Types of Graphs (Cambridge International IGCSE Mathematics)

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($y=\pm x^{3}$)

    • $y=ax^{3}+b$ or $y=ax^{3}+bx^{2}+cx$

  • Reciprocal ($y=\pm\frac{1}{x}$)

    • $y=\frac{a}{x}+b$

  • Exponential ($y=k^{\pm x}$)

    • $y=ak^{x}+b$

• You may also need to recognise the graphs $y=x^{\frac{1}{2}}$ and $y=x^{-\frac{1}{2}}$
• These are equivalent to $y=\sqrt{x}$ and $y=\frac{1}{\sqrt{x}}$

• You must also be able to recognise the three basic trigonometric graphs, covered in the Trigonometry 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 asymptote of $y=\frac{a}{x}-3$ would be $y=-3$

  • 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

How do I draw exponential growth and decay?

• The equation $y=k^{x}$ represents exponential growth when $k>1$

  • $y=k^{x}$ represents exponential decay when $0<k<1$

    • $k$ is positive but less than 1

• Both of these graphs:

  • have a horizontal asymptote at $y=0$
  • do not have a vertical asymptote
  • have a $y$-intercept of $( 0 , 1 )$

• The graph of $y=ak^{x}+b$ is a similar shape to $y=k^{x}$, but there are some differences

  • It is first stretched vertically by $a$

  • It is then shifted$b$ units upwards

    • Therefore it has a horizontal asymptote at $y=b$
    • and a $y$-intercept of $( 0 , a + b )$

• For example, a population may be modelled as $y = 400 \times ( 1 \frac{1}{2} )^{x} + 100$, where $y$ is the population and $x$ represents time

  • This is an exponential decay as $0<k<1$

  • The initial population (when $x=0$) will be 400 + 100 = 500

    • The $y$-intercept is (0, 500)

  • Over a long period of time (large $x$-value) the population will settle to 100

    • The asymptote is at $y=100$

• Exponential decay can also be identified by a negative power using index laws

  • $( 1 \frac{1}{2} )^{x = ( 2^{-1} } )^{x = 2^{-x} }$ so $y=400\times2^{-x}+100$ is the model above
  • This has the form $y=k^{-x}$ where $k>1$

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

(2) is an exponential equation with a positive coefficient so matches graph A

*(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

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