Linear Functions & Graphs (DP IB Applications & Interpretation (AI): SL): Exam Questions

The equation of a line $l_1$ is $2x-y+6=0$.

For the line $l_1$ find:

(i) the $y$-intercept

(ii) the $x$-intercept

(iii) the gradient.

A new line, $l_2,$ intersects the $x$-axis at (4, 0) and is perpendicular to $l_1$.

Find:

(i) the gradient of the line $l_2$

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

The coordinates of point $A$ are $\left(1,7\right)$ and the coordinates of point $B$ are $\left(5,5\right)$. $M$ is the midpoint of $\left[AB\right]$.

Find the coordinates of $M$.

The line $l_1$ passes through the points $A$ and $B$.

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

A new line, $l_2$, is the perpendicular bisector of $\left[\mathrm{AB}\right]$.

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

The diagram below shows the line $l_1$, which intersects the $y$-axis at A$\left(0,10\right)$  and the $x$-axis at B$\left(5,0\right)$.

Find the equation of $l_{1.}$ Give your answer in the form of $y=mx+c$.

Find $\mathrm{AB}$.

A second line, $l_2$, is parallel to $l_1$ and intersects the $x$-axis at $\mathrm{C}\left(8,0\right)$.

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

Find the coordinates of the point where $l_2$ intersects the $y$-axis.

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.

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

Determine which photocopy shop is cheaper for 220 copies.

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.

Write down an equation for the total monthly cost, $y$ dollars, for each company, where $x$ is the monthly electricity consumption in kWh.

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

Ardie’s monthly expenditure, $C\left(x\right),$ 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.

Find an expression connecting Ardie’s monthly expenditure, $C\left(x\right),$ with his monthly income, $x$.

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

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

The straight lines $l_1$ and $l_2$ are shown in the diagram below $l_1$ intercepts the $x$-axis at $\left(17,0\right)$ and the $y$-axis at $\left(0,17\right)$ and $l_2$  intercepts the $x$-axis at $\left(2,0\right)$ and the $y$-axis at $\left(0,-1\right)$.

Giving your answer in the form $y=mx+c$, find:

(i) the equation of $l_1$

(ii) the equation of $l_2$.

Find the area of the shaded region.

A line passing through the origin $\mathrm{O}$, is perpendicular to a line with equation $x+y=16.$ The two lines meet at point $\mathrm{R}$. $\mathrm{P}$ lies on $\left[\mathrm{OR}\right]$ such that $\mathrm{OP}:\mathrm{PR}=3:1$.

Find the equation of the perpendicular line and hence, the coordinates of point $\mathrm{R}$.

Find the coordinates of $\mathrm{P}$.

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 $Z$.

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 triangle PQR.

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

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

The line $l_2$ passes through $\mathrm{A}$ and has a gradient of 5.

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

Point $\mathrm{C}$ is the intersection of $l_1$ and $l_2$.

Find the coordinates of $\mathrm{C}$.

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

Find $\mathrm{AC}$.

Point $\mathrm{D}$ lies on $\left(\mathrm{AC}\right)$, and $\left(\mathrm{BD}\right)$ is perpendicular to $\left(\mathrm{AC}\right)$.

Find the coordinates of $\mathrm{D}$.

Calculate the area of the triangle ABC.

Point $\mathrm{P}$ has coordinates $\left(6,-2\right)$, and $\mathrm{R}$ has coordinates $\left(-2,2\right)$. The equation of $\left(\mathrm{PQ}\right)$ is $y=-2$ and the equation of $\left(\mathrm{QR}\right)$ is $y=2x+6$.

Find the coordinates of point $\mathrm{Q}$.

Find $\mathrm{PQ}$ and $\mathrm{QR}$.

Find the area of triangle $\mathrm{PQR}$.

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 $\mathrm{B}\left(3p,0\right)$.

Find 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 $\mathrm{C}$.

Find the area of triangle $\mathrm{OAC}$, where $\mathrm{O}$ is the origin.

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

Find the value of $x$ and the value of $y$.

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 $\mathrm{A}\left(-3,-5\right)$ and $\mathrm{B}\left(-1,-7\right)$.

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

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

Find the coordinates of $\mathrm{C}$.

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

Find the coordinates of $\mathrm{D}$.

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.

Find the gradient of the west facing slope.

The gradient of the east facing slope in the diagram is  $-\frac{3}{10}$.

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

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

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.

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%.

Let $y$ be the total charge, in dollars and including tax, made by Carpenter A for a job taking $x$ hours.

Find an equation for $y$ in terms of $x$.

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

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

Find the area of triangle $\mathrm{GFH}$.

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 $\mathrm{C}$ lies on $l_2$ and has a negative $x$-coordinate. $\mathrm{QC}=14$.

Find the coordinates of $\mathrm{C}$.

Point $\mathrm{A}$ has coordinates $\left(0,y\right)$. $l_1$ passes through points $\mathrm{A}$ and $\mathrm{B}$ and has the equation $4x-3y-3=0$. $\mathrm{AB}=10$.

Find the possible coordinates of point $\mathrm{B}$.

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

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

Point $\mathrm{C}$ has coordinates $\left(x,4\right)$ and $\mathrm{CD}=\frac{1}{2}\mathrm{AB}$.

Find the possible coordinates of point $\mathrm{D}$.

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 $\mathrm{C}$ lies on $l_2$ such that $\mathrm{MC}=12$.

Find the possible coordinates of $\mathrm{C}$.

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

Show that $\left[\mathrm{AB}\right]$ and $\left[\mathrm{CD}\right]$ are parallel.

Find:

(i) $\mathrm{AB}$

(ii) $\mathrm{CD}$.

Find the area of the quadrilateral $\mathrm{ABCD}$.

The line $l_1$ has the equation $5y-2x+1=0$. Point $\mathrm{A}$ has coordinates $\left(x,3\right)$ and is the intersection of $l_1$ and $l_2$. $l_2$ is perpendicular to $l_1$.

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

Point $\mathrm{B}$ lies on $l_2$ and is positioned such that $\mathrm{AB}=12$ and the $x$-coordinate of $\mathrm{B}$ is less than the $x$-coordinate of $\mathrm{A}$.

Find the coordinates of $\mathrm{B}$.