Advanced Functions – Working with functions

This lesson comprises eight (8) master classes focusing on:

• Graphing functions
• Linear functions
• Quadratic functions
• Cubic functions
• Polynomials
• Relations

Content:

MA-F1.1

• Use index laws and surds
• Solve quadratic equations using the quadratic formula and by completing the square
• Manipulate complex algebraic expressions involving algebraic fractions

MA-F1.2

• Define and use a function and a relation as mappings between sets, and as a rule or a formula that defines one variable quantity in terms of another
  • define a relation as any set of ordered pairs \( (x, y) \) of real numbers
  • understand the formal definition of a function as a set of ordered pairs \( (x, y) \) of real numbers such that no two ordered pairs have the same first component (or x-component)
• Use function notation, domain and range, independent and dependent variables
  • understand and use interval notation as a way of representing domain and range, eg \( [4, \infty ) \)
• Understand the concept of the graph of a function
• Identify types of functions and relations on a given domain, using a variety of methods
  • know what is meant by one-to-one, one-to-many, many-to-one and many-to-many
  • use the vertical line test to identify a function
  • determine if a function is one-to-one
• Define odd and even functions algebraically and recognise their geometric properties
• Define the sum, difference, product and quotient of functions and consider their domains and ranges where possible
• Define and use the composite function \( f(g(x)) \) of functions \( f(x) \) and \( g(x) \) where appropriate
  • identify the domain and range of a composite function
• Recognise that solving the equation \( f(x)=0 \) corresponds to finding the values of \( x \) for which the graph of \( y=f(x) \) cuts the x-axis (the x-intercepts)

MA-F1.3

• Model, analyse and solve problems involving linear functions
  • recognise that a direct variation relationship produces a straight-line graph
  • explain the geometrical significance of \( m \) and \( c \) in the equation \( f(x)=mx+c \)
  • derive the equation of a straight line passing through a fixed point \( (x_1,y_1) \) and having a given gradient m using the formula \( y−y_1=m(x−x_1) \)
  • derive the equation of a straight line passing through two points \( (x_1,y_1) \) and \( (x_2,y_2) \) by first calculating its gradient \( m \) using the formula \( m=\frac{y_2−y_1}{x_2−x_1} \)
  • understand and use the fact that parallel lines have the same gradient and that two lines with gradient \( m_1 \) and \( m_2 \) respectively are perpendicular if and only if \( m_1m_2=−1 \)
  • find the equations of straight lines, including parallel and perpendicular lines, given sufficient information
• Model, analyse and solve problems involving quadratic functions
  • recognise features of the graph of a quadratic, including its parabolic nature, turning point, axis of symmetry and intercepts
  • find the vertex and intercepts of a quadratic graph by either factorising, completing the square or solving the quadratic equation as appropriate
  • understand the role of the discriminant in relation to the position of the graph
  • find the equation of a quadratic given sufficient information
• Solve practical problems involving a pair of simultaneous linear and/or quadratic functions algebraically and graphically, with or without the aid of technology; including determining and interpreting the break-even point of a simple business problem
  • understand that solving \( f(x)=k \) corresponds to finding the values of \( x \) for which the graph \( y=f(x) \) cuts the line \( y=k \)
• Recognise cubic functions of the form: \( f(x)=kx^3 \), \( f(x)=k(x−b)^3+c \) and \( f(x)=k(x−a)(x−b)(x−c) \) where \( a \) , \( b \), \( c \) and \( k \) are constants, from their equation and/or graph and identify important features of the graph

MA-F1.4

• Define a real polynomial \( P(x) \) as the expression \( a_nx^n+a_{n−1}x^{n−1}+...+a_2x^2+a_1x+a_0 \) where \( n=0,1,2, \dots \) and \( a_0,a_1,a_2, \dots ,a_n \) are real numbers
• Identify the coefficients and the degree of a polynomial
• Identify the shape and features of graphs of polynomial functions of any degree in factored form and sketch their graphs
• Recognise that functions of the form \( f(x)=\frac{k}{x} \) represent inverse variation, identify the hyperbolic shape of their graphs and identify their asymptotes
• Define the absolute value \( |x| \) of a real number \( x \) as the distance of the number from the origin on a number line without regard to its sign
• Use and apply the notation \( |x| \)  for the absolute value of the real number \( x \) and the graph of \( y=|x| \)
  • recognise the shape and features of the graph of \( y=|ax+b| \) and hence sketch the graph
• Solve simple absolute value equations of the form \( |ax+b| =k \) both algebraically and graphically
• Given the graph of \( y=f(x) \), sketch \( y=−f(x) \) and \( y=f(−x) \) and \( y=−f(−x) \) using reflections in the x and y-axes
• Recognise features of the graphs of \( x^2+y^2=r^2 \) and \( (x−a)^2+(y−b)^2=r^2 \), including their circular shapes, their centres and their radii
  • derive the equation of a circle, centre the origin, by considering Pythagoras’ theorem and recognise that a circle is not a function
  • transform equations of the form \( x^2+y^2+ax+by+c=0 \) into the form \( (x−a)^2+(y−b)^2=r^2 \), by completing the square
  • sketch circles given their equations and find the equation of a circle from its graph
  • recognise that \( y=\sqrt{r^2−x^2} \) and \( y=\sqrt{−r^2−x^2} \) are functions, identify the semicircular shape of their graphs and sketch them