Trans Linear

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2.5 Translations of linear functions

At the end of this section you should be familiar with: The term translation. How to write the equation of the translated line, given the equation of the original line.

2.5.1 Meaning of TranslationThere are many applications in business and economics where graphs of demand, supply, cost and other functions are translated as a result of taxes, subsidies etc. It is, therefore, important to have a mathematical knowledge of translations that enables us to graph and write down the equation of the translated function. Once this knowledge is acquired, it is used to analyse various applications in subsequent chapters.A straight line is translated when it is moved, intact, to another position in the plane, that is, its slope and length do not change; however, since its vertical intercept (and, of course, the horizontal intercept) is changed, its equation will change since it is now a different line.

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Translating a straight line

Slope and length of line do not change.

Vertical intercept (and horizontal intercept) change,

thus the equation of the line changes.

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2.5.2 Equation of Translated Lines The following rules are used to write down the equation of the translated line.

Vertical translations

Rule 1: Vertical translations upwards.When a graph is translated vertically upwards by c units, replace every y variable in the original equation by (y - c).Note: This is the same as adding c to the RHS of the equation,

y - c = f(x) is the same as y = f(x) + c

Rule 2: Vertical translations downwards.When a graph is translated vertically downwards by c units, replace every y variable in the original equation by (y + c).

Note: This is the same as subtracting c from the RHS of the equation, y + c = f(x) is the same as y = f(x) - c

Worked Example 2.14 Vertical translations of linear functionsThe original linear equation is given by y = 1 + 2x.(a) Deduce the equation of the line when translated vertically upwards by two units.(b) Deduce the equation of the line when translated vertically downwards by three units.Solution:(a) When the original line is translated vertically upwards by two units, replace the y variable by (y - 2), so the equation of the translated line is,

This is illustrated in Figure 2.30.Comment: The slope of the line has not changed, but the vertical intercept, and consequently the horizontal intercept have changed. The corresponding leftward horizontal

shift is given by the equation, , (see below).

(b) When the original line is translated vertically downwards by three units, replace the y variable by (y + 3), so the equation of the translated line is,

This is also illustrated in Figure 2.30.

Comment: The slope has not changed, but the intercepts have changed. The corresponding

rightward horizontal shift is given by the equation, , (see below).



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-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

Figure 2.30 Vertical translations of line, y = 1 + 2x

Remember

Calculate the size of the horizontal shift that occurs in conjunction with a vertical translation

The size of the shift, is derived as follows:

For the line, y = mx + c, whose slope: .

Hence, when a line with a positive slope is translated:

upwards, the corresponding leftward horizontal shift is, .

downwards, the corresponding rightward horizontal shift is, .

(shifts are in opposite directions for lines with positive slopes).

when a line with a negative slope is translated:

upwards, the corresponding rightward horizontal shift is, .

downwards, the corresponding leftward horizontal shift is, .

(shifts are in same direction for lines with negative slopes).


Vertical Translationsy (y - c) moves graph upwards by c units.y (y + c) moves graph downwards by c units.


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See Worked Examples on income constraints, section 2.7 for vertical translations of lines with negative slopes.

Horizontal translations

Rule 3: Horizontal translations forward.When a graph is translated horizontally forward in a positive direction along the x-axis by c units, replace every x variable in the original equation by (x - c).

Rule 4: Horizontal translations backwards.When a graph is translated horizontally backwards in a negative direction along the x-axis by c units, replace every x variable in the original equation by (x + c).

Worked Example 2.15 Horizontal translations of linear functionsThe original linear equation is given by y = 1 + 2x(a) Deduce the equation of the line when translated horizontally forward by two units.(b) Deduce the equation of the line when translated horizontally backwards by one unit.

Solution:(a) When the original line is translated horizontally forwards by two units, replace the x variable by (x - 2), so the equation of the translated line is,

This is illustrated in Figure 2.31.(b) When the original line is translated horizontally backwards by one unit, replace the x variable by (x + 1), so the equation of the translated line is,

This is also illustrated in Figure 2.31.

-4

-3

-2

-1

0

1

2

3

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-4 -3 -2 -1 0 1 2 3 4


x

y y = 1 + 2x

y = - 3 + 2x

y = 3 + 2x

2

1


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Figure 2.31 Horizontal translations of line, y = 1 + 2x

Remember

Horizontal and vertical translations togetherA translation can be a combination of horizontal and vertical translations. The equation of the translated line can be deduced by simply using the appropriate rules given above.

Worked Example 2.16 Horizontal and vertical translations of linear functionsFind the equation of the line, y = 1 + 2x when translated down by two units and forwards by 1.5 units.

Solution:

Step 1: Translate the original line downwards by two units by replacing y with (y + 2). The equation becomes, .Step 2: Translate forwards by 1.5 units by replacing x with (x - 1.5). The equation becomes .Step 3: Simplify to get the equation of the translated line,

The translated line is illustrated in Figure 2.32.

-4

-3

-2

-1

0

1

2

3

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-4 -3 -2 -1 0 1 2 3 4


Horizontal Translationsx (x - c) moves graph forward by c units.x (x + c) moves graph backwards by c units.

y = 1 + 2x

y = - 4 + 2x

x

y

21.5


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Figure 2.32 Vertical and horizontal translation of line, y = 1 + 2x



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2.5.3 Some Applications of Translations

Application of translations will arise throughout the text. At this point some introductory examples are given.

Excise TaxAn excise tax is imposed on the supplier. The government collects a fixed amount of money for each unit of the good sold, irrespective of the selling price. For example, as a tax of £10 per bottle of brandy (in spite of the huge variation in prices). The following example analyses the effect of such a tax on the supply function.

Worked Example 2.17 The effect of an excise tax on a supply function

A supply function is given by P = 10 + 0.5Q. An excise tax of £10 per unit is imposed. Deduce the equation of the new supply function.

Solution:A tax of £10 per unit means that the price received by the supplier will be (P - 10) per unit sold. The new supply function is deduced by substituting (P - 10) into the original supply function as follows:

The supply function has been translated vertically upwards by 10 units. This is illustrated in Figure 2.33.

Figure 2.33 Effect of an excise tax (10 Euros per unit) on the supply function


p = 10 + 0.5Q

With excise tax p = 10 + 0.5Q

P

Q


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Value added Tax (VAT). This is a tax levied at each stage of the production of certain goods and services. The tax is called an ‘ad valorem’tax in that it is set at a fixed percentage of the value of the product or services. For example, the price paid by a consumer for a PC = retail price + VAT If the VAT rate for PC’s is 12%, then a PC which retails at £950 will incur a VAT of £114

. The consumer pays £950 + £114 = £1064.

In general, the demand function is translated down as a result of the imposition of VAT, since the price the consumer pays (P + VAT), hence

P = a - bQ (P + VAT) = a - bQ P = (a - VAT) - bQ

Translating cost functions

Section 2.3.2 introduced the idea of a linear cost function as expressed in equation (2.6). Some factors that translate the cost function are now analysed.

Worked Example 2.18 Translating linear cost functions

The cost function for solar panels (in units of £,000) is given as TC = 10 + 2Q.

(a) deduce the equation of the cost function when the producer receives a grant of 10 in order to cover fixed costs.

(b) deduce the equation of the cost function if no grant is given and fixed costs are increased by 150%.

Solution:

(a) With a grant of 10 (in units of £,000) to cover fixed costs, FC is replaced by (FC - 10). The equation of the total cost function becomes:

The total cost function has been translated downwards by 10 units with an unchanged slope. This is illustrated in Figure 2.35 This translation is accompanied by a shift of the TC function to the right.

(b) If fixed costs (no grant) are increased by 150%, giving a new fixed cost of

(10 + ) = 25. The total cost function now becomes:



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The total cost function has been translated upwards by 15 units with an unchanged slope. This is illustrated in Figure 2.34 .

Figure 2.34: The effect of (i) a subsidy (ii) changes in fixed costs on a total cost function



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Progress Exercise 2.6 Translations of linear functions and applications

1. The equation of a line is given by y = 50 - 0.6x(a) deduce the equation of the line when translated horizontally forward by 12 units.(b) deduce the equation of the line when translated horizontally backwards by 5 units.(c) deduce the equation of the line when translated vertically down by 6 units. What is the size and direction of the corresponding horizontal shift?

2. A supply function is given by the equation P = 10 + 3Q(a) When a tax of 1.5 per unit is imposed the supplier receives (P - 1.5) per unit.

Write down the equation of the supply function, adjusted for tax. What is the size of the vertical and the corresponding horizontal shift?(b) Plot the supply function for the good with and without tax. Compare the graphs.

3. Assume that the demand function for good X is given as:

where Y = 1000, = 10, = 14, T = 50 and A = 40.(a) Analyse the effect on the demand for good X if the price of the substitute good to

X increases in price from £10 to £30. Illustrate your answer algebraically and diagrammatically.(b) Analyse the effect on the demand for good X if the price of the complementary good to X increases in price from £14 to £35. Illustrate your answer algebraically and diagrammatically.

4. The supply function for toolboxes is given by P = 100 + 0.4Q. Deduce the equation of the new supply function when,

(a) a subsidy of £5 per unit is introduced. (b) a tax of £10 per box is imposed.

Calculate the price charged when the quantity demanded is 50, according to,(i) the original supply function. (ii) with the subsidy (iii) with the tax.

5. The fixed cost incurred in producing a good is £120, while each unit produced costs £3.50.(a) Write down the equation of the total cost function.(b) Write down the equation of the total cost function if a subsidy of £50 is given.(c) Write down the equation of the total cost function if fixed costs increase by 30%.(d) Write down the equation of the total cost function if variable costs increase by 30%.Graph (a), (b), (c) and (d) on the same diagram. Which of these functions is not a translation of the original cost function.

Progress Exercises 2.6: Solutions

1. (a) y = 50 - 0.6(x - 12) = 57.2 - 0.6x (b) y = 50 - 0.6(x + 5) = 47 - 0.6x

(c) y = 50 - 0.6x - 6 = 44 - 0.6x . Shift is leftwards


TC =10 + 2Q

TC = 25 + 2Q

Fixed costs increase by 150%

TC = 2Q Subsidy of 10


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2. (a) Price to supplier = price - tax = (b)P - 1.5. Therefore,P - 1.5 = 10 + 3Q = 11.5 + 3QCorresponding horizontal shift is,

Figure PE2.6 Q23. Q = 200 - 2P + (20) = 220 - 2P

(a) = +20 0.3 = 6. That is, 6 units are added to RHS.

Thus, Q = 226 - 2P. Shifts graph up and translated forward by .

(b) = 21 - 0.5 = - 10.5. That is, 10.5 units are subtracted from RHS. Thus, Q = 209.5 - 2P. Shifts graph down and translated leftwards by

Figure PE2.6 Q3

4. (a) A subsidy of £5: P + 5 = 100 + 0.4Q P = 95 + 0.4Q(b) A tax of £10: P - 10 = 100 + 0.4Q P = 110 + 0.4Q(i) P = 100 + 0.4(50) = £120 (ii) With subsidy: P = 95 + 0.4(50) = £115(iii) With tax: P = 110 + 0.4(50) = £130

5. (a) TC = 120 + 3.5Q (b) TC = 70 + 3.5Q (c) TC = 156 + 3.5Q(d) TC = 120 + 4.55Q (not a translation of (a).


P = 10 + 3Q

P = 11.5 + 3Q

- 3.83

- 3.33Q

P

113110104.75

226

220

209.5

P

Q

(a)

(b)

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