IBMaths: AA HLDPTopic Questions2. Functions2.1 Linear Functions & Graphs

Linear Functions & Graphs (DP IB Maths: AA HL)

Topic Questions

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2.1 Linear Functions & Graphs

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1a
Sme Calculator3 marks

The equation of a line l subscript 1 is 2 x minus y plus 6 equals 0.

For the line l subscript 1 find:

(i)
the y-intercept

(ii)
the x-intercept

(iii)
the gradient.

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1b
Sme Calculator3 marks

A new line, l subscript 2 comma intersects the x-axis at (4, 0) and is perpendicular to l subscript 1.

Find:

(i)
the gradient of the line l subscript 2

(ii)
the equation of the line l subscript 2 Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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2a
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The coordinates of point A are open parentheses 2 comma space 8 close parentheses and the coordinates of point B are open parentheses negative 8 comma space 2 close parentheses. M is the midpoint of open square brackets A B close square brackets.

Find the coordinates of M.

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2b
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The line l subscript 1passes through A and B.

Find the gradient of l subscript 1.

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2c
Sme Calculator3 marks

Find the equation of the line l subscript 1. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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3a
Sme Calculator2 marks

The coordinates of point A are open parentheses 1 comma space 7 close parentheses and the coordinates of point B are open parentheses 5 comma space 5 close parentheses. M is the midpoint of open square brackets A B close square brackets.

Find the coordinates of M.

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3b
Sme Calculator2 marks

The line l subscript 1 passes through the points A and B.

Find the equation of l subscript 1. Give your answer in the form of space y equals m x plus c.

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3c
Sme Calculator3 marks

A new line, l subscript 2, is the perpendicular bisector to l subscript 1.

Find the equation of l subscript 2. Give your answer in the form of y equals m x plus c.

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4a
Sme Calculator2 marks

Plumber A charges a fixed fee of $25 plus $15 per hour.

Using t for the number of hours a job takes, and C subscript A for the total cost of a job, in dollars, from Plumber A, write down an equation connecting t and C subscript A.

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4b
Sme Calculator2 marks

A job takes the plumber seven hours.

Calculate the total cost of the job.

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4c
Sme Calculator2 marks

Plumber B charges a fixed fee of $20 plus $16 per hour.

Using t for the number of hours a job takes, and C subscript B for the total cost of a job, in dollars, from Plumber B, write down an equation connecting t and C subscript B.

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4d
Sme Calculator3 marks

Determine which plumber would be the cheapest for a job taking six hours.

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5a
Sme Calculator2 marks

The diagram below shows the line l subscript 1, which intersects the y-axis at Aopen parentheses 0 comma 10 close parentheses  and the x-axis at Bopen parentheses 5 comma 0 close parentheses.

q5-2-1-medium-linear-functions-and-graphs

Find the equation of l subscript 1. end subscript Give your answer in the form of y equals m x plus c.

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5b
Sme Calculator2 marks

Find the length of [AB].

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5c
Sme Calculator2 marks

A second line, l subscript 2 is parallel to l subscript 1 and intersects the x-axis at Copen parentheses 8 comma 0 close parentheses.

Find the equation of l subscript 2 Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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5d
Sme Calculator1 mark

Find the coordinates where l2  intersects the y-axis.

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6a
Sme Calculator4 marks

Photocopy shop A charges $122 for 115 copies, and $190 for 200 copies.

Assuming a linear relationship, find

(i)
the price for 180 copies

(ii)
how many copies could be made for $385.20.

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6b
Sme Calculator3 marks

Photocopy shop B charges $0.82 per copy and a fixed fee of $25.50.

State which photocopy shop is cheaper to make 220 copies.

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7a
Sme Calculator2 marks

A family can be supplied with electricity by two companies that have different pricing structures:

                      Company A: Fixed fee of $25/month and $0.2 per kWh consumed.

                      Company B: Fixed fee of $10/month and $0.22 per kWh consumed.

Determine the equation of the cost function for both companies, where the total monthly cost y is a function of the monthly electricity consumption x in kWh.

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7b
Sme Calculator4 marks

Calculate the monthly energy consumption that results in the same monthly cost from both companies.

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8a
Sme Calculator2 marks

Ardie’s monthly expenditure, C space open parentheses x close parentheses comma is a linear function of his monthly income, x. Ardie’s monthly expenditure is $1000 when his monthly income is $1200 and his monthly expenditure increases by $60 for every $150 increase in his monthly income.

Write an expression connecting Ardie’s monthly expenditure, C space open parentheses x close parentheses comma with his monthly income, x.

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8b
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Calculate Ardie’s monthly expenditure when his monthly income is $1885. Give your answer to the nearest dollar.

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8c
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Find Ardie’s monthly income when his monthly expenditure is $1070. Give your answer to the nearest dollar.

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9a
Sme Calculator1 mark

The diagram below represents a mountain with a west facing slope and an east facing slope labelled W and E respectively.

Horizontal scale: 1 unit represents 100 m.

Vertical scale: 1 unit represents 100 m.

q9-2-1-medium-linear-functions-and-graphs

Find the gradient of the west facing slope.

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9b
Sme Calculator6 marks

The gradient of the east facing slope in the diagram is  negative 3 over 10.

Find the total distance to hike over the mountain in km.

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9c
Sme Calculator1 mark

Suggest a reason as to why the actual total distance hiked may be greater than the distance found in part (b).

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10a
Sme Calculator4 marks

The straight lines l subscript 1 and l subscript 2 are shown in the diagram below l subscript 1 intercepts the x-axis at open parentheses 17 comma space 0 close parentheses and the y-axis at open parentheses 0 comma 17 close parentheses and l subscript 2  intercepts the x-axis at open parentheses 2 comma space 0 close parentheses and the y-axis at open parentheses 0 comma space minus 1 close parentheses.

q10-2-1-medium-linear-functions-and-graphs

Giving your answer in the form y equals m x plus c, find:

(i)
the equation of l subscript 1

(ii)
the equation of l subscript 2.

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10b
Sme Calculator4 marks

Find the area of the shaded region.

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11a
Sme Calculator3 marks

A line passing through the origin O, is perpendicular to a line with equation x thin space plus space y space equals space 16.
The two lines meet at point R. P is a point such that OP : PR = 3 : 1.

Find the equation of the perpendicular line and hence, the co-ordinates of point R.

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11b
Sme Calculator2 marks

Find the coordinates of P.

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1a
Sme Calculator1 mark

The diagram below shows the line l with the equation 2 x minus 4 y plus 6 equals 0.

Point P has coordinates open parentheses negative 1 comma 5 close parentheses point Q has coordinates open parentheses 7 comma space 5 close parentheses and point R has coordinates open parentheses 3 comma space 3 close parentheses. M is the midpoint of [PQ].

q1-2-1-hard-linear-functions-and-graphs

Write down the coordinates of M.

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1b
Sme Calculator2 marks

Show that Q lies on l.

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1c
Sme Calculator2 marks

Show that R lies on l.

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1d
Sme Calculator2 marks

Find the area of PQR.

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2a
Sme Calculator2 marks

The line l subscript 1 passes through the points open parentheses 1 comma space 7 close parentheses and open parentheses 5 comma space 5 close parentheses.

Find the equation of l subscript 1. Give your answer in the form of y equals m x plus c.

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2b
Sme Calculator2 marks

A new line, l subscript 2, is perpendicular to l subscript 1 and passes through the point open parentheses 4 comma space 8 close parentheses.

Find the equation of l subscript 2. Give your answer in the form of y equals m x plus c.

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2c
Sme Calculator2 marks

The point Z is the intersection of l subscript 1 and l subscript 2.

Find the coordinates of Z.

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3a
Sme Calculator3 marks

Point A has coordinates open parentheses x comma space 4 close parentheses and point B has coordinates open parentheses 9 comma space y close parentheses. M is the midpoint of open square brackets A B close square brackets and has coordinates open parentheses negative 2 comma space 7 close parentheses.

Find the value of x and y.

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3b
Sme Calculator2 marks

The line l subscript 1 passes through A and B.

Find the equation of l subscript 1. Give your answer in the form a x plus b y plus d equals 0 comma where a comma space b and d are integers.

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4a
Sme Calculator2 marks

The line l subscript 1 passes through A open parentheses negative 3 comma space minus 5 close parentheses and B open parentheses negative 1 comma space minus 7 close parentheses .

Find the equation of l subscript 1. Give your answer in the form y equals m x plus c.

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4b
Sme Calculator2 marks

Point C is such that B is the midpoint of open square brackets A C close square brackets.

Find the coordinates of C.

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4c
Sme Calculator2 marks

Point D is such that C is the midpoint of open square brackets A D close square brackets.

Find the coordinates of D.

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5a
Sme Calculator3 marks

Point A has coordinates open parentheses 11 comma space 12 close parentheses and point B has coordinates open parentheses 4 comma space minus 8 close parentheses. The line l subscript 1 passes through M, the midpoint of open square brackets A B close square brackets. The gradient of l subscript 1 is minus2.

Write down the equation of l subscript 1, giving your answer in the form y equals m x plus c.

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5b
Sme Calculator1 mark

The line l subscript 2 passes through A and has a gradient of 5.

Write down the equation of l subscript 2, giving your answer in the form y equals m x plus c.

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5c
Sme Calculator2 marks

Point C is the intersection of l subscript 1 and l subscript 2.

Write down the coordinates of C, giving the x and y coordinates as fractions.

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6a
Sme Calculator2 marks

Finn borrows $3200 from his parents and decides to pay them back $c in the first month and then $m each subsequent month.

After two months Finn has paid back his parents a total of $1000, this can be expressed as m plus c space equals space 1000. After half a year he still owes his parents $1000.

Write down another equation connecting m and c.

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6b
Sme Calculator2 marks

Find the value of m and c.

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6c
Sme Calculator2 marks

Finn’s parents apply a 6.25% net interest to the $3200 total.

Calculate the number of months it takes Finn to pay back his parents.

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7a
Sme Calculator3 marks

The line l subscript 1 has the equation 5 y minus 2 x plus 1 equals 0. Point A has coordinates left parenthesis x comma space 3 right parenthesis and is the intersection of l subscript 1 and l subscript 2. l subscript 2 is perpendicular to l subscript 1.

Write down the equation of l subscript 2, giving your answer in the form y equals m x plus c.

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7b
Sme Calculator5 marks

Point B lies on l subscript 2 and is positioned such that AB is 12 units and the x-coordinate for B is less than the x-coordinate for A.

Find the coordinates of B.

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8
Sme Calculator8 marks

The points Kopen parentheses 6 comma space y close parentheses and Nopen parentheses x comma space 9 close parentheses lie on the line l subscript 1 where x, y element of straight integer numbers.

KN has a length of 5 units.

Find all the possible values for x and y.

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9a
Sme Calculator2 marks

Point A has coordinates open parentheses 2 comma space 1 close parentheses, point B has coordinates open parentheses 5 comma space 7 close parentheses, and point C has coordinates (8, 1).

Calculate the length of AC.

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9b
Sme Calculator3 marks

Point D lies on the line open parentheses A C close parentheses. The line open parentheses B D close parentheses is perpendicular to the line (AC).

Find the coordinates of D.

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9c
Sme Calculator3 marks

Calculate the area of the triangle ABC.

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10a
Sme Calculator3 marks

The distance between points Popen parentheses 20 comma space 40 close parentheses  and Qopen parentheses x comma space 20 close parentheses is equal to 25 units.

Find the possible values of x.

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10b
Sme Calculator3 marks

It is given that x less than 20.

Point R is such that point Q is the midpoint of the line segment with endpoints P and R.

Write down the coordinates of R.

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1a
Sme Calculator3 marks

A cat breeder is measuring the rate at which a kitten grows by measuring its “back-length”, which is the distance from the base of the neck to the base of the tail.
At 1 week old the kitten measured 2.42 cm. 8 weeks later the kitten’s back length had doubled.

Using a linear model, find an equation linking , the kitten’s “back-length” in centimetres, to x, the age of the kitten in weeks.

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1b
Sme Calculator1 mark

Find the kitten’s “back-length” at birth.

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1c
Sme Calculator1 mark

Use the model to find the age, in weeks, of the kitten that has a “back-length” of 7 cm.

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1d
Sme Calculator1 mark

This breed of cat is fully grown after 50 weeks.

Find the cat’s “back-length” at 50 weeks.

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1e
Sme Calculator2 marks

Comment on the suitability of the linear model used.

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2a
Sme Calculator2 marks

Point P has coordinates open parentheses 6 comma space minus 2 close parentheses, R has coordinates open parentheses negative 2 comma space 2 close parentheses. The equation of the line PQ is y space equals space minus 2 and the equation of the line QR is y equals 2 x plus 6.

Find the coordinates of point Q.

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2b
Sme Calculator3 marks

Find the distance of PQ and QR.

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2c
Sme Calculator2 marks

Find the area of triangle PQR.

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3a
Sme Calculator4 marks

Point P has coordinates (4, 2) and point Q has coordinates (1, 8). l subscript 1 is the perpendicular bisector of [PQ].

Find the equation of l subscript 1, giving your answer in the form y equals m x plus c.

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3b
Sme Calculator2 marks

l subscript 2 passes through Q and has a gradient of minus3.

Find the equation of l subscript 2, giving your answer in the form y equals m x plus c.

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3c
Sme Calculator4 marks

Point C lies on l subscript 2 and has a negative x coordinate. The length of QC is 14 units.

Find the coordinates of C.

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4a
Sme Calculator3 marks

Point A has coordinates open parentheses 0 comma space y close parentheses, l subscript 1 passes through points A and B and has the equation 4 x minus 3 y minus 3 equals 0. The length of [AB] is 10 units.

Find the possible coordinates of point B.

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4b
Sme Calculator2 marks

l subscript 2 passes through points C and D, is parallel to l subscript 1 and crosses the x-axis when x space equals 5.

Write down the equation of l subscript 2. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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4c
Sme Calculator3 marks

Point C has coordinates open parentheses x comma space 4 close parentheses and CD is half the length of AB.

Find the possible coordinates of point D.

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5a
Sme Calculator2 marks

The line l subscript 1 passes through Aleft parenthesis 10 comma negative 12 right parenthesis  and Bleft parenthesis 5 comma space 2 right parenthesisq5-2-1--very-hard-linear-functions-and-graphs

Find the equation of l subscript 1. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

.

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5b
Sme Calculator3 marks

The line l subscript 2 passes through M, the midpoint of [AB] and has a gradient of 7.

Find the equation of l subscript 2. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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5c
Sme Calculator5 marks

Point C lies on l subscript 2 such that MC has a length of 12 units.

Find the possible coordinates of C.

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6a
Sme Calculator2 marks

Carpenter A charges a fixed fee of $28 plus $22.50 per hour. The carpenter then charges tax on top of the total cost of 21%.

Defining suitable variables, write down an equation to represent the charges, including tax, made by the carpenter.

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6b
Sme Calculator4 marks

Carpenter B charges a fixed fee of $35 and $26.50 per hour. These prices already account for tax.

Find the number of hours for which both carpenters charge the same amount.

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6c
Sme Calculator2 marks

Tom needs a job doing by a carpenter that he estimates will take 8.5 hours. Carpenter C has given Tom a fixed quote of $250.

Determine which carpenter would be the cheapest option for Tom.

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7a
Sme Calculator1 mark

The line l subscript 1 has equation 6 x plus 4 y minus 21 equals 0 and crosses the x-axis at point Aopen parentheses p comma space 0 close parentheses.

Find the value of p.

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7b
Sme Calculator2 marks

The line l subscript 2 is perpendicular to l subscript 1 and crosses the x-axis at Bopen parentheses 3 p comma space 0 close parentheses.

Write down the equation of l subscript 2. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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7c
Sme Calculator5 marks

l subscript 2 crosses the y-axis at point C.

Find the area of the triangle OAC, where O is the origin. Give your answer as a fraction.

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8
Sme Calculator8 marks

The points Aopen parentheses x comma space minus 5 close parentheses  and Bopen parentheses negative 1 comma space y close parentheses  lie on the line l subscript 1 where x comma space y space element of space straight integer numbers. AB has a length of 13 units.

Find all the possible x and y values.

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9a
Sme Calculator1 mark

A quadrilateral has four vertices with coordinates  straight A left parenthesis negative 1 comma negative 1 right parenthesis comma space straight B left parenthesis 2 comma 2 right parenthesis comma space straight C left parenthesis 4 comma 2 right parenthesis comma space straight D left parenthesis negative 1 comma negative 3 right parenthesis.

Show that [AB] and [CD] are parallel.

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9b
Sme Calculator3 marks

Calculate the distance of

(i)
AB.

(ii)
CD.

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9c
Sme Calculator4 marks

Find the area of the quadrilateral ABCD.

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10
Sme Calculator6 marks

The point straight F open parentheses negative 2 comma space 4 close parentheses lies on the line l subscript 1. l subscript 1 crosses the y-axis at point straight G.

Another line, l subscript 2, is perpendicular to l subscript 1 at the point F and crosses the y-axis at the point straight H open parentheses 0 comma space minus 3 close parentheses .

Find the area of the triangle GFH. Give your answer as a fraction.

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