A **graphic** It gives us a visual representation of the relationship between two variables in an equation, therefore it provides us with a method with which we can solve them (generally in a simpler way than the algebraic method).

**Graphical representation of linear equations**

A linear equation is a mathematical equality between two algebraic expressions in which known and unknown elements (called variables) appear, and that only involves adding and subtracting one variable to the **first power**.

The graph of a linear equation with two variables is one. A common form of these equations is:. Where is the slope and the point where the line intersects the axis To represent the graph of a linear equation with two variables we use the Cartesian coordinate system, which consists of two number lines: a horizontal line (called and a vertical line (called) both intersect at the origin.

**Example: **Represent the following linear equation with two variables: y = 2x + 3

To represent, we give one of the two variables two values, with which **we get two points**:

When x = 0; y = 2 (0) + 3 = 3: P (0.3)

When x = 1; y = 2 (1) + 3 = 5: Q (1,5)

By representing and joining these points **we get a line**.

**Graphical representation of quadratic equations**

A quadratic or quadratic equation is any equation in which, once simplified, the greatest exponent of the unknown is 2. Thus, ax^{2 }+ bx + c = 0 is a quadratic equation. The graphical representation of this equation is a curve called a parabola.

**Characteristics of the parabola**

When a> 0:

- The parabola opens upwards and we say that it is
**concave up**. - The parabola always has a value of f (x) = y which is
**minimum** - The parable is
**strictly decreasing**from -∞ to the abscissa of the vertex of the parabola (which is calculated with the following formula: x = - b / 2a - And it is
**strictly growing**and from the vertex abscissa to + ∞. - The parabola always cuts the axis at point c and can cut the x axis at two points x
_{1}and x_{2}, at a point x_{1 = }x_{2 }or at no point, as these represent the roots of the equation.

When a <0:

- The parabola opens downwards and we say that it is concave downwards.
- The parabola always has a value of f (x) = y which is
**maximum**. - The parable is
**strictly growing**from -∞ to the abscissa of the vertex of the parabola - And it is
**strictly decreasing**and from the vertex abscissa to + ∞. - The parabola always cuts the y axis at point c and can cut the x axis at two x points
_{1}and x_{2}, at a point x_{1 = }x_{2}or at no point, as these represent the roots of the equation.

**Example**: Graph the following quadratic function: x^{2 }- 12x + 36 = 0

Graphic representation:

Graphic representation:

- Since a> 0 »a = 1, the parabola is concave upwards and has a minimum point.
- The vertex abscissa is:

x = - 12/2 (1) = 6

- The parabola is strictly decreasing from -∞ to the vertex abscissa and is strictly increasing from the vertex abscissa to + ∞.
- The intersection points are (corresponding to the roots of the equation): x
_{1}= x_{2}= 6; c = 3.

Solution: x^{2 }- 12x + 36 = (x - 6) (x - 6)

**Graphical representation of inequalities**

Through the graphical representation of the inequalities we can find their solutions (graphical method).

**Example: **Solve the following inequality: 2x + y ≤ 1

- Transform inequality into equality: 2x + y = 1
- Graph the line 2x + y = 1 »y = - 2x +1

For them, we give one of the two variables two values, with which we obtain two points:

When x = 0; y = 1

When x = 1; y = - 1

- By representing and joining these two points (0, 1) and (1, -1), we obtain a line

Take a point at random that satisfies the inequality. For example we take the point (0, 0) and substitute them in the inequality; if this is true, the solution is the half plane where the point is, otherwise the solution will be the other half plane.

2x + y ≤ 1

2 (0) + 0 ≤ 1

0 ≤ 1 The inequality is not satisfied!

Therefore, the **points of half plane 2 are part of the solution**.