Types of Graphs (Edexcel GCSE Maths: Higher): Revision Note

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 ($y=\pm x^2$)

    • $y=a{x}^2+bx+c$ where $a\ne 0$

  • Cubic ($y=\pm x^3$)

    • $y=a{x}^3+b$ or $y=a{x}^3+b{x}^2+cx$ where $a\ne 0$

  • Reciprocal ($y=\pm \frac{1}{x}$)

    • $y=\frac{a}{x}+b$ where $a\ne 0$

  • Exponential ($y=k^{\pm x}$) where $k>1$

    • $y=a{k}^x+b$ where $a\ne 0$

• 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 vertical asymptote remains on the y-axis

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 $\left(0,1\right)$

• The graph of $y=a{k}^x+b$ 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 $\left(0,a+b\right)$

• For example, a population may be modelled as $y=400\times {\left(\frac{1}{2}\right)}^x+100$, 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

  • ${\left(\frac{1}{2}\right)}^x={\left(2^{-1}\right)}^x=2^{-x}$ so $y=400\times 2^{-x}+100$ is the model above

  • This has the form $y=k^{-x}$ where $k>1$