Types of Graphs (Edexcel IGCSE Maths)

Spec point — 3.3I

Types of graphs

What types of graphs do I need to know?

• You need to be able to recognise, sketch, and interpret the following types of graph:

  • Linear ($y=\pm x$)

    • $y=mx+c$ or $ax+by=c$

  • Quadratic($y=\pm x^{2}$)

    • $y=ax^{2}+bx+c$

  • Cubic($y=\pm x^{3}$)

    • $y=ax^{3}+bx^{2}+cx+d$

  • Reciprocal ($y=\pm\frac{1}{x}$)

    • $y=\frac{a}{x}$
    • $y=\frac{a}{x^{2}}$

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

Where are the asymptotes on reciprocal graphs?

• 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

What does the graph of a cubic look like?

• A cubic is a function of the form $ax^{3}+bx^{2}+cx+d$

  • $a,b,c$ 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

• The exact form of a particular cubic will depend on:

  • The number (and value) of roots ($x$-axis intercepts)

  • The $y$-axis intercept

  • The sign of the coefficient of the $x^{3}$ term ($a$)

    • If $a>0$ the graph is a positive cubic ('starts' in the bottom left, 'ends' in the top right)
    • If $a<0$ the graph is a negative cubic ('starts' in the top left, 'ends' in the bottom right)

  • The turning points

• 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 $x=0$

Worked Example Match the graphs to the equations.

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

Answer:

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

Final answer: Graph A → Equation 2

Final answer: Graph B → Equation 4

Final answer: Graph C → Equation 5

Final answer: Graph D → Equation 1

Final answer: Graph E → Equation 3