y-intercepts, maximum and minimum points, and the graph.
x
Use a t-chart of values to plot points.
Relative maximum(s):__________
Relative minimum(s): __________
Absolute maximum:____________
Absolute minimum:____________
y
Roots: __________________
x
y
3.2 Homework Advanced Functions
Polynomial functions have positive, integer exponents applied to variables. They do not include
absolute values, roots, or negative exponents that are applied to variables, and they do not include
variables in the denominator.
Classify the following functions. Decide if the function is a polynomial function. If it is a polynomial
function, state its degree, type, leading coefficient and general shape.
1. f(x) = 2x – ⅔x4 + 9
2. f(x) = x + π
3. f(x) = 3x-2 + 4x-1 + 1
Polynomial?______
Degree:______ L.C.:_______
Type:______________
4. f ( x)  x 2 2  x  5
Polynomial?______
Degree:______ L.C.:_______
Type:______________
5. f ( x)  | x  5 |  3
Polynomial?______
Degree:______ L.C.:_______
Type:______________
6. f(x) = (x – 5)2 + 3
Polynomial?______
Degree:______ L.C.:_______
Type:______________
7. f(x) = – x3 + 36x2 – 3x + 7
Polynomial?______
Degree:______ L.C.:_______
Type:______________
8. f(x) = 25 – 2
Polynomial?______
Degree:______ L.C.:_______
Type:______________
Polynomial?______
Degree:______ L.C.:_______
Type:______________
Polynomial?______
Degree:______ L.C.:_______
Type:______________
Polynomial?______
Degree:______ L.C.:_______
Type:______________
9. f ( x)  2 x  5
Given the graph, describe the end behavior of the function. Also, state the ordered pairs of the real
zeros, the y-intercept, the relative maximum(s) and the relative minimum(s).
10.
11.
12.
As x +, f(x) ____
As x -, f(x) ____
Real Zeros:______________
y-intercept:______
Relative maximum(s):_______
Relative Minimum(s):_______
Absolute Maximum:_________
Absolute Minimum:_________
As x +, f(x) ____
As x -, f(x) ____
Real Zeros:______________
y-intercept:______
Relative maximum(s):_______
Relative Minimum(s):_______
Absolute Maximum:_________
Absolute Minimum:_________
Given the function, describe the end behavior.
As x +, f(x) ____
As x -, f(x) ____
Real Zeros:______________
y-intercept:______
Relative maximum(s):_______
Relative Minimum(s):_______
Absolute Maximum:_________
Absolute Minimum:_________
13. f(x) = -x3 + 1
14. f(x) = x5 + 2x2
15. f(x) = 3x8 – 4x3
16. f(x) = -x6 + 2x3 – x
As x +, f(x) ____
As x -, f(x) ____
As x +, f(x) ____
As x -, f(x) ____
As x +, f(x) ____
As x -, f(x) ____
As x +, f(x) ____
As x -, f(x) ____
Given the graph, what is the lowest degree that the function could have?
17.
18.
19.
Number of turning points:____
Lowest Degree:______
Real Zeros:______________
y-intercept:______
Relative maximum(s):_______
Relative Minimum(s):_______
Absolute Maximum:_________
Absolute Minimum:_________
As x +, f(x) ____
As x -, f(x) ____
Number of turning points:____
Lowest Degree:______
Real Zeros:______________
y-intercept:______
Relative maximum(s):_______
Relative Minimum(s):_______
Absolute Maximum:_________
Absolute Minimum:_________
As x +, f(x) ____
As x -, f(x) ____
Number of turning points:____
Lowest Degree:______
Real Zeros:______________
y-intercept:______
Relative maximum(s):_______
Relative Minimum(s):_______
Absolute Maximum:_________
Absolute Minimum:_________
As x +, f(x) ____
As x -, f(x) ____