Graph Transformations

functions year11 advanced

Graph transformations | NSW Curriculum Website

• MAV-11-03 analyses and solves algebraic and graphical problems involving transformations of functions and relations

📖 Prior Knowledge

Content	Prerequisite relationships
Equations C	- Solve quadratics by completing the square → Graph circles with centre (a,b)
Non-Linear Relationships C	- Graph circles centred at the origin → Graph circles with centre (a,b) - Graph circles with centre (a,b) (revisited content)
Functions and other Graphs	- Translate graphs of functions → Reflect graphs in the axes - Translate graphs of functions → Translate graphs - Translate graphs of functions → Dilate graphs

Graph Transformations

• Establish using graphing applications that replacing $x$ by $-x$ in the equation of a graph corresponds to the reflection of the graph in the $y$-axis, and that replacing $y$ by $-y$ corresponds to a reflection in the $x$-axis
• Establish using graphing applications that replacing $x$ by $x-a$ in the equation of a graph corresponds to a horizontal translation of a graph by $a$, that replacing $y$ by $y-b$ corresponds to a vertical translation by $b$, and recognise that the sign of $a$ and $b$ determines the direction of the translation
• Describe a horizontal dilation as a stretch from the $y$-axis and a vertical dilation as a stretch from the $x$-axis
• Establish using graphing applications that replacing $x$ by $\frac{x}{k}$ in the equation of a graph corresponds to a horizontal dilation of a graph by a factor of $k$, that replacing $y$ by $\frac{y}{l}$ corresponds to a vertical dilation by a factor of $l$, and determine the effects on the graph when the magnitude of the dilation factor lies between 0 and 1
• Establish, using graphing applications, replacing $x$ by $\frac{x}{k}$ and replacing $y$ by $\frac{y}{k}$ in the equation of a graph corresponds to a dilation by a factor of $k$, that is, an enlargement of the graph when $k>1$ and a reduction of the graph when $0<k<1$
• Apply reflections, translations and dilations to functions within the scope of the Mathematics Advanced course, excluding trigonometric functions, determine the function rule and the graph of the transformed function, the domain and range of the transformed graph, any intercepts, asymptotes and discontinuities where appropriate, and describe which transformations have been applied
• Use the principles of translations to identify the centre, radius, domain and range of graphs of circles in the form $(x-a{)}^2+(y-b{)}^2=r^2$, where $a$, $b$ and $r$ are constants
• Find the centre and radius of circles of the form $x^2+y^2+ax+by+c=0$, where $a$, $b$ and $c$ are constants, by completing the squares
• Graph circles given their equations and find the equation of a circle from its graph
• Recognise that the order in which individual transformations are applied to functions is important