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

The diagram below shows the line  with the equation .

Point P has coordinates  point Q has coordinates  and point R has coordinates . M is the midpoint of [PQ].

Write down the coordinates of M.

Show that Q lies on .

Show that R lies on 

Find the area of PQR.

The line  passes through the points  and .

Find the equation of Give your answer in the form of .

A new line, , is perpendicular to  and passes through the point .

Find the equation of . Give your answer in the form of .

The point  is the intersection of  and .

Find the coordinates of .

Point A has coordinates  and point B has coordinates . M is the midpoint of  and has coordinates .

Find the value of x and y.

The line  passes through A and B.

Find the equation of . Give your answer in the form where  and  are integers.

The line  passes through and .

Find the equation of . Give your answer in the form .

Point C is such that B is the midpoint of .

Find the coordinates of C.

Point D is such that C is the midpoint of .

Find the coordinates of D.

Point A has coordinates  and point B has coordinates . The line  passes through M, the midpoint of . The gradient of is 2.

Write down the equation of , giving your answer in the form .

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

Write down the equation of , giving your answer in the form .

Point C is the intersection of  and .

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

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

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

Write down another equation connecting m and c.

Find the value of m and c.

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.

The line  has the equation . Point A has coordinates and is the intersection of  and   is perpendicular to .

Write down the equation of , giving your answer in the form

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

Find the coordinates of B.

The points K and N lie on the line  where , .

KN has a length of 5 units.

Find all the possible values for  and .

Point A has coordinates , point B has coordinates , and point C has coordinates (8, 1).

Calculate the length of AC.

Point D lies on the line . The line is perpendicular to the line (AC).

Find the coordinates of D.

Calculate the area of the triangle ABC.

The distance between points P  and Q is equal to 25 units.

Find the possible values of .

It is given that

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.

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 , the age of the kitten in weeks.

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

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

This breed of cat is fully grown after 50 weeks.

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

Comment on the suitability of the linear model used.

Point P has coordinates , R has coordinates . The equation of the line PQ is  and the equation of the line QR is .

Find the coordinates of point Q.

Find the distance of PQ and QR.

Find the area of triangle PQR.

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

Find the equation of , giving your answer in the form .

 passes through Q and has a gradient of 3.

Find the equation of , giving your answer in the form .

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

Find the coordinates of C.

Point A has coordinates , passes through points A and B and has the equation  The length of [AB] is 10 units.

Find the possible coordinates of point B.

 passes through points C and D, is parallel to  and crosses the -axis when .

Write down the equation of . Give your answer in the form , where and  are integers.

Point C has coordinates  and CD is half the length of AB.

Find the possible coordinates of point D.

The line  passes through A  and B

Find the equation of . Give your answer in the form , where  and  are integers.

.

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

Find the equation of . Give your answer in the form , where  and  are integers.

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

Find the possible coordinates of C.

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.

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.

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.

The line  has equation  and crosses the -axis at point A.

Find the value of .

The line  is perpendicular to  and crosses the -axis at B.

Write down the equation of . Give your answer in the form , where  and  are integers.

 crosses the -axis at point C.

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

The points A  and B  lie on the line  where AB has a length of 13 units.

Find all the possible  and  values.

A quadrilateral has four vertices with coordinates  .

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

Calculate the distance of

Find the area of the quadrilateral .

The point  lies on the line  crosses the -axis at point .

Another line, , is perpendicular to  at the point F and crosses the -axis at the point  .

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