# Functions and linear functions · Years 7–10

> Functions and linear functions, Years 7–10: Reading a graph and more. Explanation, worked examples and free practice for the national curriculum in England.

In Functions and linear functions (Years 7–10) you practise 10 skills: Reading a graph · A function's table and rule · Where a function increases and decreases · What is a function? · A point on a straight line graph · Rate of change · The gradient of a straight line · The rule of a linear function · Where a line crosses the axes · Parallel lines.

A function is a rule that gives exactly one output for every number you put in. You can show the same function in three ways: as a table, as a graph and as an algebraic rule. In a table each row is a pair, the input and the output; on a graph each pair is a point.

To check whether a point is on the graph, substitute its first number into the rule and see whether you get its second number. If a vertical line meets a curve at two points, the curve is not the graph of a function: one input has two outputs.

Read a graph from left to right. Where the graph climbs, the function is increasing, and where it goes down, the function is decreasing. The graph of a linear function is a straight line, and its rate of change is constant: the change in the output divided by the change in the input.

In the rule of a straight line, y = ax + b, the number a that multiplies x is the gradient, and the number b you add is where the line crosses the y axis. Parallel lines have the same gradient.

**Tip:** In a table of a linear function, check how much the output goes up when the input goes up by 1.

## What you learn

### Reading a graph

- Example: Sophie left home on her way to school. The graph shows how many metres from home she was at each minute. (A graph of Sophie’s distance from home over time: a broken line from minute 0 to minute 10.) How many metres from home was Sophie after 9 minutes?
- Answer: 500 m
- Solution: From minute 9 on the horizontal axis, go up to the graph, then across to the vertical axis. You reach 500. Sophie was 500 metres from home.

### A function's table and rule

- Example: The table shows a function of the form y = ax . Fill in the missing value in the table. (A table of values of the function, with one value missing.)
- Answer: 18
- Solution: In each complete row, y is 3 times x , for example 3 × 3 = 9 . So the rule is y = 3x , and in the missing row: y = 3 × 6 = 18

### Where a function increases and decreases

- Example: (A graph of a function: a broken line of 4 stretches, from the point (−5, 3) to the point (3, −2).) On which stretch is the function increasing?
- Options: 2 to 3 2 < x < 3 · −3 to −1 −3 < x < −1 · −5 to −3 −5 < x < −3 · −1 to 2 −1 < x < 2
- Answer: 2 to 3
- Solution: The graph climbs on only one stretch, from the point (2, −3) to the point (3, −2) . On the other stretches it goes down. The stretch: 2 < x < 3

### What is a function?

- Example: Does the table show a function? (A table of pairs of values.)
- Options: No · Yes
- Answer: No
- Solution: The value x = 0 appears twice, with two different values of y : (0, −3) and (0, −9) . So the table does not show a function.

### A point on a straight line graph

- Example: Is the point (−3, 15) on the graph of y = −3x + 6 ?
- Options: No · Yes
- Answer: Yes
- Solution: Substitute x = −3 into the rule: y = −3 × (−3) + 6 = 15 You get exactly the y coordinate of the point, so the point is on the graph.

### Rate of change

- Example: Is the rate of change of the function in the table constant? (A table of values of the function.)
- Options: No · Yes
- Answer: No
- Solution: For each two neighbouring rows, the change in the second value divided by the change in the first value: 3 ÷ 1 = 3 9 ÷ 1 = 9 6 ÷ 1 = 6 Not every pair gives the same number, so the rate is not constant.

### The gradient of a straight line

- Example: What is the gradient of the line? (A line on a grid, with two marked points: (−5, 0) and (−2, 3).)
- Answer: 1
- Solution: From the point A(−5, 0) to the point B(−2, 3) you go 3 steps to the right and up 3 steps. (The line, with a dashed step from the first point to the right and then to the second point.) The gradient is the change up or down divided by the number of steps to the right: 3 ÷ 3 = 1

### The rule of a linear function

- Example: Write the rule of the function whose graph is shown. (A line on a grid, with two marked points: (0, 0) and (2, −6).)
- Answer: −3x
- Solution: From the point (0, 0) to the point (2, −6) you go 2 steps to the right and down 6 steps, so the gradient is −3. The line goes through the origin, so you add nothing: y = −3x

### Where a line crosses the axes

- Example: Where does the line y = −4x + 9 cross the y axis?
- Answer: (0, 9)
- Solution: Substitute x = 0 : y = −4 × 0 + 9 = 9 The line crosses the y axis at (0, 9)

### Parallel lines

- Example: Which line is parallel to the line y = −4x + 8 ?
- Options: gradient −3, add 8 y = −3x + 8 · gradient −8, add 8 y = −8x + 8 · gradient −4, subtract 6 y = −4x − 6 · gradient 4, subtract 6 y = 4x − 6
- Answer: gradient −4, subtract 6
- Solution: The gradient of the line is −4. Only the line y = −4x − 6 has the same gradient, −4, so it is parallel to it. The other lines have a different gradient.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 7 topics](https://kixplo.com/uk/en/maths/year-7/)