Non-linear relationships - intermediate

This lesson comprises four (4) master classes focusing on:

• Functional notation
• Graphing parabolas
• Graphing exponential curves

Content:

MA5-NLI-C-01

Examine the connection between algebraic and graphical representations of quadratics and exponentials

• Represent quadratic and exponential relationships using graphing applications
• Construct a table of values to graph non-linear relationships involving quadratics and exponentials
• Explain that quadratic relationships are represented by parabolas
• Identify graphs and equations of parabolas and exponential curves
• Recognise quadratics and exponentials in real-life contexts

MA5-NLI-C-02

Graph and examine quadratic relationships

• Graph and compare parabolas of the form \( y=kx^2 \) and \( y=kx^2+c \) using graphing applications
• Identify and describe the key features of parabolas of the form \( y=kx^2 \) and \( y=kx^2+c \) including the vertex, x- and y-intercepts, axis of symmetry and concavity

Graph and examine exponential relationships

• Graph and compare exponential curves of the form \( y=a^x \) using graphing applications
• Describe features of exponential curves including the y-intercept, asymptote and the nature of the curve for very large and very small values of \( x \)

Distinguish between linear, quadratic and exponential relationships by examining their graphical representations

• Associate graphs of straight lines, parabolas and exponential curves with the appropriate equations
• Recognise non-linear relationships in real-life contexts and solve related problems
• Use graphing applications to solve a pair of simultaneous equations where one equation is non-linear and interpret the solution

MA5-FNC-P-01

Define relations and functions, and use function notation

• Define a relation over 2 sets as an association between the elements of one set to the elements of another set
• Notate relations as a set of ordered pairs \( (x,y) \) of real numbers
• Define a function as a relation over 2 sets where each element of the first set is associated with exactly one element in the second set
• Apply the vertical line test on a graph to decide whether it represents a function
• Use the notation \( y=f(x) \) when expressing a function
• Use the notation \( y=f(c) \) to determine the value of \( y=f(x) \) when \( x=c \)

Find the domain and range of a function and graph functions

• Define the domain as the set of all allowable values of \( x \)
• Define the range as the set of possible y-values as \( x \) varies over the domain of the function
• Determine and describe the domain and range for a variety of functions
• Use graphing applications to graph and compare functions of the form \( y=f(x) \), \( y=f(x)+c \), \( y=f(x-b) \) and \( y=f(ax) \), and describe their transformations

Graph regions corresponding to linear inequalities in one and 2 variables

• Graph linear inequalities of the form \( ax+by > c \), testing whether the points satisfy the given inequality and shading appropriate regions