Algebraic Representation

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The prime minister is probably one of the busiest people in the United Kingdom and he can't be everywhere at the same time. On an occasion when he can't be present for an event, he will send a representative. The person is not the prime minister but is kind of like a substitute. This is similar to what we see in algebraic expressions and equations. The variables used are a representation of the real value. We call this algebraic representation.

Anything relating to algebra involves using letters to represent something.

In this article, we will explore the meaning of algebraic representation, representing geometrical transformations, formulae and functions algebraically, and some examples of their application.

Algebraic representation meaning

Algebraic representation involves the use of variables, numbers and symbols to represent quantities in an equation or expression.

Algebraic representations describe what is happening without having to say it in words. Algebraic representations can be applied to different things. It can be applied to geometrical transformations, to build formulae, and represent functions.

Some examples of algebraic representations are $2 y + 5 = x$ , $f (x) = 2 y$ and $F o r c e = m \times a$ .

Algebraic representation of transformations

A transformation has to do with the geometrical change of a mathematical object. The object can be a geometrical shape that undergoes a transformation in its position or size. The different forms of transformations are translation, reflection, rotation, enlargement or a combination of these. To get in-depth knowledge on transformations, check out our article on Transformations.

For the algebraic representation of transformations, we are dealing with geometrical shapes being transformed in the $x$ and $y$ -axis.

Translation

Translation has to do with the movement of a shape either up, down, left or right, or a composition of these. The shape is just being slid from one position to another. The size and the shape remain the same but the position changes. To know more about translation, check out our article on Translations.

Consider the image below.

Algebraic Representation Graph showing translation transformation StudySmarter

Graph showing translation transformation - StudySmarter Original

In this image, we see a rectangle $A B C D$ and a rectangle $A^{'} B^{'} C^{'} D^{'}$ . We will consider this last rectangle the resulting shape, the image, by applying a translation to the first rectangle. Note that the shape and size of the shape remain the same but the position is different.

So how can we use algebraic representation in translation? Let's establish some rules. The rules show how the coordinates change when a shape is translated.

Four different movements can occur here; the shape can move up, down, left and right on the $x$ and $y$ -axis.

Let's start with the pair $(x, y)$ and let $a$ represent the number of units the shape is to move in the $x$ axis. If the shape is to move in the right direction in the $x$ -axis, it will be $x + a$ . If it is to move left, it will be $x - a$ .

Let $b$ represent the number of units the shape is to move in the $y$ axis. If the image is to move up, it will be $y + b$ . If it is to move down, it will be $y - b$ .

So we have,

Translation	Rules
Move right $a$ units	$(x, y) \to (x + a, y)$
Move left $a$ units	$(x, y) \to (x - a, y)$
Move up $b$ units	$(x, y) \to (x, y + b)$
Move down $b$ units	$(x, y) \to (x, y - b)$

Reflection

Reflection is the flip of a shape across a line. The line is called the line of reflection, or line of symmetry. Reflection is also referred to as the mirror image of a shape. In this transformation, the position's shape is changed, but its size and shape remain the same. See the image below.

Graph showing reflection transformation - StudySmarter Original

In the image above, we can see $△ A' B' C'$ and $△ A B C$ . We will consider the first triangle the image of the second by applying a reflection over the y-axis.

There are rules that show how the coordinates change when a shape is reflected. The way the coordinates change depends on if the shape moves up, down, left or right. We will see two simple reflections over the axes.

A shape is reflected either over the $x$ -axis or the $y$ -axis, and when this happens the signs of the coordinates change. So, if you have the coordinates of a shape $(x, y)$ and it is being reflected over the $x$ -axis, you should multiply the $y$ -coordinate by $- 1$ . If it is being reflected over the $y$ -axis, you should multiply the $x$ -axis by $- 1$ .

Reflection	Rules
Move over $x$ -axis	Multiply the $y$ coordinate by $- 1$ : $(x, y) \to (x, - y)$
Move over $y$ -axis	Multiply the $x$ coordinate by $- 1$ : $(x, y) \to (- x, y)$

Rotation

Rotation has to do with turning a figure around a fixed point or an axis. When a shape is rotated around an axis, the coordinates will change. Once again in this transformation, the position's shape is changed, but its size and shape remain the same. See the graph below.

Algebraic Representation Showing rotation transformation StudySmarter Graph showing rotation transformation - StudySmarter Original

Some rules will guide you to know how the coordinates will change. The rules are shown in the table below.

Rotation	Rules
$90 °$ clockwise	Switch the coordinates and multiply the one on the right by $- 1$ : $(x, y) \to (y, - x)$
$90 °$ counter-clockwise	Switch the coordinate and multiply the one on the left by $- 1$ : $(x, y) \to (- y, x)$
$180 °$ clockwise or counter-clockwise	Multiply both coordinates by $- 1$ : $(x, y) \to (- x, - y)$

Algebraic representation and formulae

Formulae are equations that show a relationship between quantities. An example of a formula is

$F = m a$ .

This is the formula for calculating force, from Physics, where $F$ is force, $m$ is mass, and $a$ is acceleration.

In the formula, F is directly proportional to a and m is introduced as the constant of proportionality.

$F \propto a F = m a$

$\propto$ is the symbol for proportionality.

Another example of a formula is the area of a circle. The area of a circle is directly proportional to the square of the radius.

$A r e a \propto r^{2}$ ,

where $r$ is the radius.

Here, $π$ is introduced as the constant of proportionality and the formula becomes.

$A r e a = {πr}^{2}$

We can see the algebraic representation in the formula. The unknown quantities are represented with variables. We can come across a situation where a formula will be a combination of variables and numbers. In this case, we will simplify them the same way we will simplify an algebraic expression.

We will take some examples later.

Algebraic representation of a function

A function is an expression that shows the relationship between input and output. In a function, for every input, there is one and only one output that corresponds to it.

Most times, a function is represented with a lowercase letter, $f$ , or any other letter to represent it. If $x$ is the input and $y$ is the output, then a function $f$ is written as:

$f (x) = y$ .

The expression above is an algebraic representation of that function. The variables $x$ and $y$ are a representation of an actual number. To get the output, you'll have to substitute different values for x.

If $f (x) = y$ and $x = 2$ , we will have, for instance:

$f (x) = y f (2) = 2$

This means that the input $x$ is 2 and the output $y$ is 2.

If $f (x) = y$ , $x = 2$ and $y = x + 2$ , we will have:

$f (x) = y f (x) = x + 2 f (2) = 2 + 2 f (2) = 4$

This means that the input $x$ is 2 and the output $y$ is 4. The expression $y = x + 2$ is called the rule of the function.

Examples of algebraic representations

We will now take some examples to give us a clearer picture of what we've been talking about.

Let's take an example of a translation transformation.

Triangle ABC has vertices $A (0, 0), B (3, 4), C (1, - 2)$ . Find the vertices of $A' B' C'$ after translation of 4 units to the right and 2 units down. Graph the triangle and its translated image.

Solution

We have the coordinates of a triangle given to us and we are told that translation is to occur 4 units to the right and 2 units down.

If you recall the rules we talked about in the translation section, moving $a$ units to the right is: $(x, y) \to (x + a, y)$ and moving $b$ units down is: $(x, y) \to (x, y - b)$ .

If we combine the two rules to fit our question, it will be:

$(x, y) \to (x + a, y - b)$ .

Let's put this on a table.

$△ A B C$	$(x, y) \to (x + a, y - b)$	$△ A' B' C'$
$A (0, 0)$	$(0 + 4, 0 - 2)$	$(4, - 2)$
$B (3, 4)$	$(3 + 4, 4 - 2)$	$(7, 2)$
$C (1, - 2)$	$(1 + 4, - 2 - 2)$	$(5, - 4)$

We've done the calculation in the table and the coordinates of the translated triangle are:

A' (4, - 2), B' (7, 2), C' (5, - 4)

If we plot this, we will have the graph below.

Algebraic Representation Graph showing translation transformation StudySmarter

The graph shows the $△ A B C$ and the translated $△ A' B' C'$ .

Let's take an example of a reflection transformation.

A rectangle $A B C D$ has vertices $A (4, 2), B (2, 2), C (2, 6), D (4, 6)$ . Find the vertices of the rectangle $A' B' C' D'$ after a reflection over the x-axis. Graph the rectangle and its reflected image.

Solution

We have the coordinates of a rectangle given to us as $A (4, 2), B (2, 2), C (2, 6), D (4, 6)$ . The rectangle is to be reflected over the x-axis.

Recall that when a figure is to be reflected over the x-axis, you multiply the y coordinate by -1. So we have:

$(x, y) \to (x, - y)$ .

Let's do the calculation in a table.

Rectangle $A B C D$	$(x, y) \to (x, - y)$	$A' B' C' D'$
$A (4, 2)$	$(4, 2 (- 1))$	$A' (4, - 2)$
$B (2, 2)$	$(2, 2 (- 1))$	$B' (2, - 2)$
$C (2, 6)$	$(2, 6 (- 1))$	$C' (2, - 6)$
$D (4, 6)$	$(4, 6 (- 1))$	$D' (4, - 6)$

We've done the calculations and the co-ordinates of the reflected rectangle are:

A' (4, - 2), B' (2, - 2), C' (2, - 6), D' (4, - 6)

Now, let's plot the graph.

Algebraic Representation Graph showing reflection transformation StudySmarter

The graph above shows the rectangle $A B C D$ and its reflected image $A' B' C' D'$

Let's take an example of a rotation transformation.

A quadrilateral has its vertices $A (3, - 2), B (5, - 3), C (4, 4), D (0, 0)$ . Find the vertices of $A' B' C' D'$ after a $90 °$ counter-clockwise rotation. Plot the graph of the quadrilateral and its rotated image.

Solution

We are given the coordinates of a quadrilateral $A (3, - 2), B (5, - 3), C (4, 4), D (0, 0)$ . The quadrilateral is to be rotated in the $90 °$ counter-clockwise direction.

Recall that the rule for $90 °$ counter-clockwise rotation is that you switch the coordinates and multiply the left by $- 1$ .

$(x, y) \to (- y, x)$

Let's do the calculation in a table.

Quadrilateral $A B C D$	$(x, y) \to (- y, x)$	$A' B' C' D'$
$A (3, - 4)$	$(- 4 (- 1), 3)$	$A' (4, 3)$
$B (5, - 3)$	$(- 3 (- 1), 5)$	$B' (3, 5)$
$C (4, 4)$	$(4 (- 1), 4)$	$C' (- 4, 4)$
$D (0, 0)$	$(0 (- 1), 0)$	$D' (0, 0)$

We've done the calculations and the vertices for the rotated quadrilateral are $A' (4, 3), B' (3, 5), C' (- 4, 4), D' (0, 0)$ .

Now let's plot the graph

Algebraic Representation Graph showing rotation transformation StudySmarter

The graph shows the quadrilateral $A B C D$ and its rotated image $A' B' C' D'$

Let's take some examples of algebraic representations in formulae.

Find the area of the rectangle below.

Solution

To find the area of a rectangle, we need a formula. The formula is:

$A = L \times B$

where $A$ is the Area

$L$ is the length

$B$ is the breadth

From the figure above,

$L = y c m B = (y + 4) c m$

Let's substitute the formula

$A = L \times B A = y \times (y + 4)$

We can take out the multiplication sign and it will still mean the same thing.

$A = y (y + 4)$

We can further simplify by using the $y$ outside to multiply each term in the bracket and we will have:

$A = y^{2} + 4 y$

The area of the rectangle is:

$A = y^{2} + 4 y$

Let's take another example.

The formula for calculating simple interest is $S . I = P R T$ . $S . I$ represents simple interest. Find the simple interest when $P = £ 200, R = 2 %$ and $T = 2 y e a r s$ .

Solution

The question asks us to find simple interest. We are given the formula for simple interest:

$S . I = P \times R \times T$

We are also given the values for the variables in the formula.

$P = £ 200 R = 2 % T = 2 y e a r s$

What we need to do now is substitute the values given in the formula.

$S . I = P \times R \times T S . I = 200 \times \frac{2}{100} \times 2$

Notice that we are dividing the value for R by 100. This is because it is in percentage.

$S . I = 200 \times \frac{2}{100} \times 2 S . I = 200 \times 0.02 \times 2 S . I = 8$

The simple interest is $£ 8$ .

Let's take some examples on algebraic representation and formula.

Given $f (x) = 2 x + 5$ , evaluate the following.

$f (3)$
$f (7)$

Solution

a. $f (3)$

The function given is

f (x) = 2 x + 5

and we asked to evaluate

f (3)

. This means that

x = 3

and we will substitute this value in the function.

$f (x) = 2 x + 5 f (3) = 2 (3) + 5 f (3) = 6 + 5 f (3) = 11$

b. $f (7)$

We will do the same thing we did above.

Here, $x = 7$ . We will substitute this value for $x$ in the function.

$f (x) = 2 x + 5 f (7) = 2 (7) + 5 f (7) = 14 + 5 f (7) = 19$

Algebraic Representation - Key takeaways

Algebraic representation involves the use of variables, numbers and symbols to represent quantities in an equation or expression.
Algebraic representation can be applied to different things. It can be applied to transformations, formulae and functions.
Anything relating to algebra involves using letters to represent something.

Frequently Asked Questions about Algebraic Representation

Algebraic representation involves the use of variables, numbers and symbols to represent quantities in an equation or expression.

An example of algebraic representation is

2x + 5 =20. The variable 'x' is used to represent an actual value.

An algebraic representation is written using a variable to represent an actual value.

The algebraic representation for translation are a set of rules that shows how the coordinates of an image change when translated.

Let a represent the number of units an image will move when translated.

The rules are below.

Move right a units : (x, y) - (x + a, y)

Move left a units : (x, y) - (x -a, y)

Move up b units : (x, y) - (x, y + b)

Move down b units : (x, y) - (x, y - b)

The algebraic representation of a function is written below:

f(x) = y

where x is the input

y is the output

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An algebraic representation involves the use of variables, numbers and symbols to represent quantities in an equation or expression.

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