#NumberAndAlgebra #Path #Adv #Ext >[!info]- [Non-linear relationships C (Path) | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-5/fa633cd425) >- MA5-NLI-P-01 interprets and compares non-linear relationships and their transformations, both algebraically and graphically ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ----------------------------------- | ----------------------------------------------------- | ------------------------------------------------------- | | [[Right-angled Triangles]] | - Pythagoras' theorem | - equation of a circle | | [[Equations C]] | - Factorising quadratics and completing the square | - sketching parabolas | | [[Non-linear Relationships B]] | - Basic transformations of parabolas and exponentials | - further transformations of parabolas and exponentials | | [[Variation and Rates of Change A]] | - hyperbolas in inverse variation | - transformations of hyperbolas | ## [[Graph parabolas and describe their features and transformations.pdf]] - 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 $y=ax^2+bx+c$ identifying and applying features of parabolas and their equations without graphing software ## [[Graph exponentials and describe their features and transformations.pdf]] - 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 $a≠1$), and compare and describe any relevant features ## [[Graph hyperbolas and describe their features and transformations.pdf]] - 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.pdf]] - 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.pdf]] - 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.pdf]] - 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 $n≥2$, 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 $n≥2$, and describe the transformations from $y=kx^n$