Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 2) Find the gradient of the line through (0, 1) and (2, 7).
Answer
3: the rise is 6 and the run is 2. - Solve 2x − 6 = 0.
Answer
x = 3. - Work out 3 × (−2) + 5.
Answer
−1.
Words used in this lesson
- gradient
- the steepness of a line, written m. For example: m = 2 in y = 2x + 1.
- y-intercept
- the point where a line crosses the y-axis, (0, c). For example: (0, −1) for y = 2x − 1.
- x-intercept
- the point where a line crosses the x-axis, where y = 0. For example: (4, 0) for y = 0.5x − 2.
- equation of a line
- a rule that every point on the line follows. For example: y = 3x + 2.
- parallel
- never meeting, always the same distance apart. For example: y = 2x + 1 and y = 2x − 5.
- horizontal
- going straight across, with gradient 0. For example: y = 3.
What do the numbers in a fare do?
In Lesson 1 a taxi in Lagos charged 500 naira before any distance and 400 naira for each km, so the fare in naira for x km is 400x + 500. Imagine drawing that fare against distance. Which of the two numbers, 400 or 500, sets how steep the line is? Which one sets where it starts on the vertical axis?
Predict first: if you made the 500 bigger, would the line get steeper, or would it move? Then set the two sliders below to find out. Try a negative m, and m = 0.
y = 2x − 1 crosses the y-axis at (0, −1) and rises 2 for every 1 across. It crosses the x-axis at x = 1/2.
Show me the steps
- m = 2, c = −1: y = 2x − 1. It crosses the y-axis at (0, −1) and rises 2 for each 1 across.
- m = 2, c = 3: y = 2x + 3. The same steepness, slid up by 4.
- m = −1, c = 2: y = −x + 2. The gradient is negative, so the line falls, and it crosses the y-axis at (0, 2).
- m = 0, c = 3: y = 3. No x term, so the line is horizontal, 3 above the x-axis, and it never crosses the x-axis.
- m = 3, c = 0: y = 3x. A line through the origin, where both intercepts are the same point.
- m = −3, c = −2: y = −3x − 2. It falls steeply and crosses the x-axis at x = −2/3.
In y = mx + c, m is the gradient and c is where the line crosses the y-axis. The point (0, c) is the y-intercept. From there, every step of 1 across moves the line m up (or down, if m is negative).
The order does not matter: in y = 5 + 2x the gradient is 2 and c is 5. To find where the line crosses the x-axis, put y = 0 and solve: 0 = mx + c gives x = −c ÷ m, when m is not 0.
The fare line: y = 400x + 500 has m = 400, so each extra km adds 400 naira, and c = 500 is the fare at 0 km. Making the 500 bigger moves the line up. It does not make it steeper.
Two special cases
When m = 0 the line is horizontal and the equation is y = c. For example y = 3 is every point with y equal to 3. A vertical line is different: x = 4 is every point with x equal to 4. It has no gradient, because its run is 0, so there is no m to put in y = mx + c. It cannot be written that way.
Building a line from a step
The gradient is a rise over a run, so it is a recipe for a step. Start at (0, c), move across by the run and up by the rise, and repeat. A gradient of 1/2 means across 2 and up 1; a gradient of −2/3 means across 3 and down 2.
y = x/2 + 1: start at (0, 1). Each step is across 2, up 1. After 2 steps you are at (4, 3).
Show me the steps
- y = x/2 + 1: start at (0, 1). The gradient 1/2 means across 2, up 1: (2, 2), (4, 3), (6, 4).
- y = 3x/2 − 2: start at (0, −2). The gradient 3/2 means across 2, up 3: (2, 1), (4, 4), (6, 7).
- y = −2x/3 + 2: start at (0, 2). The gradient −2/3 means across 3, down 2: (3, 0), (6, −2), (9, −4).
- y = −x/4 + 3: start at (0, 3). The gradient −1/4 means across 4, down 1: (4, 2), (8, 1), (12, 0).
A common mistake: “c is where the line starts.” A line has no start: it goes on both ways. c is the height at which it crosses the y-axis, which is only the place we begin to count from.
Worked example: graph a line from m and c
Draw y = −2x + 3.
- c = 3, so the line crosses the y-axis at (0, 3). Plot it.
- m = −2 means down 2 for every 1 across: (1, 1), (2, −1), (3, −3).
- Check with the equation: when x = 3, y = −2 × 3 + 3 = −3, matching the last point.
Your go. The first step is given. Draw y = 0.5x − 2 and find where it crosses the x-axis.
- c = −2, so plot (0, −2).
- The gradient 0.5 is 1/2: across 2, up 1. Plot two more points, then find where y = 0 and check it in the equation.
Check the answer
The points are (2, −1) and (4, 0), so the line crosses the x-axis at x = 4. Check: 0.5 × 4 − 2 = 0.
Check before you go on
1. The line y = 3x − 5 is drawn. What do the 3 and the −5 tell you?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Write down the gradient of the line y = 4 − 3x. Use a minus sign if it is negative.
3. A line has gradient −3 and crosses the y-axis at (0, 6). What is y when x = 4?
Level 2 · Building
4. A tank holds V = 80 − 4t litres after t minutes. After how many minutes is it empty?
5. A line crosses the y-axis at (0, −2) and passes through (3, 4). What is its gradient?
6. Where does the line y = 0.25x + 3 cross the x-axis? Give the value of x.
Level 3 · Stretch
7. A line crosses the y-axis at (0, −2) and the x-axis at (5, 0). What is its gradient? Give it as a decimal.
8. The line y = kx + 3 passes through the point (2, −5). Find k.
Something to think about
Two lines, y = 2x + 1 and y = 2x − 5, are drawn on one grid. What do you notice about them? Where does each cross the y-axis, and how far apart are they, measured straight up the page?
One way to think about it
They have the same gradient, so they never meet: they are parallel. They cross the y-axis at 1 and at −5, which is 6 apart, and at every x the vertical gap is the same 6. Changing c only slides a line up or down, which is why parallel lines share m.
Exit ticket
9. Why can the line x = 4 not be written in the form y = mx + c?