Types of Graphs

You need to be able to recognise the following lines:
- Straight lines
  - y = mx + c
  - Such as y = 3x + 2, y = 5x - 1, ...
  - Two important ones are y = x and y = -x
- Horizontal lines
  - y = c
  - Such as y = 4, y = -10, ...
- Vertical lines
  - x = k
  - Such as x = 2, x = -1, ...
You need to be able to recognise quadratic graphs
- y = x²
- y = -x²
- y = ax² + bx + c
You also need to be able to recognise reciprocal graphs
- $y = \frac{1}{x}$
- $y = \frac{a}{x}$

The equation of a quadratic graph is y = ax² + bx + c
A quadratic graph has either a u-shape or an n-shape
- This type of shape is called a parabola
u-shapes are called positive quadratics
- because the number in front of x² is positive
  - For example, y = 2x² + 3x + 4
n-shapes are called negative quadratics
- because the number in front of x² is negative
  - For example, y = -3x² + 2x + 4
You can plot quadratic graphs using a table of values

Positive and negative quadratic graphs

The point where the graph turns is called the vertex
- Positive quadratics have a minimum point
  - The bottom of the u-shape
- Negative quadratics have a maximum point
  - The top of the n-shape
Quadratic graphs always have a vertical line of symmetry down the middle
- The equation of the vertical line of symmetry is x = k
  - k is the x-coordinate of the minimum or maximum point
Quadratic graphs do not have to cross the x-axis
- If they do, two x-intercepts are created, called roots
  - If the curve just touches the x-axis, only 1 root is created
- Roots are symmetric about the vertical line of symmetry
Quadratic graphs always have one y-intercept

The equation of a reciprocal graph is
- You cannot substitute in x = 0 (division by zero is not allowed)
  - $x \neq 0$
- You should not include x = 0 in a table of values
The shape of is shown below (for a positive value of a)
- It has two two curved branches
  - The branches are L-shaped
  - The L is more rectangular when a is smaller
- The branches never connect!
A negative value of a reflects the shape in the y-axis