Types of Graphs (Edexcel IGCSE Maths)

Spec point — 3.3I

Types of graphs

What graphs do I need to know?

• 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 need to be able to recognise simple cubic graphs

  • y = x³
  • y = -x³
  • y = ax³ + bx² + x + c

• You also need to be able to recognise reciprocal graphs

  • $y=\frac{1}{x}$, where $x\neq0$

What does a quadratic graph look like?

• 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

What does a cubic graph look like?

• The equation of a cubic graph is y = ax³ + bx² + cx + d

• A cubic graph can have two points where it changes direction (turning points)

• A positive cubic goes uphill(from the bottom left to the top right)

  • The number in front of x³ is positive

    • For example, y = x³ - 3x² + 2x + 1

• A negative cubic goes downhill (from the top left to the bottom right)

  • The number in front of x³ is negative

    • For example, y = -x³ + 2x² - x + 5

• You can plot cubic graphs using a table of values

What does a reciprocal graph look like?

• The equation of the basic reciprocal graph is $y=\frac{1}{x}$

  • You cannot substitute in x = 0 (division by zero is not allowed)

    • $x\neq0$

  • You should not include x = 0 in a table of values

• The shape of $y=\frac{1}{x}$ is shown below

  • It has two curved branches
  • The branches never connect!

Worked Example In each of the cases below, state the letter of the graph that corresponds to the equation given.

(a) $y=x+5$

Answer:

This is a straight-line graph, y = mx + c The graph is a straight line going uphill and crosses the x-axis above (0,0)

Final answer: Graph D

(b) $y=-x^{2}+3x+2$

Answer:

This is a quadratic graph, y = ax² + bx +* c *(a = -1, b = 3, c = 2)

The number in front of* x*² is negative so it has an n-shape

Final answer: Graph C

(c)$\begin{gathered} \\ \end{gathered}$ $y=\frac{1}{x}$ Answer: This is the reciprocal graph, $y=\frac{1}{x}$ It has two curved shaped branches and no y-value when x = 0

Final answer: Graph B