Before viewing this page, it would be helpful to learn Linear Graphs.

Gradient = | Rise |

Run |

where
**Rise** is the vertical height
**Run** is the horizontal height.

*Civil engineers use this formula to work out the gradient of roads. Carpenters use this formula to calculate the slope of a step.*

The world's most dangerous road is the Yungas Road in Bolivia in South America. More than 200 people are killed each year. Not only is it steep in parts but it is also in an area of high blinding rainfall. Calculate the gradient (steepness) of a part of the road with a rise of 215 metres and a run of 500 metres.

**Answer:**

Gradient = | Rise | = | 215 | = 0.43 |

Run | 500 |

Gradient = m = | y_{2} – y_{1} |

x_{2} – x_{1} |

where (x_{1}, y_{1}) and (x_{2}, y_{2}) are two points on a linear or straight-line graph.

*Notice that the Rise is the difference between the y-values and the Run is the difference between the x-values.*

What is the gradient between the points (2,1) and (4,5)?

**Answer:**

(x_{1}, y_{1}) = (2,1)

(x_{2}, y_{2}) = (4,5)

Gradient = | y_{2} – y_{1} | = | 5 – 1 | = | 4 | = 2 |

x_{2} – x_{1} | 4 – 2 | 2 |

What is the gradient between the points (0,2) and (3,1)?

**Answer:**

(x_{1}, y_{1}) = (0,2)

(x_{2}, y_{2}) = (3,1)

Gradient = | y_{2} – y_{1} | = | 1 – 2 | = | –1 | = –^{1}⁄_{3} |

x_{2} – x_{1} | 3 – 0 | 3 |

*Notice that when the gradient is negative, the graph slopes backwards.*

**Q1.** (5,6) and (7,12)
**Q2.** (3,6) and (5,2)

**Answers**
**A1.** 3
**A2.** –2

Lines that are **parallel** have the **same gradient, m _{1} = m_{2}**

The graphs of y = 3x (red line) and y = 3x + 2 (blue line) are shown in the diagram. What do you notice about their gradients (slopes)?

**Answer:**

Same gradient

**Perpendicular** lines are **at right angles to each other**.

When the **gradients of perpendicular lines are multiplied**, the resulting product is **–1**.

**m _{1} × m_{2} = –1**

The graphs of y = 2x + 3 (blue line) and y = –^{1}⁄_{2} x + 5 (red line) are shown in the diagram. What do you notice about these graphs?

**Answer:**

Perpendicular (at right angles) to each other

If the gradient of a line is 2, what is the gradient of a line that is at right angles to it?

**Answer:**

–^{1}⁄_{2}

(2 multiplied by –^{1}⁄_{2} equals –1)

If the gradient of a line is 4, find the gradient of a line that is:
**Q1.** parallel
**Q2.** perpendicular.

**Answers**
**A1.** 4
**A2.** –^{1}⁄_{4}

The x-intercept is the point at which a graph crosses the x-axis.

The **y-coordinate at the x-intercept is 0**.

Find the x-intercept of the graph of y = x + 3.

**Answer:**

y = x – 3

0 = x – 3

x = 3

The coordinates of the x-intercept are (3,0)

The y-intercept is the point at which a graph crosses the y-axis.

The **x-coordinate at the y-intercept is 0**.

Find the y-intercept of the graph of y = x + 3.

**Answer:**

y = x – 3

y = 0 – 3

y = –3

The coordinates of the x-intercept are (0,–3)

**Q1.** Find the x-intercept of y = 5x + 3.
**Q2.** Find the y-intercept of y = 4x – 8.

**Answers**
**A1.** 3
**A2.** 2

The midpoint is the point that is **halfway** between 2 given points.

Midpoint = | (x_{1} + x_{2}) | , | (y_{1} + y_{2}) |

2 | 2 |

What is the midpoint between the points (2,1) and (4,5)?

**Answer:**

(x_{1}, y_{1}) = (2,1)

(x_{2}, y_{2}) = (4,5)

Midpoint = | (x_{1} + x_{2}) | , | (y_{1} + y_{2}) |

2 | 2 |

Midpoint = | (2 + 4) | , | (1 + 5) |

2 | 2 |

Midpoint = (3,3)

**Q1.** (2,5) and (6, 10)
**Q2.** (–3, –5) and (–7, 9)

**Answers**
**A1.** (4 , 7.5)
**A2.** (–5, 2)

The distance between 2 points on a graph is really the **hypotenuse** of a right-angled triangle.

**Pythagoras's Rule** is used to find the distance between 2 points on a graph.

**Distance = √(x _{2} – x_{1}) ^{2} + (y_{2} – y_{1}) ^{2}**

What is the distance between the points (1,5) and (4,9)?

**Answer:**

(x_{1}, y_{1}) = (1,5)

(x_{2}, y_{2}) = (4,9)

Distance = √(x_{2} – x_{1}) ^{2} + (y_{2} – y_{1}) ^{2}

= √(4 – 1) ^{2} + (9 – 5) ^{2}

= √25

= 5

**Q1.** (2,5) and (7,17)
**Q2.** (2,1) and (4,5)

**Answers**
**A1.** 13
**A2.** 4.5