Polynomial Functions

The general form of Polynomial Functions will be as the form.

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+....+a_{2}x^{2}+a_{1}x+a_{0}$

But need to remember that $a_{n}$ , $a_{n-1}$ ......$a_{1}$ and $a_{0}$ they can all be 0

The graph of Polynomial Functions can be looking as a U-shaped graph called a parabola.

One important feature of graphing the polynomial even functions graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric if you draw a vertical line drawn through the vertex, called the axis of symmetry.

quadratic function

A quadratic function is a common kind of polynomial function. When $f(x)=ax^{2}+bx+c$ and $a\neq 0$ it will be a quadratic function.

Different form of quadratic function

1. General form $f(x)=ax^{2}+bx+c$
2. Factor form $f(x)=(x-h)(x-o)$
3. Vertex form $f(x)=a(x-h)^{2}+k$

Turning General form to Vertex form

$f(x)=ax^{2}+bx+c$

$f(x)=a(x^{2}+{bx \over a}+{c \over a})$

$f(x)=a(x^{2}+{b \over a}^{2}+{b \over 2a}^{2}-{b \over 2a}^{2}+{c \over a})$

$f(x)=a({(x+{b \over a}}^{2}-{b^{2} \over 4a^{2}}+{c \over a})$

$f(x)=a{(x+{b \over 2a}}^{2}-{b^{2} \over 4a}+c$

$f(x)=a{(x+{b \over 2a})}^{2}-{b^{2}-4ac \over 4a}$

$a(x-h)^{2}+k=a{(x+{b \over 2a})}^{2}-{b^{2}-4ac \over 4a}$

$h=-{b \over 2a}$

$k=-{b^{2}-4ac \over 4a}$

properties of quadratic function

1. The graph of a quadratic function is always a parabola that either opens upward or downward
2. The domain of a quadratic function is all real numbers
3. The vertex is the lowest point when the parabola opens upwards; while the vertex is the highest point when the parabola opens downward.
4. In general form $f(x)=ax^{2}+bx+c$ when a > 0 will have a minimum and facing up.
5. In general form $f(x)=ax^{2}+bx+c$ when a < 0 will have a maximum and facing down.

discriminant

When $f(x)=ax^{2}+bx+c$ and $a\neq 0$ the discriminant of quadratic function will be $b^{2}-4ac$. The properties of discriminant

1. $b^{2}-4ac>0$ Then There will be 2 Real and Distinct Roots.
2. $b^{2}-4ac=0$ Then There will be 2 Real and Equal Roots.
3. $b^{2}-4ac\geq 0$ Then There will be Real Roots.
4. $b^{2}-4ac<0$ Then There will be Complex or Imaginary Roots.

Identiy of quadratic function

1. When $b^{2}-4ac=0$ and a > 0, y will always bigger or equal to 0
2. When $b^{2}-4ac=0$ and a < 0, y will always smaller or equal to 0
3. When $b^{2}-4ac<0$ and a > 0, y will always bigger then 0
4. When $b^{2}-4ac<0$ and a < 0, y will always smaller then 0