Functions and graphs - (2.1-2.4)

Recognise the intuitive concepts of functions, domains and co-domains, independent and dependent variables

  • Graph
  • Theory
  • Examples
    In each of the following, y is a function of x. Find the largest domain of real numbers for each function.
    (a) \(y = {(x - 2)^2}\)
    (b) \(y = \frac{x}{{{{(2x + 1)}^2}}}\)
  • Solutions

    (a) Consider \(y = {(x - 2)^2}\).
    For any real number x, the corresponding value of y is real according to the expression.
    The largest domain of real numbers for the function is the set of all real numbers.

    (b) Consider \(y = \frac{x}{{{{(2x + 1)}^2}}}\).
    For any real number x except \(x = - \frac{1}{2}\), the corresponding value of y is real according to the expression.
    The largest domain of real numbers for the function is the set of all real numbers except \(\underline{\underline { - \frac{1}{2}}} \).

Recognise the notation of functions and use tabular, algebraic and graphical methods to represent functions

  • Graph
  • Theory
  • Examples
  • Solutions

Understand the features of the graphs of quadratic functions

  • Graph

  • Theory
  • Examples
  • Solutions

Find the maximum and minimum values of quadratic functions by the algebraic method

  • Graph

  • Theory
  • Examples
    Find the maximum / minimum value of each of the following functions.
    (a) \(y = - 2{x^2} + 7x - 6\)
    (b) \(y = 3{x^2} - 8x + 5\)
  • Solutions

    \(\begin{array}{c}a)\qquad y = - 2{x^2} + 7x - 6\\ = - 2({x^2} - \frac{7}{2}x + 3)\\ = - 2{[{x^2} - \frac{7}{2}x + {(\frac{7}{4})^2} - {(\frac{7}{4})^2} + 3]^{^{^{}}}}\\ = - 2{[{(x - \frac{7}{4})^2} - \frac{{49}}{{16}} + 3]^{^{^{}}}}\\ = - 2{[{(x - \frac{7}{4})^2} - \frac{1}{{16}}]^{^{^{}}}}\\ = - 2{(x - \frac{7}{4})^2} + \frac{1}{8}\end{array}\)

    For any real number x,
    \( - {\rm{ }}2{(x - \frac{7}{4})^2} \le 0\)
    \(y = - 2{(x - \frac{7}{4})^2} + \frac{1}{8} \le \frac{1}{8}\)
    Therefore, the maximum value of y is \(\frac{1}{8}\).

    \(\begin{array}{c}b)\qquad y = 3{x^2} - 8x + 5\\ = 3({x^2} - \frac{8}{3}x + \frac{5}{3})\\ = 3{[{x^2} - \frac{8}{3}x + {(\frac{4}{3})^2} - {(\frac{4}{3})^2} + \frac{5}{3}]^{^{^{}}}}\\ = 3{[{(x - \frac{4}{3})^2} - \frac{{16}}{9} + \frac{5}{3}]^{^{^{}}}}\\ = 3{[{(x - \frac{4}{3})^2} - \frac{1}{9}]^{^{^{}}}}\\ = 3{(x - \frac{4}{3})^2} - \frac{1}{3}\end{array}\)

    For any real number x,
    \(3{(x - \frac{4}{3})^2} \ge 0\)
    \(y = 3{(x - \frac{4}{3})^2} - \frac{1}{3} \ge - \frac{1}{3}\)
    Therefore, the minimum value of y is \( - \frac{1}{3}\).