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4: Linear Functions - Mathematics


Recall that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data.

  • 4.0: Introduction to Linear Functions
    Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour. A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.
  • 4.1: Linear Functions
    The ordered pairs given by a linear function represent points on a line. Linear functions can be represented in words, function notation, tabular form, and graphical form. The rate of change of a linear function is also known as the slope. An equation in the slope-intercept form of a line includes the slope and the initial value of the function. The initial value, or y-intercept, is the output value when the input of a linear function is zero.
  • 4.2: Modeling with Linear Functions
    We can use the same problem strategies that we would use for any type of function. When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. Draw a diagram, where appropriate. Check for reasonableness of the answer. Linear models may be built by identifying or calculating the slope and using the y-intercept. The x-intercept may be found by setting y=0, which is setting the expression mx+b equal to 0.
  • 4.3: Fitting Linear Models to Data
    Scatter plots show the relationship between two sets of data. Scatter plots may represent linear or non-linear models. The line of best fit may be estimated or calculated, using a calculator or statistical software. Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. The correlation coefficient, r , indicates the degree of linear relationship between data.

4. Linear DEs of Order 1

For linear DEs of order 1, the integrating factor is:

The solution for the DE is given by multiplying y by the integrating factor (on the left) and multiplying Q by the integrating factor (on the right) and integrating the right side with respect to x, as follows:

Example 1

Now for the integrating factor:

We need to apply the following formula: `ye^(intPdx)=int(Qe^(intPdx))dx`

For the left hand side of the formula, we have

For the right hand of the formula, Q = 7 and the IF = x -3 , so:

Applying the outer integral:

Now, applying the whole formula, `ye^(intPdx)=intQe^(intPdx)dx`, we have

Multiplying throughout by x 3 gives the general solution for `y`.:

Is it correct? Differentiate this answer to make sure it produces the differential equation in the question.

Here is the solution graph of our answer for Example 1 (I've used K = 0.5 ).

Example 2

`"IF"=e^(intPdx)` `=e^(int cot x dx)` `=e^(ln sin x)` `=sin x`

Apply the formula, `ye^(intPdx)=intQe^(intPdx)dx` to obtain:

The integral needs a simple substitution: `u = sin x`, `du = cos x dx`

Divide throughout by sin x to get the required general solution of the DE:

Here is the solution graph of our answer for Example 2 (I've used K = 0.1 ) .

It is a composite trigonometric curve, where the main shape is the cosecant curve, and the "wiggles" are due to the addition of the (sin x)/2 part:

Typical solution graph using `K=0.1`.

Example 3

Dividing throughout by dx to get the equation in the required form, we get:

Now the integrating factor in this example is

Using `ye^(intPdx)=intQe^(intPdx)dx+K`, we have:

Here is the solution graph for Example 3 (I've used K = 5 ).

It was necessary to zoom out (a lot) to see what is going on in this graph.

Typical solution graph using `K=5`.

Example 4

We need to get the equation in the form of a linear DE of order 1.

Expand the bracket and divide throughout by dx:

Solving directly, using Scientific Notebook

Scientific Notebook cannot solve our original question:

We have to rearrange it in terms of `(dy)/(dx)` and then solve it using

Compute menu &rarr Solve ODE. &rarr Exact


Multiple Choice Questions
Question. 1 The solution of which of the following equations is neither a fraction nor an integer?
(a) -3x + 2=5x + 2 (b)4x-18=2 (c)4x + 7 = x + 2 (d)5x-8 = x +4
Solution. For option (c)

Question. 2 The solution of the equation ax + b = 0 is

Solution.

Question. 3 If 8x – 3 = 25 + 17x, then x is
(a) a fraction (b) an integer
(c) a rational number (d) Cannot be solved
Solution. (c) Given, 8x-3 = 25+17x

Question. 4 The shifting of a number from one side of an equation to other is called
(a) transposition (b) distributivity
(c) commutativity (d) associativity .
Solution. (a) The shifting of a number from one side of an equation to other side is called transposition.
e.g. x +a = 0is the equation, x = -a
Here, number ‘a’ shifts from left hand side to right hand side.

Question. 5 If (frac < 5x >< 3 >)-4 =(frac < 2x >< 5 >) , then the numerical value of 2x – 7 is
(a)(frac < 19 >< 13 >) (b)(frac < -13 >< 19 >)
(c)0 (d)(frac < 13 >< 19 >)
Solution.(b)

Question. 6 The value of x, for which the expressions 3x – 4 and 2x + 1 become equal, is
(a) -3 (b) 0
(c) 5 x (d) 1
Solution. (c) Given expressions 3x – 4 and 2x + 1 are equal.
Then, 3x-4 = 2x + 1
3x- 2x = 1 + 4 [transposing 2x to LHS and -4 to RHS]
x = 5
Hence, the value of x is 5.

Question. 7 If a and b are positive integers, then the solution of the equation ax = b has to be always
(a) positive (b) negative (c) one (d) zero
Solution. (a) If ax = b, then x = (frac < b >< a >)
Since, a and b are positive integers. So,(frac < b >< a >) is also positive integer, Hence, the solution of the given equation has to be always positive.

Question. 8 Linear equation in one variable has
(a) only one variable with any power
(b) only one term with a variable
(c) only one variable with power 1
(d) only constant term
Solution. (c) Linear equation in one variable has only one variable with power 1.
e.g. 3x + 1 = 0,2y – 3 = 7 and z + 9 = – 2 are the linear equations in one variable.

Question. 9 Which of the following is a linear expression?
(a) (< x >^< 2 >) +1 (b) y + (< y >^< 2 >)
(c) 4 (d) 1 + z
Solution. (d) We know that, the algebraic expression in one variable having the highest power of the variable as 1, is known as the linear expression.
Here, 1 + z is the only linear expression, as the power of the variable z is 1.

Question.10 A linear equation in one variable has
(a) only one solution (b) two solutions
(c) more than two solutions (d) no solution
Solution. (a) A linear equation in one variable has only one solution.
e.g. Solution of the linear equation ax + b = 0 is unique, i.e. x = (frac < -b >< a >)

Question. 11 The value of S in (frac < 1 >< 3 >) + S = (frac < 2 >< 5 >) is
(a)(frac < 4 >< 5 >) (b)(frac < 1 >< 15 >)
(c)10 (d)0
Solution.(b) Given, (frac < 1 >< 3 >) + S = (frac < 2 >< 5 >)

Question.12 If –(frac < 4 >< 3 >) y = –(frac < 3 >< 4 >) then y is equal to
(a)(-< left[ frac < 3 > < 4 > ight] >^< 2 >) (b)(-< left[ frac < 4 > < 3 > ight] >^< 2 >)
(c)(< left[ frac < 3 > < 4 > ight] >^< 2 >) (d)(< left[ frac < 4 > < 3 > ight] >^< 2 >)
Solution.

Question. 13 The digit in the ten’s place of a two-digit number is 3 more than the digit in the unit’s place. Let the digit at unit’s place be b. Then, the number is
(a) 11b+30 (b) 10b+ 30
(c) 11 b + 3 (d) 10b + 3
Solution. (a) Let digit at unit’s place be b.
Then, digit at ten’s place = (3 + b)
Number = 10 (3 + b) + b – 30 + 10b + b = 11b + 30

Question. 14 Arpita’s present age is thrice of Shilpa. If Shilpa’s age three years ago was x, then Arpita’s present age is
(a) 3 (x – 3) (b)3x + 3
(c) 3x – 9 (d) 3(x + 3)
Solution. (d) Given, Shilpa’s age three years ago = x
Then, Shilpa’s present age = (x + 3)
Arpita’s present age = 3 x Shilpa’s present age = 3 (x + 3)

Question. 15 The sum of three consecutive multiples of 7 is 357. Find the smallest multiple.(a) 112 (b) 126 (c) 119 (d) 116
Solution.

Fill in the Blanks
In questions 16 to 32, fill in the blanks to make each statement true.
Question. 16 In a linear equation, the——— power of the variable appearing in the equation is one.
Solution. highest
e.g. x + 3 = O and x + 2 = 4 are the linear equations.

Question. 17 The solution of the equation 3x – 4 = 1 – 2x is————- .
Solution.

Question. 18 The solution of the equation 2y = 5y-(frac < 18 >< 5 >) is————.
Solution.

Question. 19 Any value of the variable, which makes both sides of an equation equal, is known as a———–of the equation.
Solution. e.g. x + 2 = 3 => x = 3-2 = 1 [transposing 2 to RHS]
Hence, x = 1 satisfies the equation and it is a solution of the equation.

Question. 20 9x – ……………….. = – 21 has the solution (- 2).
Solution. 3
Let 9x-m= -21 has the solution (-2).

Question. 21 Three consecutive numbers whose sum is 12 are——–,————-and———.
Solution.

Question. 22 The share of A when Rs 25 are divided between A and B, so that A gets Rs 8 more than B, is——–.
Solution.

Question. 23 A term of an equation can be transposed to the other side by changing its—-.
Solution. sign
e.g. x + a = 0 is a linear equation. .
=> x = -a
Hence, the term of an equation can be transposed to the other side by changing its sign.

Question. 24 On subtracting 8 from x, the result is 2. The value of x is——–.
Solution.

Question. 25 ( frac < x >< 5 >) + 30 = 18 has the solution as——–.
Solution.

Question. 26 When a number is divided by 8, the result is -3. The number is——–.
Solution.

Question. 27 When 9 is subtracted from the product of p and 4, the result is 11. The value of p is—-.
Solution.

Question. 28 If ( frac < 2 >< 5 >) x-2=5-( frac < 3 >< 5 >) x,then x=——-.
Solution.

Question. 29 After 18 years, Swarnim will be 4 times as old as he is now. His present age is——–.
Solution.

Question. 30 Convert the statement ‘adding 15 to 4 times x is 39’ into an equation.
Solution. 4x+ 15=39
To convert the given statement into an equation, first x is multiplied by 4 and then 15 is added to get the result 39. i.e. 4x + 15=39

Question. 31 The denominator of a rational number is greater than the numerator by 10. If the numerator is increased by 1 and the denominator is decreased by 1, then expression for new denominator is——.
Solution.

Question. 32 The sum of two consecutive multiples of 10 is 210. The smaller multiple is——-.
Solution.

True/False
In questions 33 to 48, state whether the statements are True or False.
Question. 33 3 years ago, the age of boy was y years. His age 2 years ago was (y — 2) years.
Solution. False
Given, 3 yr ago, age of boy = y yr
Then, present age of boy = (y + 3)yr
2 yr ago, age of boy = y + 3-2 = (y + 1)yr

Question. 34 Shikha’s present age is p years. Reemu’s present age is 4 times the present age of Shikha. After 5 years, Reemu’s age will be 15p years.
Solution. False
Given, Shikha’s present age = pyr
Then, Reemu’s present age = 4 x (Shikha’s present age) = 4pyr After 5 yr, Reemu’s age = (4p+5)yr

Question. 35 In a 2-digit number, the unit’s place digit is x. If the sum of digits be 9, then the number is (10x – 9).
Solution. False
Given, unit’s digit = x
and sum of digits = 9
Ten’s digit = 9 – x
Hence, the number = 10 (9 -x)+x = 90 -10x + x = 90 – 9x

Question. 36 Sum of the ages of Anju and her mother is 65 years. If Anju’s present age is y years, then her mother’s age before 5 years is (60 – y) years.
Solution. True
Given, Anju’s present age = y yr
Then, Anju’s mother age = (65 – y)yr
Before 5 yr, Anju’s mother age = 65 – y – 5 = (60 – y)yr

Question. 37 The number of boys and girls in a class are in the ratio 5 : 4. If the number of boys is 9 more than the number of girls, then number of boys is 9.
Solution. False
Let the number of boys be 5x and the number of girls be 4x.
According to the question, – 5x – 4x = 9 => x = 9
Hence, number of boys = 5 x 9 = 45

Question. 38 A and B are together 90 years old. Five years ago, A was thrice as old as B was. Hence, the ages of A and B five years back would be (x – 5) years and (85 – x) years, respectively.
Solution. True
Let the age of A be x yr.
Then, age of S = (90 – x) yr
Five years ago, the age of A = (x- 5) yr
The age of B= 90-x-5 = (85-x)yr
Hence, the ages of A and 8 five years back would be (x – 5) yr and (85 – x) yr, respectively.

Question. 39 Two different equations can never have the same answer.
Solution. False
Two different equations may have the same answer.
e.g.2x + 1 = 2 and 2x – 5 = – 4 are the two linear equations whose solution is (frac < 1 >< 2 >)

Question. 40 In the equation 3x – 3 = 9, transposing – 3 to RHS, we get 3x = 9.
Solution. False
Given, 3x – 3 = 9
=> 3x = 9 + 3 [transposing -3 to RHS]
=> 3x = 12

Question. 41 In the equation 2x = 4 – x, transposing – x to LHS, we get x = 4.
Solution. False
Given, 2x = 4-x
=> 2x + x = 4 [transposing -x to LHS]
=> 3x = 4

Question. 42

Solution.

Question. 43

Solution.

Question. 44 If 6x = 18, then 18x = 54.
Solution.

Question. 45 If (frac < x > < 11 >) , then x=(frac < 11 > < 15 >).
Solution.

Question. 46 If x is an even number, then the next even number is 2(x +1).
Solution. False
Given, x is an even number. Then, the next even number is (x + 2).

Question. 47 If the sum of two consecutive numbers is 93 and one of them is x, then the other number is 93 – x.
Solution. True
Given, one of the consecutive number = x
Then, the next consecutive number = x + 1

Question. 48 Two numbers differ by 40. When each number is increased by 8, the bigger becomes thrice the lesser number. If one number is x, then the other number is (40 – x).
Solution.

In Questions 49 to 78, solve the following.
Question. 49 (frac < 3x-8 > < 2x >=1).
Solution.

Question. 50 (frac < 5x > < 2x-1 >=2).
Solution.

Question. 51 (frac < 2x-3 > < 4x+5 >=frac < 1 >< 3 >) .
Solution.

Question. 52 (frac < 8> < x >=frac < 5 >< x-1 >) .
Solution.

Question. 53 (frac < 5(1-x)+3(1+x) > < 1-2x >=8 ) .
Solution.

Question. 54 (frac < 0.2x+5 > < 3.5x-3 >=frac < 2 >< 5 >)
Solution.

Question. 55 ( frac < y-(4-3y) > < 2y-(3y+4y) >=frac < 1 >< 5 >)
Solution.

Question. 56 (frac < x > < 5 >=frac < x-1 >< 6 >)
Solution.

Question. 57 0.4(3x-1)=0.5x +1
Solution.

Question. 58 8x-7-3x=6x-2x-3
Solution.

Question. 59 10x-5-7x=5x+15-8
Solution.

Question. 60 4t-3-(3t+1)=5t-4
Solution.

Question. 61 5(x-1)-2(x+8)=0
Solution.

Question. 62 (frac < x > < 2 >-frac < 1 > < 4 >left( x-frac < 1 > < 3 > ight) =frac < 1 > < 6 >left( x+1 ight) +frac < 1 >< 12 >)
Solution.

Question. 63 (frac < 1 > < 2 >left( x+1 ight) +frac < 1 > < 3 >left( x-1 ight) =frac < 5 > < 12 >left( x-2 ight))
Solution.

Question. 64 (frac < x+1 > < 4 >=frac < x-2 >< 3 >)
Solution.

Question. 65 (frac < 2x-1 > < 5 >=frac < 3x+1 >< 3 >)
Solution. Given (frac < 2x-1 > < 5 >=frac < 3x+1 >< 3 >)

Question. 66 1-(x-2)-[(x-3)-(x-1)]=0
Solution.

Question. 67 (3x-frac < x-2 > < 3 >=4-frac < x-1 >< 4 >)
Solution.

Question. 68 ( frac < 3t+5 > < 4 >-1=frac < 4t-3 >< 5 >)
Solution.

Question. 69 ( frac < 2y-3 > < 4 >-frac < 3y-5 > < 2 >=y+frac < 3 >< 4 >)
Solution.

Question. 70 0.25(4x-5)=0.75x +8
Solution.

Question. 71 (frac < 9-3y > < 1-9y >=frac < 8 >< 5 >)
Solution.

Question. 72 ( frac < 3x+2 > < 2x-3 >=-frac < 3 >< 4 >)
Solution.

Question. 73 ( frac < 5x+1 > < 2x >=-frac < 1 >< 3 >)
Solution.

Question. 74 (frac < 3t-2 > < 3 >+frac < 2t+3 > < 2 >=t+frac < 7 >< 6 >)
Solution.

Question. 75 ( m-frac < m-1 > < 2 >=1-frac < m-2 >< 3 >)
Solution. Given ( m-frac < m-1 > < 2 >=1-frac < m-2 >< 3 >)

Question. 76 4 (3p + 2) – 5 (6p – 1) = 2 (p – 8) – 6 (7p – 4)
Solution.

Question. 77 3(5x-2)+2(9x-11)=4(8x-7)-111
Solution.

Question. 78 0.16 (5x-2)=0.4x +7
Solution.

Question. 79 Radha takes some flowers in a basket and visits three temples one-by-one. At each temple, she offers one half of the flowers from the basket. If she is left with 3 flowers at the end, then find the number of flowers she had in the beginning.
Solution.

Question. 80 Rs 13500 are to be distributed among Salma, Kiran and Jenifer in such a way that Salma gets Rs 1000 more than Kiran and Jenifer gets Rs 500 more than Kiran. Find the money received by Jenifer.
Solution.

Question. 81 The volume of water in a tank is twice of that in the other. If we draw out 25 litres from the first and add it to the other, the volumes of the water in each tank will be the same. Find the volumes of water in each tank.
Solution.

Question. 82 Anushka and Aarushi are friends. They have equal amount of money in their pockets. Anushka gave 1/3 of her money to Aarushi as her birthday gift. Then, Aarushi gave a party at a restaurant and cleared the bill by paying half of the total money with her. If the remaining money in Aarushi’s pocket is Rs 1600, then find the sum gifted by Anushka.
Solution.

Question. 83 Kaustubh had 60 flowers. He offered some flowers in temple and found that the ratio of the number of remaining flowers to that of flowers in the beginning is 3 : 5. Find the number of flowers offered by him in the temple.
Solution.

Question. 84 The sum of three consecutive even natural numbers is 48. Find the greatest of these numbers.
Solution.

Question. 85 The sum of three consecutive odd natural numbers is 69. Find the prime number out of these numbers.
Solution.

Question. 86 The sum of three consecutive numbers is 156. Find the number which is a multiple of 13 out of these numbers.
Solution.

Question. 87 Find a number whose fifth part increased by 30 is equal to its fourth part decreased by 30.
Solution.

Question. 88 Divide 54 into two parts, such that one part is 2/7 of the other.
Solution.

Question. 89 Sum of the digits of a two-digit number is 11. The given number is less than the number obtained by interchanging the digits by 9. Find the number.
Solution.

Question. 90 Two equal sides of a triangle are each 4 m less than three times the third side. Find the dimensions of the triangle, if its perimeter is 55 m.
Solution.

Question. 91 After 12 years, Kanwar shall be 3 times as old as he was 4 years ago. Find his present age.
Solution.

Question. 92 Anima left one-half of her property to her daughter, one-third to her son and donated the rest to an educational institute. If the donation was worth Rs 100000, how much money did Anima have?
Solution.

Question. 93 If 1/2 is subtracted from a number and the difference is multiplied by 4, the result is 5. What is the number?
Solution.

Question. 94 The sum of four consecutive integers is 266. What are the integers?
Solution.

Question. 95 Hamid has three boxes of different fruits. Box A weighs (2frac < 1 >< 2 >) kg more than box B and Box C weighs (10frac < 1 >< 4 >)kg more than box B. The total weight of the three boxes is (48frac < 3 >< 4 >) kg. How many kilograms does box A weigh?
Solution.

Question. 96 The perimeter of a rectangle is 240 cm. If its length is increased by 10% and its breadth is decreased by 20%, then we get the same perimeter. Find the length and breadth of the rectangle.
Solution.

Question. 97 The age of A is five years more than that of B. 5 years ago, the ratio of their ages was 3 :2. Find their present ages.
Solution.

Question. 98 If numerator is 2 less than denominator of a rational number and when 1 is subtracted from numerator and denominator both, the rational number in its simplest form is 1/2. What is the rational number?
Solution.

Question. 99 In a two-digit number, digit in unit’s place is twice the digit in ten’s place. If 27 is added to it, digits are reversed. Find the number.
Solution.

Question. 100 A man was engaged as typist for the month of February in 2009. He was paid Rs 500 per day, but Rs 100 per day were deducted for the days he remained absent. He received Rs 9100 as salary for the month. For how many days did he work?
Solution.

Question. 101 A steamer goes downstream and covers the distance between two ports in 3 hours. It covers distance in 5 hours, when it goes upstream. If the stream flows at 3 km/h, then find what is the speed of the steamer upstream?
Solution.

Question. 102 A lady went to a bank with Rs 100000. She asked the cashier to give her Rs 500 and Rs 1000 currency notes in return. She got 175 currency notes in all. Find the number of each kind of currency notes.
Solution.

Question. 103 There are 40 passengers in a bus, some with Rs 3 tickets and remaining with Rs 10 tickets. The total collection from these passengers is Rs 295. Find how many passengers have tickets worth Rs 3?
Solution.

Question. 104 Denominator of a number is 4 less than its numerator. If 6 is added to the numerator, it becomes thrice the denominator. Find the fraction.
Solution.

Question. 105 An employee works in a company on a contract of 30 days on the condition that he will receive Rs 120 for each day he works and he will be fined Rs 10 for each day he is absent. If he receives Rs 2300 in all, for how many days did he remain absent?
Solution.

Question. 106 Kusum buys some chocolates at the rate of Rs 10 per chocolate. She also buys an equal number of candies at the rate of Rs 5 per candy. She makes a 20% profit on chocolates and 8% profit on candies. At the end of the day, all chocolates arid’ candies are sold out and her profit is Rs 240. Find the number of chocolates purchased.
Solution.

Question. 107 A steamer goes downstream and covers the distance between two ports in 5 hours, while it covers the same distance upstream in 6 hours. If the speed of the stream is 1 km/h, then find the speed of the steamer in still water.
Solution.

Question. 108 Distance between two places A and B is 210 km. Two cars start simultaneously from A and B in opposite directions and distance between them after 3 hours is 54 km. If speed of one car is less than that of other by 8 km/h, then find the speed of each.
Solution.

Question. 109 A carpenter charged Rs 2500 for making a bed. The cost of materials used is Rs 1100 and the labour charge is Rs 200 per hour. For how many hours did the carpenter work?
Solution.

Question. 110 For what value of x is the perimeter of shape 77 cm?

Solution.

Question. 111 For what value of x is the perimeter of shape 186 cm?

Solution.

Question. 112 On dividing Rs 200 between A and B, such that twice of A’s share is less than 3 times B’s share by 200, what is B’s share?
Solution.

Question. 113 Madhulika thought of a number, doubled it and added 20 to it. On dividing the resulting number by 25, she gets 4. What is the number?
Solution.


Equations and Expressions: Level 4

The key idea of equations and expressions at level 4 is that linear relationships between variables can be represented by a linear equation.

  • a linear equation can involve number all four number operations (not including exponents)
  • a linear equation can be solved in order to find an unknown variable in a particular situation

Linear equations take the form:
y = mx + c.
At level 4 m is restricted to whole numbers (i.e. numbers without decimal points) and c is restricted to integers (i.e. can be either positive or negative whole numbers). When linear equations are represented on a graph, they form a straight line with the value of “m” corresponding to the slope and “c” representing the point where the line crosses the y axis.

Simple linear equations describe the relationship between two related variables, x and y. Any value of x has a corresponding value of y, and the linear relationship uniquely links the values of these two sets of variables. As the value of x varies, so does the value of y. Because the values of x and y are uniquely linked, a linear equation can be solved in order to find an unknown variable in a particular situation.

When working with equations that balance, such as linear equations, both sides of the equals sign need to be preserved for the equation to remain correct. For example, when the same number is subtracted from each side the equation remains accurate. For example 3p – 6 = 18 so 3p = 24 (adding six to both sides) so p = 8 (dividing both sides by three).

This key idea develops from the key idea of equations and expressions at level 3 that equations show relationships of equality between parts on either side of the equal sign.

This key idea is extended to the key idea of equations and expressions at level 5 that some types of relationships between variables can be represented by a quadratic equation.


Transformation of a Linear Function Worksheets

This ensemble of transformation worksheets is targeted to help high school learners gain an understanding of the transformation of a linear function and its graph. Find the indicated transformed function g(x) from its parent function f(x). Basic knowledge of translation such as horizontal shift and vertical shift reflection, horizontal / vertical stretches and horizontal / vertical compressions are required to solve these transformation worksheet pdfs. A number of free printable worksheets are available for practice.

Printing Help - Please do not print worksheets with grids directly from the browser. Kindly download them and print.

Look at the parent function y = f(x) and shift it right, left, up or down and draw the translated graph g(x). Use the answer key to verify the vertical or horizontal shifts.

In this set of pdf transformation worksheets, for every linear function f(x), apply the translation and find the new translated function g(x). Follow the relevant rules f(x) + c / f(x) - c to make vertical shifts of c units up/down and f(x + c) / f(x - c) to make horizontal shifts of c units left/right.

Find the reflection of each linear function f(x). A reflection over the x- axis should display a negative sign in front of the entire function i.e. -f(x). Negate the independent variable x in f(x), for a mirror image over the y-axis.

Find the vertical stretch or compression by multiplying the function f(x) by the given factor and the horizontal stretch or compression by multiplying the independent variable x by the reciprocal of the given factor. Use the relevant rules to make the correct transformations.

In this set of printable transformation worksheets for high school, test your comprehension on translation of graphs. Each grid has two graphs, the original graph f(x) and the translated graph g(x). Find the correct vertical or horizontal shift.


Domain And Range For Linear Functions

Thus for the quadratic function f x x2 f x x 2 the domain is the set of all real numbers and the range is only non-negative real numbers. Domains and ranges are sets.

Domain Co Domain And Range Of Function Functions Math Function Math Relation

Open Domain and Range of Linear Functions in the GeoGebra program.

Domain and range for linear functions. The range however will depend on the vertex of the absolute value function the minimum or the maximum. R 0 R as f x 1 x f x 1 x. The domain of a function is all the possible input values for which the function is defined and the range is all possible output values.

Or you can use the calculator below to determine the domain and range of ANY equation. In its simplest form the domain is all the values that go into a function and the range is all the values that come out. So the range is 0 0.

Which functions have the same domain and range as the linear function. Domain And Range Worksheet 1 Worksheets Solving Quadratic Equations Graphing Worksheets. Domain and range worksheet.

F xx5 - - - here there is no restriction you can put in any value for x and a value will pop out. The range of a function is defined as a set of solutions to the equation for a given input. There is only one range for a given function.

These Algebra 1 Domain and Range Worksheets will produce problems for identifying whether graphed sets are functions or not. Y x 2 4 x 1. R 0 R f.

Just like our previous examples a quadratic function will always have a domain of all x values. The domain is all real numbers because an absolute value is still a linear function. Domain and Range 0 0 Example 2.

Lets explore both of these scenarios. Some of the worksheets for this concept are Functions domain and range review date block Performance based learning and assessment task Name class date 2 6 Domain and range of a function Name date ms Work Work. The number of buses needed to transport the members on each trip is a function of the number of members who went on each trip.

These Domain and Range Worksheets are a good resource for students in the 9th Grade through the 12th Grade. Sometimes the domain is restricted depending on the nature of the function. Find the domain and range of the quadratic function.

State the domain and range for each graph and then tell if the graph is a function write yes or no. Domain range function worksheet pdf. Sometimes however the domain and range of a linear function may be restricted based on the information it represents.

In other words the range is the output or y value of a function. This function consists of only the ordered pairs 523 724 865 and 1056. Domain and range of functions worksheet - To notice the image more obviously in this article you are able to click on the wanted image to watch the picture in its original dimensions or in full.

The range is simply y 2. Domain and Range of Function Types. Aparking lot is to be 400 feet wide and 370 feet deep.

How many standard-sized cars fit in this lot considering a double-loaded w4 two way traffic set up an angle of 90 and a sw of 9. Domains and ranges are sets. What is the domain of this situation.

Linear equations Quadratics and Absolute Value Functions. For this reason we can conclude that the domain of any function is all real numbers. Given a real-world situation that can be modeled by a linear function or a graph of a linear.

A person can also look at Domain And Range Of Functions Worksheet image gallery that many of us get prepared to locate the image you are interested in. You can select the types of functions and non-functions to be graphed. Linear Function Domain And Range Linear Function Domain And Range - Displaying top 8 worksheets found for this concept.

The domain and range of linear functions is all real numbers or negative infinity to positive infinity. If you are still confused you might consider posting your question on our message board or reading another websites lesson on domain and range to get another point of view. The summary of domain and range is the following.

The student council sent its members on four field trips during the school year. We define a function f. 2 Show answers Another question on Mathematics.

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Contents

A linear function is a polynomial function in which the variable x has degree at most one: [2]

Such a function is called linear because its graph, the set of all points ( x , f ( x ) ) in the Cartesian plane, is a line. The coefficient a is called the slope of the function and of the line (see below).

The slope of a nonvertical line is a number that measures how steeply the line is slanted (rise-over-run). If the line is the graph of the linear function f ( x ) = a x + b , this slope is given by the constant a .

from which one can immediately see the slope a and the initial value f ( 0 ) = b , which is the y-intercept of the graph y = f ( x ) .

Example Edit

Note that the graph includes points with negative values of x or y, which have no meaning in terms of the original variables (unless we imagine selling meat to the butcher). Thus we should restrict our function f ( x ) to the domain 0 ≤ x ≤ 2 .

Also, we could choose y as the independent variable, and compute x by the inverse linear function: x = g ( y ) = − 1 2 y + 2 <2>>y+2> over the domain 0 ≤ y ≤ 4 .

If the coefficient of the variable is not zero ( a ≠ 0 ), then a linear function is represented by a degree 1 polynomial (also called a linear polynomial), otherwise it is a constant function – also a polynomial function, but of zero degree.

A straight line, when drawn in a different kind of coordinate system may represent other functions.

For example, it may represent an exponential function when its values are expressed in the logarithmic scale. It means that when log(g(x)) is a linear function of x , the function g is exponential. With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function.

If both arguments and values of a function are in the logarithmic scale (i.e., when log(y) is a linear function of log(x) ), then the straight line represents a power law:

On the other hand, the graph of a linear function in terms of polar coordinates:


Ex 4.4 Class 9 Maths Question 1.
Give the geometric representations of y – 3 as an equation
(i) in one variable.
(ii) in two variables.
Solution:
The given linear equation is
y = 3 …(i)
(i) The representation of the solution on the number line is shown in the figure below, where y = 3 is treated as an equation in one variable.

(ii) We know that y = 3 can be written as
0. x + y = 3
Which is a linear equation in the variables x and y. This is represented by a line. Now, all the values of x are permissible because 0.x is always 0. However, y must satisfy the equation y = 3.
Note that, the graph AB is a line parallel to the x-axis and at a distance of 3 units of the upper side of it.

Ex 4.4 Class 9 Maths Question 2.
Give the geometric representations of 2x + 9 = 0 as an equation
(i) in one variable
(ii) in two variables
Solution:
Given linear equation is 2x + 9 = 0 i.e., x = –
(i) If x = – is treated as an equation in one variable, then it has a unique
solution x = – .
So, it is a point on the number line as shown below :

(ii) Given equation 2x + 9 = 0 can be written as 2x + 0 . y + 9 = 0, which is a linear equation in two variables x and y.
We can write the given equation as,

When y=1, then = –
When y=2, then = –
When y=3, then = –

Now Playing the points – , – – paper and joining then ,we get a line PQ as a solution of 2x+9=0

Thus, the graph PQ is a line parallel to the y-axis at a distance of units in the direction of negative X-axis.

We hope the NCERT Solutions for Class 9 Maths Chapter 4 Linear Equations in Two Variables Ex 4.4 help you. If you have any query regarding NCERT Solutions for Class 9 Maths Chapter 4 Linear Equations in Two Variables Exx 4.4, drop a comment below and we will get back to you at the earliest.


Solving a Linear Equation

The way to solve a linear equation is to rewrite it in such a form that on the one side of the equality sign we end up with one term only containing x, and on the other side we have one term which is a constant. To achieve this we can perform several operations. Fist of all we can add or subtract a number on both sides of the equation. We must make sure that we perform the action on both sides such that the equality is preserved. Also we can multiply both sides with a number, or divide by a number. Again we must make sure that we perform the same action on both sides of the equality sign.

Our first step would be subtracting 3x on both sides to get:

Then we subtract 4 on both sides:

Finally, we divide both sides by 4 to get our answer:

To check if this answer is indeed correct we can fill it in on both sides of the equation. If the answer is correct we should get two equal answers:

So indeed both sides equal 1/2 if we choose x = - 1/2, which means that the lines intersect at the point (-1/2 , 1/2) in the coordinate system.

Lines of the Equations of the Example


7th & 8th Grade Math - Comparing Linear Functions

In fall of 2008, Sally Keyes (math coach), Kamaljit Sangha (7/8 math teacher/department leader) and Cecilio Dimas (7/8 math teacher) developed our first lesson on cost-analysis. By the end of fall 2008, all eleven 7th grade classes, regular and accelerated, had been taught the DVD Plan lesson. The foundation of this lesson is constructing, communicating, and evaluating student-generated tables while making comparisons between three different financial plans. Students had been given three different DVD rental plans and asked to analyze each one to see if they could determine when the 3 different DVD plans cost the same amount of money, if ever. Using this idea as an anchor problem taken from our original pre-assessment with the MARS task "Gym," we were devising avenues for our students to explore and understand specific multiple representations of breaking points. We believe that being able to understand multiple representations for the breaking even points in a written explanation, a table, a graph, and an algebraic rule is critical to success in algebra. The design of our year-long lesson study is to address each of these multiple representations: verbal, tabular, graphical, and algebraic generalization. Our perception is that students tend to see the graph as the “last thing” with no real connection to the mathematics of the situation or to other representations and that was our reason to put it after the verbal and tabular representations.

-Comparisons of different deals: do these representations make mathematical sense and do they match the mathematics of the three plans?
-Verbal descriptions
-Tables graphs
-Algebraic rules


Watch the video: 4 - Πράξεις με πολυώνυμα Πρόσθεση (November 2021).