Algebra, Year 9: straight-line graphs

gradient

HintIt measures how steep the line is.

The whyThe m is the number multiplying x. Each time x goes up by 1, y changes by m, so a bigger m means a steeper line and a negative m means a line that falls.

y = 3x + 4

HintEvery term on both sides has to be divided by the number in front of the y.

The whyDividing each term by 2 gives y = 3x + 4, so the gradient is 3 and the line crosses the y-axis at 4. The gradient and intercept can only be read off once the equation starts "y =".

2

HintThe number multiplying x tells you how the height changes for each step across.

The whyThe intercept −1 gives the starting point on the y-axis, and the gradient 2 says "up 2 for every 1 across", leading to (1, 1), (2, 3) and so on. Two facts from the equation are enough to draw the whole line.

y = −2x + 3

HintLook at the sign of the number multiplying x.

The whyA negative gradient means y falls as x increases, so the line slopes down to the right. y = 2x + 3 climbs, and y = 3 has gradient 0, so it is horizontal.

The gradient of the line

HintOn a road sign for a hill the same idea is shown as a percentage or as "1 in 10".

The whyGradient = change in y ÷ change in x. A line that rises 6 while going 2 across has gradient 3: it gains 3 for each single step to the right.

4

HintCompare how far the line rises with how far it travels across between the two points.

The whyThe y-values rise by 14 − 2 = 12 while the x-values go across by 4 − 1 = 3, and 12 ÷ 3 gives the gradient. Always divide the change in y by the change in x, taking both changes in the same order.

Each extra hour costs £4

HintA gradient is "vertical units per horizontal unit"; read the two axis labels.

The whyThe gradient is the change in cost divided by the change in time, so its unit is pounds per hour. Every gradient on a real-life graph is a rate: so much of the vertical quantity for each one of the horizontal quantity.

Plan B charges more per GB: £5 against £2

HintThe steeper line climbs further up the cost axis for each gigabyte used.

The whyEach gradient is a price per gigabyte, so comparing steepness compares rates. It does not tell you which plan is cheaper overall: that also depends on any fixed monthly charge, which is where each line starts.

It gets 2 cm shorter every hour

HintThe sign tells you the direction of the change; the size tells you how quickly.

The whyA negative gradient means the vertical quantity falls as the horizontal one increases. Here the height drops by 2 cm for each hour of burning, so the line slopes down to the right.

y = −2x + 6

HintOne of those two facts is the steepness and the other is where the line starts on the vertical axis.

The whyFalling 2 for every 1 across is a gradient of −2, and crossing the y-axis at 6 is an intercept of 6. They go into y = mx + c as m and c.

No: 2 × 3 + 3 = 9, not 10

HintPut the x-value into the rule and see what comes out.

The whyA point lies on a line exactly when its coordinates make the equation true. When x = 3 the line is at height 9, so (3, 9) is on the line and (3, 10) sits one unit above it.

y = x/2 (the same as y = 0.5x)

HintFind the rise for a single unit across, and ask what height the line has when x is 0.

The whyUp 1 for every 2 across is a gradient of 1 ÷ 2 = 1/2. The line goes through (0, 0), so the intercept is 0 and nothing is added on.

x = 2 and y = 5

HintLook for the one point that sits on both lines.

The whyThe crossing point is on both lines, so its coordinates fit both rules: 2 × 2 + 1 = 5 and 2 + 3 = 5. No other point does, which is why the pair has exactly one solution.

5

HintGo across at that height until you meet the line, then read down to the x-axis.

The whySolving 2x − 3 = 7 means finding the x that makes y equal 7 on the line y = 2x − 3. The line reaches that height when x = 5, and 2 × 5 − 3 = 7 confirms it.

After 5 visits both gyms have cost £20

HintAt a crossing, the two graphs share one pair of readings.

The whyBefore 5 visits Gym B's line is lower, so B is cheaper; after 5 visits Gym A's line is lower. The crossing point is where the better choice switches.

Substitute x = 2.5 into both equations and see whether each gives 6.5

HintA point on a line makes the rule for that line true, and this point claims to be on two of them.

The why3 × 2.5 − 1 = 6.5 and 2.5 + 4 = 6.5, so the point fits both rules and the reading was exact. Graph readings are only as accurate as the drawing, so a substitution check is worth a few seconds.

It depends on how accurately the lines are drawn and read

HintThink about the thickness of a pencil line and the size of the grid squares.

The whyA crossing at x = 2.33… can only be read as "about 2.3" from a drawing. To be sure of an exact answer, check the reading by substitution.