Straight-line graphs – Year 8 maths worksheet with answers to download as PDF

Points \(A(-4,3)\) and \(B(2,-5)\) are plotted on a coordinate grid.

1. In which quadrant is each point?
2. Find the midpoint of \(AB\).
3. Find the horizontal and vertical distance between \(A\) and \(B\).
4. Find the length of \(AB\) to 1 decimal place.

Complete a table of values for each equation for \(x=-2,-1,0,1,2\).

1. \(y=2x+1\)
2. \(y=3-x\)
3. \(y=x^2\) is not a line, but complete it anyway and say what shape the points make.
4. Which of the three equations gives a straight line?

Use gradient \(=\dfrac{\text{change in }y}{\text{change in }x}\).

1. Find the gradient of the line through \((1,2)\) and \((4,11)\).
2. Find the gradient of the line through \((-2,5)\) and \((2,-3)\).
3. Find the gradient of the line through \((0,4)\) and \((6,4)\).
4. A line has gradient \(\dfrac{3}{4}\) and passes through \((0,1)\) and \((8,k)\). Find \(k\).

Write down the gradient and the \(y\)-intercept of each line.

1. \(y=4x-3\)
2. \(y=-\frac12x+6\)
3. \(y=2-5x\)
4. \(2y=6x+8\) (rearrange first)
5. \(x+y=7\) (rearrange first)

Find the equation of each line in the form \(y=mx+c\).

1. Gradient 3, passes through \((0,-2)\).
2. Gradient \(-2\), passes through \((1,5)\).
3. Passes through \((0,1)\) and \((2,7)\).
4. Passes through \((1,4)\) and \((3,10)\).

Look at these equations: \(y=2x+3\), \(y=2x-5\), \(y=-x+3\), \(2y=4x+1\), \(y=7\), \(x=-2\).

1. Which lines are parallel to \(y=2x+3\)?
2. Which line is horizontal and which is vertical?
3. Write the equation of a line parallel to \(y=-x+3\) that passes through \((0,6)\).
4. Write the equation of a line parallel to \(y=2x+3\) through \((1,0)\).

A taxi company charges £3 plus £2 for each mile. The fare \(F\) pounds for \(m\) miles is given by \(F=2m+3\).

1. Check the table values using the formula.
2. What are the gradient and the \(F\)-intercept? What do they mean in this context?
3. Use the formula to find the fare for 9.5 miles.
4. Sam paid £21. How far did he travel?
5. A second company charges \(F=1.5m+5\). For what distance are the two fares equal?

A water tank contains 60 litres. Water is added at a steady rate of 15 litres per minute, so after \(t\) minutes the volume is \(V=15t+60\).

1. How much water is in the tank after 4 minutes?
2. How long until it contains 150 litres?
3. What does the gradient represent? What does the intercept represent?
4. The tank holds 300 litres. How long until it is full?

The graph shows the monthly cost \(C\) (£) for \(d\) gigabytes (GB) of data on plan A: \(C=2d+8\). Plan B costs a fixed £14 whatever the data.

1. Find the cost on plan A for 5 GB.
2. For what amount of data are the plans equal?
3. Which plan is cheaper for 2 GB? For 8 GB?
4. Explain what the gradient of plan B tells you.

The formula \(F=1.8C+32\) changes Celsius to Fahrenheit.

1. Find \(F\) when \(C=0\), \(C=10\) and \(C=100\).
2. Write down the gradient and the intercept.
3. Use the formula to convert \(-40\)°C and comment on the result.
4. At what temperature are the two scales equal? Solve \(C=1.8C+32\).

A bike hire shop charges £8 for the first hour and £4 for each extra hour. Write \(C=4h+4\) for \(h\) hours (\(h\ge1\)).

1. Check the formula against the table.
2. Find the cost of a 7-hour hire.
3. Maya paid £44. How many hours did she hire the bike?
4. A rival shop charges \(C=5h\). Where do the two lines cross?

Hamid saves £5 each week and started with £12. The graph of his savings \(S=5w+12\) is a straight line.

1. What are the gradient and \(y\)-intercept in context?
2. After how many weeks does he have £62?
3. His friend Lola starts with £40 and saves £3 each week. Write her equation.
4. When do they have the same amount?

A walker climbs a hill. After \(t\) hours her height above sea level is \(h=150t+300\) metres.

1. What is her starting height?
2. How fast does she climb in metres per hour?
3. At what time has she climbed 450 m from the start?
4. The summit is at 1 200 m. How long does she take to reach it?

A town centre car park charges £1.50 per hour plus a £2 entry fee. The cost is \(C=1.5h+2\) for \(h\) hours.

1. Check the table using the formula.
2. Find the cost of parking for 5 hours.
3. A shopper pays £12.50. How long did she park?
4. The other car park charges £2.50 per hour with no entry fee. When do the two car parks cost the same?

Two lines have equations \(y=2x+1\) and \(y=-x+7\).

1. Complete a table of values for both lines for \(x=0,1,2,3\).
2. Find the coordinates of the point \(P\) where they cross.
3. Check your answer by solving \(2x+1=-x+7\).
4. Use the lines to describe a real situation, such as a shop and its charges, in which this crossing point would matter.

The line \(y=-\dfrac{3}{2}x+6\) crosses the \(y\)-axis at \(A\) and the \(x\)-axis at \(B\). \(O\) is the origin.

1. Write down the coordinates of \(A\).
2. Find the coordinates of \(B\) by setting \(y=0\).
3. Find the area of triangle \(OAB\).
4. Find the length of \(AB\) to 1 decimal place.
5. Write the equation of the line through \(O\) parallel to \(AB\).

Two friends cycle. Tom's distance from home is \(d=12t\) (km, \(t\) in hours). Zara starts 6 km ahead and goes at 8 km/h, so \(d=8t+6\).

1. How far apart are they at the start?
2. Find when Tom catches up with Zara.
3. How far from home are they then?
4. Who is ahead after 3 hours, and by how much?
5. Explain what the two gradients mean.

Four pupils found the equation of the line through \((2,7)\) and \((4,13)\).

1. Work out the gradient.
2. Find the correct equation.
3. Which pupil is correct?
4. Find the mistake each of the others made.

Two taxi firms charge as follows: Firm A: £4 plus £1.50 per mile. Firm B: £2.50 plus £2 per mile.

1. Write the equation of each fare.
2. Find where the two fares are equal.
3. Which firm is cheaper for a 5-mile journey?
4. Draw (describe) the graphs and explain which is cheaper after the crossing point.

Two lines are perpendicular if the product of their gradients is \(-1\). Line \(L_1\) is \(y=2x+1\).

1. Find the gradient of a line perpendicular to \(L_1\).
2. Find the equation of the perpendicular through \((0,5)\).
3. Find where this line meets \(L_1\).
4. Find the shortest distance from \((0,5)\) to \(L_1\) to 1 decimal place.