#Algebra #Year12 #Standard >[!info]- [Algebraic relationships | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-standard-11-12-2024/content/n12-tba2/faf5e47f4a) >- MST-12-S2-01 represents the relationships between quantities algebraically and graphically to solve problems and make predictions in practical contexts ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ----------------------------------------------------------------- | -------------------------------------- | ---------------------------------------------------------- | | [[Linear Relationships C]] | - solving simultaneous equations | - *repeated content*<br>- break-even analysis | | [[Non-linear Relationships B]] | - quadratic and exponential graphs | - *repeated content*<br>- quadratic and exponential models | | [[Variation and Rates of Change A]] | - inverse variation | - *repeated content* | | [[Non-Linear Relationships C]] | - hyperbolas | - *repeated content* | | [[Stage 6 Standard 2/Linear Relationships\|Linear Relationships]] | - graphing lines<br>- direct variation | - non-linear graphing<br>- inverse variation | ## Simultaneous linear equations - Graph two linear equations and identify the point of intersection, with and without using graphing applications - Solve a pair of simultaneous linear equations using graphical and algebraic methods - Develop a pair of simultaneous linear equations to model a practical situation - Solve practical problems that involve simultaneous linear equations - Use simultaneous equations to model and analyse the break-even point where cost and revenue are represented by linear equations - Identify the break-even point and solve problems involving profit and loss using a spreadsheet ## Exponential relationships - Recognise that an exponential relationship can be represented by an equation, a table of values, a set of ordered pairs or a graph, and move flexibly between these representations - Graph an exponential relationship of the form $y=a^x$ and $y=a^{-x}$ , where $a>0$, with and without using graphing applications - Construct and analyse an exponential model of the form $y=ka^x$ and $y=ka^{-x}$, where $a>0$ and $k$\ is a constant, to solve practical growth or decay problems - Interpret the meaning of the intercept of an exponential graph in a variety of contexts - Explain the limitations of exponential models in practical contexts ## Quadratic relationships - Recognise that a quadratic relationship can be represented by an equation, a table of values, a set of ordered pairs or a graph, and move flexibly between these representations - Recognise the parabolic shape of a quadratic relationship, noting its vertex and axis of symmetry - Graph a quadratic relationship of the form $y=ax^2+bx+c$ using graphing applications - Identify the $x$-intercepts and $y$-intercept of a parabola from a graph - Determine the axis of symmetry and vertex of a parabola using the midpoint of the $x$-intercepts shown on a graph - Analyse a given graph of a quadratic relationship and use it to solve practical problems - Interpret the meaning of the intercepts and vertex of a parabola in a variety of contexts - Explain the limitations of a model of a quadratic relationship in a practical context and consider the values for $x$ and $y$ for which the quadratic model is valid ## Reciprocal relationships - Recognise that inverse variation is represented by a reciprocal relationship of the form $y=\frac{k}{x}$, where $k$ is the constant of variation - Identify the hyperbolic shape of a reciprocal graph - Graph a reciprocal relationship of the form $y=\frac{k}{x}$, with and without using graphing applications - Construct a reciprocal model of the form $y=\frac{k}{x}$ to analyse inverse variation problems - Solve inverse variation problems graphically, or algebraically given the value of two of the variables in $y=\frac{k}{x}$ - Explain the limitations of reciprocal models in practical inverse variation problems