# Graphics 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, ax2 + 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 x1 and x2, at a point x1 = x2 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 points1 and x2, at a point x1 = x2 or at no point, as these represent the roots of the equation.

Example: Graph the following quadratic function: x2 - 12x + 36 = 0

#### 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): x1 = x2 = 6; c = 3.

Solution: x2 - 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.