Non-linear relationships - advanced

This lesson comprises three (3) master classes focusing on:

• Graphing parabolas and their transformations
• Graphing exponentials and their transformations
• Graphing hyperbolas and their transformations
• Graphing circles and their transformations

Content:

MA5-NLI-P-01

Graph parabolas and describe their features and transformations

• Use graphing applications to compare parabolas of the form \( y=kx^2 \), \( y=kx^2+c,\ y=k(x-b)^2 \) and \( y=k(x-b)^2+c \), and describe their features and transformations
• Find x- and y-intercepts algebraically, where appropriate, for the graph of \( y=ax^2+bx+c \), given \( a \), \( b \) and \( c \)
• Determine the equation of the axis of symmetry of a parabola using either the formula \( x=-\frac{b}{2a} \) or the midpoint of the x-intercepts
• Find the coordinates of a parabola’s vertex using a variety of methods
• Graph quadratic relationships of the form \( y=ax^2+bx+c \) by identifying and applying features of parabolas and their equations without graphing software

Graph exponentials and describe their features and transformations

• Use graphing applications to graph exponential relationships of the form \(y=k(a)^x+c \) and \(y=k(a)^{-x}+c \) for integer values of \( k \), \( a \) and \( c \) (where \( a>0 \) and \( a \ne 1 \)  ), and compare and describe any relevant features

Graph hyperbolas and describe their features and transformations

• Use graphing applications to graph, compare and describe hyperbolic relationships of the form \( y=\frac{k}{x} \) for integer values of \( k \)
• Use graphing applications to graph and describe a variety of hyperbolas, including where the equation is given in the form \( y=\frac{k}{x}+c \) or \( y=\frac{k}{x-b} \) for integer values of \( k \), \( b \) and \( c \)

Graph circles and describe their features and transformations

• Derive the equation of a circle \( x^2+y^2=r^2 \) with centre \( (0, 0) \) and radius \( r \) using the distance formula
• Identify and describe equations that represent circles with centre at the origin and radius of the circle \( r \)
• Graph circles of the form \( x^2+y^2=r^2 \), where \( r \) is the radius of the circle using graphing applications
• Establish the equation of the circle with centre \( (a, b) \) and radius \( r \), and graph equations of the form \( (x-a)^2+(y-b)^2=r^2 \)
• Find the centre and radius of a circle with the equation in the form \( x^2+y^2+ax+by+c=0 \) by completing the square

Distinguish between different types of graphs by examining their algebraic and graphical representations and solve problems

• Identify and describe features of different types of graphs based on their equations
• Identify a possible equation from a graph and verify using graphing applications
• Find points where a line intersects with a parabola, hyperbola or circle, both graphically and algebraically

Graph and compare polynomial curves and describe their features and transformations

• Use graphing applications to graph and compare features of cubic equations of the form \( y=ax^3+c \), where \( a \) and \( c \) are integers
• Use graphing applications to graph a variety of equations of the form \( y=kx^n \), where \( n \) is an integer and \( n \ge 2 \), and describe the effect on the shape of the curve where \( n \) is an odd or an even number
• Use graphing applications to graph curves of the form \( y=kx^n+c \) and \( y=k(x-b)^n \) where \( n \) is an integer and \( n \ge 2 \) and describe the transformations from \( y=kx^n \)