Section 2.3 Real Zeros of Polynomial Functions

There are two ways to divide a polynomial function.
One of them is long division. Long division is harder than synthetic division.

Here is an example of long division.

The second way to divide a polynomial function is known as synthetic division. Synthetic division is easier than long division.

Here are two examples of synthetic division. The zero of this equation would be -4.

To make synthetic division easier, we use the rational zero test (also known as the rational root theorem). This relates the possible rational zeros of a polynomial to the leading coefficient and to the constant term of the polynomial.

Example 7 on page 166 in the book.

Find the rational zeros of f(x) = x^3+x+1

Because the leading coefficient is 1, the possible rational zeros are simply on the factors of the constant term.

Possible Rational Zeros: ±1

f(1) = (1)^3+1+1 = 3

f(-1) (-1)^3+(-1)+1 = -1

The polynomial has no rational zeros.

2.2 Polynomial Functions of Higher Degree

Graph of Polynomial Functions:

The graph of a polynomial function is continuous, meaning it has no breaks. The graph also only has smooth, rounded turns, polynomial functions cannot have sharp turns.

A.

Continuous Graph

The Leading Coefficient Test:

Whether the graph of a polynomial eventually rises or falls can be determined by the funtion's degree (even or odd) and by its leading coefficient.

When n is odd:

the graph falls to the left and rises to the right.

the graph rises to the left and falls to the right.

When n is even:

the graph rises to the left and right.

the graph falls to the left and right.

Zeros of Polynomial Functions:

It can be shown that for a polynomial function f of degree n, the following statements are true.

1. The graph of f has at most n real zeros.

2. The function f has at most n-1 relative extrema (relative minimums or maximums).

Real Zeros of Polynomial Functions:

1. x=a is a zero of the function f

2. x=a is a solution of the polynomial equation f(x)=0

3. (x-a) is a factor of the polynomial f(x).

4. (a,0) is an x-intercept of the graph of f.

Intermediate Value Theorem:

Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a)f(b), then in the interval [a,b], f takes on every falue between f(a) and f(b).

Section 2.1 Quadratic Functions

Polynomial Function: f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

• the coefficients (a_n, a_n-1, ..., a₁, a_{0 ) are all real numbers}
• _{polynomial functons are classified by degrees}

Degree name example 0 constant y=1 1 linear f(x)= 2x+5 2 quadratic h(x)=x^2-x-4 3 cubic y=x^3 4 quartic g(x)=x^4-2x-1 Quadratic Function:

• graphing quadratics

the parent function is , a

nything you add to it will transform the

graph. by changing the coefficient from 1 to 2

it vertically stretches the graph by a unit of two.

Sandard Form of a Quadratic Function:

, the point (h,k) is the vertex of the parabola. the coefficient a shows whether it is opening up or down.(up a>0, down a<0)

• to write a quadratic in standard form, complete the square

Parabola to Sandard From:

• The vertex is also point (h,k), which is (1,1)

is standard form so is the new equation.

solve for a to find the full formula, do this by plugging in a point for f(x) and x.

Maximum and Minimum: use the formula , to find the minimum and maximum points

example:

b=3 and a=-2

the maximum point is .75, it is a maximum because the parabola is facing down.(remember when a is less then 0 it faces down)

Section 1.4 Combinations of Functions

The domain of an Arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the case of the quotient f(x)/g(x) - g(x) ≠ 0.

1. Sum: (f + g)(x) = f(x) + g(x)

2. Difference: (f – g)(x) = f(x) – g(x)

3. Product: (fg)(x) = f(x) ∙ g(x)

4. Quotient: (f/g)(x) = f(x)/g(x), g(x) ≠ 0

ex. (f + g)(x) = f(x) + g(x)

= (2x + 1) + (x² + 2x – 1)

= x² + 4x

Composition of Functions

(f ○ g)(x) = f(g(x))

ex. f(x) = x + 2 g(x) = 4 – x²

(f ○ g)(x) = f(g(x))

= f(4 – x²)

= (4 – x²) + 2

= –x² + 6

Section 1.5- Inverses

Finding the inverse of functions are based on composition

ex. Find inverse of F(x)

F(x)=2x-3

F^-1(x)= x+3/2

(FoF^-1)(x)= F(F^-1(x))

= F (x+3/2)

= 2(x+3/2)

= x+3-3

= x

OR

(F^-1 o F)(x)= F^-1 (F(x))

= F^-1 (2x-3)

= 2x-3+3/2

= 2x/2

= x

*You should get x regardless of which way you solve it

- two functions, f and g, are inverses of each other if and only if....

(F o G) (x)= (G o F)(x)= x

*graphs are reflections by Y=X

One-to-one function- passes horizontal line test and has inverse that is a function

-A function is a one-to-one function if and only if F(a)=F(b)----> a=b

Section 1.4 - Arithmetic Operations

This section is about how to add, subtract, multiply, and divide functions.

Addition, subtraction, multiplication, and division are all called arithmetic functions.

1. Adding Functions

When adding functions, you have to add them as two separate quantities

ex. f(x) = (x + 1)
g(x) = (x - 1)

( f + g)(x) = (x + 1) + (x - 1)
= 2x

2. Subtracting Functions

The same principal applies to subtracting functions.

ex. f(x) = (2x - 5)
g(x) = (1 - x)

(f - g)(x) = (2x - 5) - (1 - x)
= 3x - 6

3. Multiplying Functions

Multiplying functions requires the use of FOIL.

ex. f(x) = (3x - 2)
g(x) = (5x + 7)

(fg)(x) = (3x - 2)(5x + 7)
=

4. Dividing Functions

If you are asked to divide two functions, you will also have to state the domain.

ex. f(x) =

g(x) =

(f/g)(x) =

Domain:

Section 1.3 - Shifting, Reflecting, and Stretching Graphs

Section 1.3 covers the shifting, reflecting, and stretching of graphs.

These parent functions provide the basis for a strong understanding of the section:

(This graph uses c=4, though
c is an unknown value until

specified)

Shifting, Reflecting, and Stretching Graphs

1. Adding or subtracting values from the y-coordinate shifts the graph vertically

Adding c shifts the graph up the value of c

Subtracting c shifts the graph down the value of c

Ex:

The graph is shifted up 2 units

2. Adding or subtracting values from the x-coordinate shifts the graph horizontally

Adding c shifts the graph left the value of c

Subtracting c shifts the graph right the value of c

Ex:

The graph shifts 5 units right

3. Multiplying the y-coordinate stretches or compresses the graph vertically

y=c(f(x))

If the value of c is greater than one, the graph is stretched vertically
If the value of c is less than one, the graph is compressed vertically

Ex: If the absolute value of x in multiplied by 4, the graph will be vertically stretched by a factor of 4.

4. Multiplying the x-coordinate stretches or compresses the graph horizontally

y=f(c(x))
If the value of c is greater than one, the graph is compressed horizontally

If the value of c is less than one, the graph is stretched horizontally
Ex: If the square root of x is multiplied by 3, the graph will be horizontally compressed by a factor of 3.

5. Multipying the y-coordinate by a negative number reflects the graph in the x-axis
y=-f(x)
The x values do not change
The y values are opposite of what they were before

Ex:
y=x+2
y=-(x+2)

y=-(x+2) = y=-f(x)

The y values are now opposite of what they were before while the x values do not change

6. Multiplying the x-coordinate by a negative number reflects the graph in the y-axis
y=f(-x)
The x values are opposite of what they were before
The y values do not change
Ex:

y=5x-2

y=5(-1)-2

y=5(-1)-2 = y=f(-x)

The x values are now opposite of what they were before while the y values do not change