# Linear Graphs

**Linear graphs** are so-named because they are **straight lines**.

**Coordinates** are read from the **origin (0,0)**. They are in order of **x-coordinate (horizontal)** and **y-coordinate (vertical)**.

Because they are straight lines, **only 3 points** are needed. Two points are needed to draw the line, and the third point is used to check for correctness.

Graphs **do not** always pass through the origin (0,0).

**Linear Equations** are written in the form of:

*y* = *mx* + *c*

where
*m* is the gradient (slope)
*c* is the y-intercept (point where the graph cuts the y-axis)

*m* and *c* are both constants (fixed numbers).

**Examples** are:

- In the equation y = 3x + 2, the gradient is 3 and the y-intercept is 2.
- In the equation y = 4x – 5, the gradient is 4 and the y-intercept is –5.
- In the equation y =
^{1}⁄_{2}x + 6, the gradient is^{1}⁄_{2}or 0.5 and the y-intercept is 6. - In the equation y = x, the gradient is 1 and the y-intercept is 0.

More information about **gradient** and the **y-intercept** is at Gradient, Midpoint And Distance.

## Example One - Buying Motorbike Race Tickets

Motorbike race tickets cost $50 each. Draw a table of values and a linear graph showing the cost of the tickets.

**Answer:**
*Because this graph is a straight line, only 3 points are needed - two points for drawing the line, and the third for checking correctness.
Choose x = 0, 1, 2 unless the question states otherwise.*

Number of tickets | 0 | 1 | 2 |

Cost of tickets | $ 0 | $ 50 | $ 100 |

## Example Two - Graph of y = x

(a) In the linear equation y = x, what is the gradient?

(b) What is the y-intercept?

(c) Draw a table of values and the graph of the linear equation y = x

**Answer:**

(a) Gradient = 1

(b) y-intercept = 0

(c) *Because this graph is a straight line, only 3 points are needed - two points for drawing the line, and the third for checking correctness.
Choose x = 0, 1, 2 unless the question states otherwise.*

x | 0 | 1 | 2 |

y = x | 0 | 1 | 2 |

Working | y = x y = 0 | y = x y = 1 | y = x y = 2 |

Coordinates | (0,0) | (1,1) | (2,2) |

## Example Three - Graph of y = 2x

(a) In the linear equation y = 2x, what is the gradient?

(b) What is the y-intercept?

(c) Draw a table of values and the graph of the linear equation y = 2x

**Answer:**

(a) Gradient = 2

(b) y-intercept = 0

(c) *Because this graph is a straight line, only 3 points are needed - two points for drawing the line, and the third for checking correctness.
Choose x = 0, 1, 2 unless the question states otherwise.*

x | 0 | 1 | 2 |

y = 2x | 0 | 2 | 4 |

Working | y = 2x y = 2 × 0 y = 0 | y = 2x y = 2 × 1 y = 2 | y = 2x y = 2 × 2 y = 4 |

Coordinates | (0,0) | (1,2) | (2,4) |

## Example Four - Graph of y = 2x + 3

(a) In the linear equation y = 2x + 3, what is the gradient?

(b) What is the y-intercept?

(c) Draw a table of values and the graph of the linear equation y = 2x + 3

**Answer:**

(a) Gradient = 2

(b) y-intercept = 3

(c)

x | 0 | 1 | 2 |

y = 2x + 3 | 3 | 5 | 7 |

Working | y = 2x + 3 y = 2 × 0 + 3 y = 3 | y = 2x + 3 y = 2 × 1 + 3 y = 5 | y = 2x + 3 y = 2 × 2 + 3 y = 7 |

Coordinates | (0,3) | (1,5) | (2,7) |

## Questions

What do you notice about the **slope** of the graphs of y = **2**x and y = **2**x + 3?

## Example Five - Graph of y = ^{1}⁄_{2} x + 1

(a) In the linear equation y = ^{1}⁄_{2} x + 1, what is the gradient?

(b) What is the y-intercept?

(c) Draw a table of values and the graph of the linear equation y = ^{1}⁄_{2} x + 1

**Answer:**

(a) Gradient = ^{1}⁄_{2}

(*Notice that the graph is steeper than y = 1x but not as steep as y = 2x and y = 2x + 3*)

(b) y-intercept = 1

(c)

x | 0 | 1 | 2 |

y = ^{1}⁄_{2} x + 1 | 1 | 1^{1}⁄_{2} | 2 |

Working | y = ^{1}⁄_{2}x + 1y = ^{1}⁄_{2} × 0 + 1y = 1 |
y = ^{1}⁄_{2}x + 1y = ^{1}⁄_{2} × 1 + 1y = 1 ^{1}⁄_{2} |
y = ^{1}⁄_{2}x + 1y = ^{1}⁄_{2} × 2 + 1y = 2 |

Coordinates | (0,1) | (1,1^{1}⁄_{2}) | (2,2) |

## Questions

What do you notice about the **coefficient of x** (the number multiplied by x) and the **steepness of the graph**?

What do you notice about the **constant** (the number at the end of the equation) and the **point where the graph intersects with the y-axis**?

## Example Six - Graph of y = 3x – 1

(a) In the linear equation y = 3x – 1, what is the gradient?

(b) What is the y-intercept?

(c) Graph the equation y = 3x – 1 where **–2 < x < 2**

**Answer:**

(a) Gradient = 3 (*Notice that the graph is steeper than y = 1x, y = 2x and y = 2x + 3*)

(b) y-intercept = –1

(c)

*The question specified that 3 values of x must be between –2 and 2. Choose x-values of –2, 0, 2.*

x | –2 | 1 | 2 |

y = 3x – 1 | –7 | 1 | 3 |

Working | y = 3x – 1 y = 3 × (–2) – 1 y = –6 – 1 y = –7 |
y = 3x – 1 y = 3 × 0 – 1 y = 0 – 1 y = –1 |
y = 3x – 1 y = 3 × 2 – 1 y = 4 – 1 y = 3 |

Coordinates | (–2,–7) | (0,–1) | (2,3) |

## Question

Draw a table of values and then graph the equation **y = 4x – 3** where **–3 < x < 3**
*(Remember that your graph will be correct if it is a straight line.)*

## Example Seven - Graph of y = –2x + 5 (Negative Gradient)

(a) In the linear equation y = –2x + 5, what is the gradient?

(b) What is the y-intercept?

(c) Graph the equation y = –2x + 5

**Answer:**

(a) Gradient = –2 (*Notice that the graph is backwards.*)

(b) y-intercept = 5

(c)

x | 0 | 1 | 2 |

y = –2x + 5 | 5 | 3 | 1 |

Working | y = –2x + 5 y = (–2) × 0 + 5 y = 0 + 5 y = 5 |
y = –2x + 5 y = (–2) × 1 + 5 y = (–2) + 5 y = 3 |
y = –2x + 5 y = (–2) × 2 + 5 y = (–4) + 5 y = 1 |

Coordinates | (0,5) | (1,3) | (2,1) |