# Functions on a Cartesian Plane

## Key Questions

There is a procedure to graph a function.

#### Explanation:

• Define the domain and codomain

• Find the intersection between function and x-axes:
solve $f \left(x\right) = 0$

• Calculate the first derivative and its intersection with x-axes:
$f ' \left(x\right) = 0$. This points are called extrema, geometrically represent the points where the tangent of the function is horizontal. This mean that the function reach its minimum or maximum or stationary points.

• Calculate the second derivative and its intersection with x-axes:
$f ' ' \left(x\right) = 0$. This points (inflection point) are points on a curve at which the curve changes from being concave to convex or vice versa.
if $f ' ' \left(x\right) > 0$ the function is convex (is smiling)
if $f ' ' \left(x\right) < 0$ the function is concave (is sad)

See explanation below

#### Explanation:

$\left(x , y\right)$ is a pair of real numbers. The meaning is:

$\left(x , y\right)$ is an ordered pair of numbers belonging to $\mathbb{R} \times \mathbb{R} = {\mathbb{R}}^{2}$. The first pair memeber belongs to the first set $\mathbb{R}$ and the second belongs to second $\mathbb{R}$. Althoug in this case is the same set $\mathbb{R}$. Could be in other cases $\mathbb{R} \times \mathbb{Z}$ or $\mathbb{Q} \times \mathbb{R}$

$\left(x , y\right)$ has the meaning of an aplication from $\mathbb{R}$ to $\mathbb{R}$ in which to every element x, the aplication asingns the y element.

$\left(x , y\right)$ has the meaning of plane's point coordinates. The first x is the horizontal coodinate (abscisa) and second is the vertical coordinate (ordenate). Both are coordinates.

$\left(x , y\right)$ has the meaning of a complex number: x is the real part and y is the imaginary part: $x + y i$

$\left(x , y\right)$ has the meaning of a plane's vector from origin of coordinates

etc...

You will see that meaning of $\left(x , y\right)$ could be whatever of above depending of context, but if you think a little bit, all meanings are quite similar

Hope this helps