Linear Functions & Graphs (DP IB Analysis & Approaches (AA): SL): Exam Questions

The diagram below shows the line $l$ with the equation $2x-4y+6=0$.

Point P has coordinates $\left(-1,5\right)$ point Q has coordinates $\left(7,5\right)$ and point R has coordinates $\left(3,3\right)$. M is the midpoint of [PQ].

Write down the coordinates of M.

Show that Q lies on l.

Show that R lies on l.

Find the area of PQR.

The line $l_1$ passes through the points $\left(1,7\right)$ and $\left(5,5\right)$.

Find the equation of $l_1.$ Give your answer in the form of $y=mx+c$.

A new line, $l_2$, is perpendicular to $l_1$ and passes through the point $\left(4,8\right)$.

Find the equation of $l_2$. Give your answer in the form of $y=mx+c$.

The point $Z$ is the intersection of $l_1$ and $l_2$.

Find the coordinates of $Z$.

Point A has coordinates $\left(x,4\right)$ and point B has coordinates $\left(9,y\right)$. M is the midpoint of $\left[AB\right]$ and has coordinates $\left(-2,7\right)$.

Find the value of x and y.

The line $l_1$ passes through A and B.

Find the equation of $l_1$. Give your answer in the form $ax+by+d=0,$ where $a,b$ and $d$ are integers.

The line $l_1$ passes through $A\left(-3,-5\right)$ and $B\left(-1,-7\right)$ .

Find the equation of $l_1$. Give your answer in the form $y=mx+c$.

Point C is such that B is the midpoint of $\left[AC\right]$.

Find the coordinates of C.

Point D is such that C is the midpoint of $\left[AD\right]$.

Find the coordinates of D.

Point A has coordinates $\left(11,12\right)$ and point B has coordinates $\left(4,-8\right)$. The line $l_1$ passes through M, the midpoint of $\left[AB\right]$. The gradient of $l_1$ is $-$2.

Write down the equation of $l_1$, giving your answer in the form $y=mx+c$.

The line $l_2$ passes through A and has a gradient of 5.

Write down the equation of $l_2$, giving your answer in the form $y=mx+c$.

Point C is the intersection of $l_1$ and $l_2$.

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

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 $m+c=1000$. 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 $l_1$ has the equation $5y-2x+1=0$. Point A has coordinates $(x,3)$ and is the intersection of $l_1$ and $l_2.$ $l_2$ is perpendicular to $l_1$.

Write down the equation of $l_2$, giving your answer in the form $y=mx+c.$

Point B lies on $l_2$ 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.

The points K$\left(6,y\right)$ and N$\left(x,9\right)$ lie on the line $l_1$ where $x$, $y\in \mathrm{\mathbb{Z}}$.

KN has a length of 5 units.

Find all the possible values for $x$ and $y$.

Point A has coordinates $\left(2,1\right)$, point B has coordinates $\left(5,7\right)$, and point C has coordinates (8, 1).

Calculate the length of AC.

Point D lies on the line $\left(AC\right)$. The line $\left(BD\right)$ is perpendicular to the line (AC).

Find the coordinates of D.

Calculate the area of the triangle ABC.

The distance between points P$\left(20,40\right)$  and Q$\left(x,20\right)$ is equal to 25 units.

Find the possible values of $x$.

It is given that $x<20.$

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.

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 $y$, the kitten’s “back-length” in centimetres, to $x$, 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 $\left(6,-2\right)$, R has coordinates $\left(-2,2\right)$. The equation of the line PQ is $y=-2$ and the equation of the line QR is $y=2x+6$.

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). $l_1$ is the perpendicular bisector of [PQ].

Find the equation of $l_1$, giving your answer in the form $y=mx+c$.

$l_2$ passes through Q and has a gradient of $-$3.

Find the equation of $l_2$, giving your answer in the form $y=mx+c$.

Point C lies on $l_2$ and has a negative $x$ coordinate. The length of QC is 14 units.

Find the coordinates of C.

Point A has coordinates $\left(0,y\right)$, $l_1$ passes through points A and B and has the equation $4x-3y-3=0.$ The length of [AB] is 10 units.

Find the possible coordinates of point B.

$l_2$ passes through points C and D, is parallel to $l_1$ and crosses the $x$-axis when $x=5$.

Write down the equation of $l_2$. Give your answer in the form $ax+by+d=0$, where $a,b$ and $d$ are integers.

Point C has coordinates $\left(x,4\right)$ and CD is half the length of AB.

Find the possible coordinates of point D.

The line $l_1$ passes through A$(10,-12)$  and B$(5,2)$

Find the equation of $l_1$. Give your answer in the form $ax+by+d=0$, where $a,b$ and $d$ are integers.

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The line $l_2$ passes through M, the midpoint of [AB] and has a gradient of 7.

Find the equation of $l_2$. Give your answer in the form $ax+by+d=0$, where $a,b$ and $d$ are integers.

Point C lies on $l_2$ 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 $l_1$ has equation $6x+4y-21=0$ and crosses the $x$-axis at point A$\left(p,0\right)$.

Find the value of $p$.

The line $l_2$ is perpendicular to $l_1$ and crosses the $x$-axis at B$\left(3p,0\right)$.

Write down the equation of $l_2$. Give your answer in the form $ax+by+d=0$, where $a,b$ and $d$ are integers.

$l_2$ 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.

The points A$\left(x,-5\right)$  and B$\left(-1,y\right)$  lie on the line $l_1$ where $x,y\in \mathrm{\mathbb{Z}}.$ AB has a length of 13 units.

Find all the possible $x$ and $y$ values.

A quadrilateral has four vertices with coordinates  $\mathrm{A}(-1,-1),\mathrm{B}(2,2),\mathrm{C}(4,2),\mathrm{D}(-1,-3)$.

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

Calculate the distance of

(i) $\mathrm{AB}.$

(ii) $\mathrm{CD}.$

Find the area of the quadrilateral ABCD.

The point $\mathrm{F}\left(-2,4\right)$ lies on the line $l_1.$ $l_1$ crosses the $y$-axis at point $\mathrm{G}$.

Another line, $l_2$, is perpendicular to $l_1$ at the point F and crosses the $y$-axis at the point $\mathrm{H}\left(0,-3\right)$ .

Find the area of the triangle $\mathrm{GFH}$. Give your answer as a fraction.