2.1 Linear Functions & Graphs

The coordinates of pointA{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="A" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">are2, 8{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 2 comma space 8 close parentheses" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">and the coordinates of pointBare-8, 2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses negative 8 comma space 2 close parentheses" style="vertical-align:-6px;height:22px;width:53px" loading="lazy">.M{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="M" style="vertical-align:-6px;height:22px;width:18px" loading="lazy">is the midpoint ofAB{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open square brackets A B close square brackets" style="vertical-align:-6px;height:22px;width:41px" loading="lazy">.

Find the coordinates ofM{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="M" style="vertical-align:-6px;height:22px;width:18px" loading="lazy">.

l1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">The line passes throughA{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="A" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">andB{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="B" style="vertical-align:-6px;height:22px;width:14px" loading="lazy">.

Find the gradient ofl1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">.

Find the equation of the linel1.{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1." style="vertical-align:-12px;height:28px;width:19px" loading="lazy">Give your answer in the formax+by+d=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a x plus b y plus d equals 0" style="vertical-align:-6px;height:22px;width:108px" loading="lazy">, wherea, b{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a comma space b" style="vertical-align:-6px;height:22px;width:29px" loading="lazy">andd{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="d" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">are integers.

The coordinates of pointA{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="A" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">are1, 7{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 1 comma space 7 close parentheses" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">and the coordinates of pointB{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="B" style="vertical-align:-6px;height:22px;width:14px" loading="lazy">are5, 5{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 5 comma space 5 close parentheses" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">.M{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="M" style="vertical-align:-6px;height:22px;width:18px" loading="lazy">is the midpoint ofAB{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open square brackets A B close square brackets" style="vertical-align:-6px;height:22px;width:41px" loading="lazy">.

Find the coordinates ofM{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="M" style="vertical-align:-6px;height:22px;width:18px" loading="lazy">.

The linel1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">passes through the pointsA{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="A" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">andB{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="B" style="vertical-align:-6px;height:22px;width:14px" loading="lazy">.

Find the equation ofl1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">. Give your answer in the form of y=mx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y equals m x plus c" style="vertical-align:-6px;height:22px;width:82px" loading="lazy">.

A new line,l2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">, is the perpendicular bisector tol1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">.

Find the equation ofl2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">. Give your answer in the form ofy=mx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals m x plus c" style="vertical-align:-6px;height:22px;width:78px" loading="lazy">.

水管工的指控fixed fee of $25 plus $15 per hour.

Usingt{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="t" style="vertical-align:-6px;height:22px;width:7px" loading="lazy">for the number of hours a job takes, andCA{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="C subscript A" style="vertical-align:-12px;height:28px;width:24px" loading="lazy">for the total cost of a job, in dollars, from Plumber A, write down an equation connectingt{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="t" style="vertical-align:-6px;height:22px;width:7px" loading="lazy">andCA{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="C subscript A" style="vertical-align:-12px;height:28px;width:24px" loading="lazy">.

A job takes the plumber seven hours.

Calculate the total cost of the job.

管道工B收取固定费用+ $ 20美元16 per hour.

Usingt{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="t" style="vertical-align:-6px;height:22px;width:7px" loading="lazy">for the number of hours a job takes, andCB{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="C subscript B" style="vertical-align:-12px;height:28px;width:24px" loading="lazy">for the total cost of a job, in dollars, from Plumber B, write down an equation connectingt{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="t" style="vertical-align:-6px;height:22px;width:7px" loading="lazy">andCB{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="C subscript B" style="vertical-align:-12px;height:28px;width:24px" loading="lazy">.

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

The diagram below shows the linel1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">, which intersects they{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis at A0,10{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 0 comma 10 close parentheses" style="vertical-align:-6px;height:22px;width:45px" loading="lazy">and thex{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis at B5,0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 5 comma 0 close parentheses" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">.

Find the equation ofl1.{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1. end subscript" style="vertical-align:-12px;height:28px;width:18px" loading="lazy">Give your answer in the form ofy=mx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals m x plus c" style="vertical-align:-6px;height:22px;width:78px" loading="lazy">.

Find the length of [AB].

A second line,l2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">is parallel tol1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">and intersects thex{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis at C8,0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 8 comma 0 close parentheses" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">.

Find the equation ofl2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">Give your answer in the formax+by+d=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a x plus b y plus d equals 0" style="vertical-align:-6px;height:22px;width:108px" loading="lazy">, wherea, b{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a comma space b" style="vertical-align:-6px;height:22px;width:29px" loading="lazy">andd{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="d" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">are integers.

Find the coordinates wherel2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">intersects they{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis.

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

Assuming a linear relationship, find

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.

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 costy{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">is a function of the monthly electricity consumptionx{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">in kWh.

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

Ardie’s monthly expenditure,C x,{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="C space open parentheses x close parentheses comma" style="vertical-align:-6px;height:22px;width:45px" loading="lazy">is a linear function of his monthly income,x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">. 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 x,{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="C space open parentheses x close parentheses comma" style="vertical-align:-6px;height:22px;width:45px" loading="lazy">with his monthly income,x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">.

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 diagram below represents a mountain with a west facing slope and an east facing slope labelledW{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="W" style="vertical-align:-6px;height:22px;width:19px" loading="lazy">andE{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="E" style="vertical-align:-6px;height:22px;width:13px" loading="lazy">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-310{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="negative 3 over 10" style="vertical-align:-17px;height:47px;width:40px" loading="lazy">.

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).

The straight linesl1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">andl2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">如下图所示吗l1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">intercepts thex{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis at17, 0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 17 comma space 0 close parentheses" style="vertical-align:-6px;height:22px;width:49px" loading="lazy">and they{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis at0,17{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 0 comma 17 close parentheses" style="vertical-align:-6px;height:22px;width:45px" loading="lazy">andl2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">intercepts thex{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis at2, 0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 2 comma space 0 close parentheses" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">and they{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis at0, -1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 0 comma space minus 1 close parentheses" style="vertical-align:-6px;height:22px;width:57px" loading="lazy">.

Giving your answer in the formy=mx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals m x plus c" style="vertical-align:-6px;height:22px;width:78px" loading="lazy">, find:

the equation ofl1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 1" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">

the equation ofl2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="l subscript 2" style="vertical-align:-12px;height:28px;width:14px" loading="lazy">.

Find the area of the shaded region.

A line passing through the originO{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="O" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">, is perpendicular to a line with equationx + y = 16.{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x thin space plus space y space equals space 16." style="vertical-align:-6px;height:22px;width:94px" loading="lazy">
The two lines meet at pointR{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="R" style="vertical-align:-6px;height:22px;width:14px" loading="lazy">.P{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="P" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">is a point such thatOP : PR= 3 : 1.

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

Find the coordinates ofP.