**Plane ****Axis of Symmetry** – We often find the symmetries of the objects around us. Symmetry is like the exact reflection or mirror image of a line, shape, or object. A line of symmetry can be defined as an axis or an imaginary line that passes through the center of an object and divides it into two identical parts, i.e. it can be vertical, horizontal or diagonal. In addition, there are many forms of this symmetry, there are folding, rotating, and axes of symmetry.

## Understanding the Axis of Symmetry in Flat Shapes

There are three main types of symmetry operations, namely reflection, rotation, and translation.

- Reflection (mirroring) is the operation of reflecting an object in a line as a mirror plane.
- Rotation is an operation to rotate an object with a point as the center. An example is an equilateral triangle having rotational symmetry with a rotation angle of 120 degrees.
- Translation is an operation to transform objects from one region to another with a vector. More complicated symmetries are combinations of these operations. Symmetry is widely used in various scientific disciplines such as geometry, mathematics, physics, biology, chemistry, art and so on.

Even if two objects with high similarity appeared together, they should be different. For example, if you rotate an equilateral triangle about its center by 120 degrees, the triangle will appear the same as before the rotation. This rotation in Euclidean geometry results in an unrecognized change. In fact, each corner of an equilateral triangle modeled as a molecule exhibits a different symmetrical behavior.

The object with the most symmetry is empty space because each part can be rotated, reflected or translated without changing. The most common type of symmetry is left-right or mirror-image symmetry which is symbolized by T. This symbol is used to reflect along the vertical axis.

An equilateral triangle exhibits three axes of reflection symmetry, and a rotational symmetry. Rotating the triangle about the center of the triangle by 120 or 240 degrees shows no change. An object that only exhibits rotational symmetry behavior but does not have reflection symmetry is the swastika.

Felix Klein, the German geometer, gave a very influential statement in *the Erlangen program* in 1872, namely symmetry as a combination and organization of principles in geometry. This gave rise to the group’s new concern in geometry and the slogan transformation of geometry (an aspect of new mathematics, but highly controversial in modern mathematical practices).

The axis of symmetry itself is a line that divides an object or shape into two symmetrical parts in such a way that it will appear that the object on one side will be similar to the mirror image of the other side. This line can divide an object into two parts, four parts, and so on.

In mathematics, the axis of symmetry and optimum value are two things that are usually used in solving quadratic equations and functions. The axis of symmetry itself is the shadow line that divides the two plane shapes equally, while the optimum value is the optimum and minimum value of an equation.

To understand the axis of symmetry and the optimum value in quadratic function equations, see the explanation in this article. Make sure you read it to the end, okay!

## Geometry in Dimensions

Geometry in two dimensions is a form that is two dimensions, which means that the building only involves length and width.

- A square, which is a two-dimensional flat shape formed by four ribs (a) that are the same length and has four angles, all of which are right angles. This shape is also known as a square.
- Rectangle, which is a two-dimensional flat shape formed by two pairs of sides, each of which is the same length and parallel to its partner, and has four angles, all of which are right angles.
- A triangle, that is a polygon with three ends and three vertices. It is one of the basic shapes in geometry. The triangle with vertices A, B, and C is denoted △ABC. In Euclidean geometry, every three points, when non-collinear, define a unique triangle and, at the same time, a unique plane (i.e., two-dimensional Euclidean space). In other words, there is only one plane containing the triangle, and each triangle is contained in several planes. If the entire geometry is just Euclidean planes, there is only one plane and all the triangles are contained within it. However, in higher dimensional Euclidean space, this is no longer true.
- Trapezoid, which is a two-dimensional flat shape formed by four edges, two of which are parallel to each other, but not the same length. A trapezoid is a type of quadrilateral that has special characteristics.
- A parallelogram or parallelogram (English:
*parallelogram*) is a two-dimensional plane shape formed by two pairs of edges, each of which is the same length and parallel to its partner, and has two pairs of angles, each of which is equal to the angle opposite it. A parallelogram is a derivative of a quadrilateral which has special characteristics. A parallelogram with four edges of the same length is called a rhombus. - Circle, which is a shape consisting of all points in a plane that are a certain distance from a certain point, the center; the equivalent is the curve traced by a point moving in a plane, so that its distance from a given point is constant. The distance between any point of the circle and the center is called the radius. Specifically, a circle is a simple closed curve that divides the plane into two regions, namely the interior and exterior. In everyday usage, the term “circle” can be used interchangeably to refer to the boundaries of the image, or the entire image including its interior; in strict technical usage, the circle is just the boundary and the whole image is called a disc. Circles can also be defined as a special type of ellipse; the two foci coincide and the eccentricity is 0,
- A regular ellipse or oval is a figure that resembles a circle that has been extended in one direction. An ellipse is an example of a conic section and can be defined as the locus of all points in a plane that have the same sum of distances from two predetermined fixed points (called foci). In Indonesian, regular ellipses or ovals are also known as equivalent terms, namely ellipses (or just ovals), oblong circles, and ellipses.

## How to Determine the Symmetry Axis of Plane and Optimum Value

### 1. Axis of Symmetry

Symmetry means balance between left and right or top and bottom. An example of an object or shape of symmetry: if we draw a line from the head, nose, mouth and navel downwards, it will be seen that the right and left sides are the same. There is a right eye and a left eye, a right ear and a left ear, a right hand and a left hand, a right foot and a left foot, and so on.

If a shape when folded the sides coincide exactly, the shape has fold symmetry. The fold marks are called the fold axes of symmetry. The number of fold symmetries is equal to the number of axes of symmetry. A shape that can be folded into two equal parts and coincides exactly is called a symmetrical shape.

Meanwhile, quoted from a book entitled *Mathematics Genius Class 5 SD* written by Sulis Sutrisna, the axis of symmetry is a line drawn in a flat plane, so that it can divide that plane into two equal and congruent parts. The axis of symmetry in the graph of a quadratic function functions as a line of reflection from a point on the graph of the quadratic function.

Symmetry features, namely:

- Has fold symmetry.
- It can be folded or divided into two equal parts.

How to determine the axis of symmetry? Here’s an example of how to determine the axis of symmetry.

### 2. Optimum Value

According to Yuliansyah in a book entitled *Mathematics Teaching Materials Supporting Book* , the optimum value is the maximum value (maximum) or the smallest value (minimum). The maximum and/or minimum values are commonly known as the objective form, objective function, objective function, or objective function.

The optimum value in the quadratic function can be found using the following calculation formula.

y = -D/4a

## Number of Symmetry Axis Plane Shape

The number of axes of symmetry for a flat shape is as follows.

Two-dimentional figure |
Symmetry Axis |

Circle | infinite |

Parallelogram | do not have |

Trapezoid isosceles | 1 |

Right angled trapezoids and any trapezoids | do not have |

Rectangle | 2 |

Equilateral triangle | 3 |

Isosceles triangle | 1 |

Any triangle | do not have |

Rectangle | 4 |

Cut the rice cake | 2 |

Kite | 1 |

Do you still remember the number of axes of symmetry from the shape above? One thing that must be understood is that the axis of symmetry definitely divides a flat shape into two, three, four and so on. That way, you will know better without having to memorize the number of axes of symmetry of a flat shape.

**Parabolic Axis of Symmetry**

The graph of a quadratic function will look like a parabola. The parabola’s axis of symmetry is the vertical line that divides the parabola into two congruent or equal parts. The axis of symmetry always passes through the apex of the parabola. The x-coordinate of the vertex is the equation of the parabola’s axis of symmetry.

We can identify a line of symmetry graphically by simply finding the point farthest from the curve of the parabola. This is called the apex, the point where the two lines are connected. If the parabola is a hill, the highest point on the hill will represent the apex of the parabola. If the parabola is a valley, the lowest point in the valley will represent the apex of the parabola. However, if you go through a quadratic function, there is a formula you should know.

**Parabola Symmetry Axis Formula**

For a quadratic function in standard form, y = ax² + bx + c, the axis of symmetry is the vertical line.

**Examples of Symmetry Axis** and Optimum Value Problems

To understand how to determine the axis of symmetry and the optimum value, see the example problem and how to solve it below.

### Example Question 1

Find the axis of symmetry from the graph y = x² − 6x + 5!

**Solution:**

As already mentioned, for a quadratic function y = ax² + bx + c, the axis of symmetry is the vertical line** **

a = 1, b = −6 and c = 5.

If entered into the formula will be as follows.

### Example Problem 2

If you know the quadratic function: f(x) = 4x 2 8x + 3, what is the axis of symmetry, the optimum value and the optimum point of the function?

**Solution:**

Given the quadratic function: f(x) = 4x 2 8x + 3

**Determine the axis of symmetry (x value):**

The axis of symmetry can be calculated by the formula x = – b/2a:

x= -b/2a

x = – (-8)/2 (4)

x = 1

**Finding the optimum value:**

The optimum value can be determined by calculating y = -D/4a or entering a value for x. Here’s how to find the optimum value by entering the value of x.

f (x) = – b2-4ac/4a

f(1) = -8^2-4(4) (3)/ 4(4)

y = -1

**Determining the optimum point:**

The optimum point is the point that lies at one of the extreme points (corner points) of the settlement area. The optimum point can be determined after the x and y values have been found.

The optimum point of the equation f(x) = 4x 2 8x + 3 is (1,-1).

### Example Problem 3

A fashion company wants to produce x-cut trousers. The required production costs are expressed by the function f(x) = 3×2 –30x+175 in hundreds of thousands of rupiah. What is the minimum cost required to produce shirts?

**Solution:**

It is known that the function f(x) = 3×2 –30x+175, value a = 3, which means a > 0, so the parabola opens upwards. So, the function f(x) = 3x 2–30x+175 has a minimum value.

**Determining the axis of symmetry (x value):**

Finding the value of x using the equation x = -b/2a will yield:

x = -b/2a

x = – (-30)/ 2. 3

x = 30/6

x = 5.

Value of x is 5.

**Determining the optimum value:**

The optimum value in this case the minimum cost of the function f(x) = 3x 2 – 30x + 175 can be calculated by plugging the value of x into the function. The result is as follows.

f(x) = 3x 2-30x+175

f(5) = 3. 5^2 – 30(5) + 175

y = 100 (in hundreds of thousands of rupiah).

The minimum cost to produce x pants is IDR 10,000,000.

### Example Problem 4

Check out the picture below!

The line which is the axis of symmetry is …

A. p line

B. q line

C. r line

D. s line

Discussion:

Answer B.

The line q is the axis of symmetry, which is the line that divides a figure into two equal parts.

### Example Problem 5

Check out the picture below!

The number of axes of symmetry in the figure is . . .

A. 8

B. 4

C. 2

D. 1

Discussion:

Answer A.

The axis of symmetry is the line that divides a shape into two equal parts.

### Example Problem 6

Determine the axis of symmetry and the optimum value of f(x) = -2x² + 3x + 4.

**Discussion:**

f(x) = -2x² + 3x + 4

a = -2

b = 3

c = 4

D = b² – 4ac = 3² – 4(-2)(4) = 9 + 32 = 41

**Symmetry axis:**

x _{p} = -b/(2a) = -3/2(-2) = 3/4

**Optimum value:**

y _{p} = -D/(4a) = -41/4(-2) = 41/8

### Example Problem 7

Determine the equation of the axis of symmetry and the optimum value of the following quadratic function.

y = – x2 ^{+} 2x

**Discussion:**

It is known:

a = –1 , b = 2 and c = 0

The equation of the axis of symmetry of the function is x = – b/2.a

x = –2/2. (–1) = –2/–2

x = 1

So, the equation of the axis of symmetry is x = 1.

For the value x = 1, the function value is y = – 1 ^{2} + 2. 1 = –1 + 2 = 1

Because a = –1 < 0 (negative), the maximum value of the function is 1.

It is called the maximum (largest) value because there is no value for the function that is greater than 1. It

can be seen for this problem by calculating the value of the function for x = –b /2a , the computation is simpler.

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Such is the meaning of the axis of symmetry and also the formula for finding the axis of symmetry from a parabolic graph. If you are interested in learning material about this axis of symmetry or various other forms of material, you can study with sinaumedia.