Big Unorganized Math Notes

Turns out making specific pages for each note has been bad for me actually making notes, so here I’m just gonna spam all the notes without much explaining. I’ll spin em off into their own notes later, or not, we’ll see lmao

Polynomial functions

Graph Behavior

When the degree is even and the leading coefficient is positive

$$x\to -\infty ,y\to \infty$$ $$x\to \infty ,y\to \infty$$

Graph

When the degree is even and the leading coefficient is negative

$$x\to -\infty ,y\to -\infty$$ $$x\to \infty ,y\to -\infty$$

Graph

When the degree is odd and the leading coefficient is positive

$$x\to -\infty ,y\to -\infty$$ $$x\to \infty ,y\to \infty$$

Graph

When the degree is odd and the leading coefficient is negative

$$x\to -\infty ,y\to \infty$$ $$x\to \infty ,y\to -\infty$$

Graph

Because of this you can determine the the behavior just by looking at the function

Zeroes of a Polynomial function

All x-intercepts are zeroes (also known as roots) of a polynomial function, but not all zeroes of a polynomial function are x-intercepts.

Also all zeroes of function are a part of its factor! If $x=a$ is a zero of $f(x)$ then $(x-a)$ is a factor of $f(x)$.

To find the zeroes of a function, you set $f(x)$ to be 0, and then solve by factoring (You probably will need the Important Formulas).

$$\begin{array}{c}f(x)=2x^2+11x-6\\ 2x^2+11x-6=0\\ (x+6)(2x-1)=0\\ x=-6,\frac{1}{2}\end{array}$$

There can’t be more zeroes than the degree of a function.

If you use the Important Formulas and get an imaginary number, it’s still a valid zero, even if it’s not a valid x-intercept.

So a function with a degree of 8, can have up to 8 zeroes, but they can be less.

Multiplicities

A repeated zero is one that’s repeated, usually denoted by raising it to a certain power.

For example:

$$g(x)=x^3-3x-2=(x-2)(x+1{)}^2$$

As a result, the zeroes of this function are $x=2,x-1$. You don’t write the duplicated zero twice, however you describe it by having a multiplicity of 2 (or 3, or however many times that zero shows up)

For example:

$$h(x)=x^3+9x^2+27x+27=(x+3{)}^3$$

The zero for this would be $x=-3$ with a multiplicity of 3.

When the multiplicity is even, the graph bounces off the x-axis but does not cross it.

When the multiplicity is odd, then the graph crosses the x-axis.

Characteristics of a graph

Let’s step through how to find the following characteristics of the following function and graph

$$f(x)=-2x(x-3{)}^2+8$$

Graph

• Determine if the graph opens up or down
• Find any x-intercepts
• Find any y-intercepts
• Find the Vertex
• Determine if it is maximum or minimum
• Graph the function by hand
• or decreasing
• Find the domain
• Find the range

Determine if the graph opens up or down

This is determined by whether or not the leading coefficient is positive or negative

Find any x-intercepts

Find any y-intercepts

Find the Vertex

Determine if it is maximum or minimum

Graph the function by hand

Determine the intervals on which the function is increasing and/or decreasing

Find the domain

Find the range

Synthetic division

given this equation

$$3x^3+17x^2+12x-6$$

You can find all the real zeros by first either checking it by calc and picking one of the x intercepts, or listing all the possible zeroes by dividing the factors o the leading coefficient by the factors of the constant! Once you have one zero use synthetic division to keep reducing it until it’s a quadratic equation!

$$\begin{array}{crrrc}& 3& 17& 12& -6\\ \frac{1}{3}& & 1& 6& 6\\ & & & & \\ & 3& 18& 18& 0\end{array}$$