Types of Graphs (Edexcel IGCSE Maths A (Modular))

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Maths

You must also be able to recognise the three basic trigonometric graphs, covered in a separate section

An asymptote is a line on a graph that a curve becomes closer to but never touches
- These may be horizontal or vertical
The reciprocal graph, $y = \frac{a}{x}$ (where $a$ is a constant)
- does not have a y-intercept
- and does not have any roots
This graph has two asymptotes
- A horizontal asymptote at the x-axis: $y = 0$
  - This is the limiting value when the value of x gets very large (or very negative)
- A vertical asymptote at the y-axis: $x = 0$
  - This is the value that causes the denominator to be zero

The reciprocal graph, $y = \frac{a}{x} + b$ (where $a$ and $b$ are both constants)
- is the same shape as $y = \frac{a}{x}$
- but is shifted upwards by $b$ units
  - $y = \frac{a}{x} - 3$ would be $y = \frac{a}{x}$ shifted down by 3 units
- This means the horizontal asymptote also shifts up by $b$ units
  - The vertical asymptote remains on the y-axis
The graph of $y = \frac{1}{x^{2}}$ is similar to $y = \frac{1}{x}$ but has two key differences
- $y = \frac{1}{x^{2}}$ is steeper than $y = \frac{1}{x}$
- $y = \frac{1}{x^{2}}$ is always positive, even when $x$ is negative

A cubic is a function of the form $a x^{3} + b x^{2} + c x + d$
- $a, b, c$ and $d$ are constants
- It is sometimes referred to as a polynomial of degree (order) 3
In general the graph of a cubic will take one of the four forms
- All are smooth curves

Cubics can have two turning points
- a maximum point and a minimum point
However, note that the graphs of $y = x^{3}$ and $y = - x^{3}$ :
- Do not have a maximum or minimum (turning points)
- Only cross the $x$ -axis once, at $x = 0$

Match the graphs to the equations.

(1) $y = 0.6 x + 2$ , (2) $y = 3^{x}$ , (3) $y = - 0.7 x^{3}$ , (4) $y = \frac{4}{x}$ , (5) $y = - x^{2} + 3 x + 2$

Starting with the equations,

(1) is a linear equation (y = mx + c) so matches the only straight line, graph D

(3) is a cubic equation with a negative coefficient so matches graph E

(4) is a reciprocal equation with a positive coefficient so matches graph B

(5) is a quadratic equation with a negative coefficient so matches graph C

(2) is the only equation not yet used, so by elimination it is graph A

Graph A → Equation 2

Graph B → Equation 4

Graph C → Equation 5

Graph D → Equation 1

Graph E → Equation 3

the (exam) results speak for themselves:

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