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).
P(x) = -2x3-2x2-x-20, c=10 / 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

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:

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 / Standard Form
Constant
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

Now, repeat:

f(x) = −2x / f(x) = −x3 / f(x) = −x3 + 1

What happened to the leading coefficient? What happened to the graph?

Sketch again:

f(x) = 2x2 / f(x) = 2x2 + 3 / f(x) = x4

Now, repeat:

f(x) = −2x2 / f(x) = −2x2 + 3 / f(x) = −x4

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)”

Odd degree function / an> 0 / an< 0
x → −∞
x → +∞
Even degree function / an> 0 / an< 0
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)
As x +, f(x)_____
As x -, f(x)_____
Real Zeros: ______
Relative Maximum(s):______
Relative Minimum(s):______
Absolute Maximum:______
Absolute Minimum:______
Number of Turning Points:______
Lowest Degree:______/ b)
As x +, f(x)_____
As x -, f(x)_____
Real Zeros: ______
Relative Maximum(s):______
Relative Minimum(s):______
Absolute Maximum:______
Absolute Minimum:______
Number of Turning Points:______
Lowest Degree:______

Example: Sketch the graph of .

Label x- and y-intercepts, maximum and minimum points, and the graph.

Use a t-chart of values to plot points.

Relative maximum(s):______

Relative minimum(s): ______

Absolute maximum:______

Absolute minimum:______

Roots: ______

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
Polynomial?______
Degree:______L.C.:______
Type:______/ 2. f(x) = x + π
Polynomial?______
Degree:______L.C.:______
Type:______/ 3. f(x) = 3x-2 + 4x-1 + 1
Polynomial?______
Degree:______L.C.:______
Type:______
4.
Polynomial?______
Degree:______L.C.:______
Type:______/ 5.
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:______/ 9.
Polynomial?______
Degree:______L.C.:______
Type:______

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.

As x +, f(x) ____
As x -, f(x) ____
Real Zeros:______
y-intercept:______
Relative maximum(s):______
Relative Minimum(s):______
Absolute Maximum:______
Absolute Minimum:______/ 11.

As x +, f(x) ____
As x -, f(x) ____
Real Zeros:______
y-intercept:______
Relative maximum(s):______
Relative Minimum(s):______
Absolute Maximum:______
Absolute Minimum:______/ 12.
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.

13. f(x) = -x3 + 1
As x +, f(x) ____
As x -, f(x) ____ / 14. f(x) = x5 + 2x2
As x +, f(x) ____
As x -, f(x) ____ / 15. f(x) = 3x8 – 4x3
As x +, f(x) ____
As x -, f(x) ____ / 16. f(x) = -x6 + 2x3 – x
As x +, f(x) ____
As x -, f(x) ____

Given the graph, what is the lowest degree that the function could have?

17.

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) ____ / 18.

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) ____ / 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) ____