Algebraic Relationships

algebra year12 standard

Algebraic relationships | NSW Curriculum Website

• MST-12-S2-01 represents the relationships between quantities algebraically and graphically to solve problems and make predictions in practical contexts

📖 Prior Knowledge

Content	Prerequisite relationships
Equations	- Solve equations from formulas → Solve variation problems using an equation
Linear Relationships	- Find the intersection of two lines → Solve simultaneous equations graphically
Equations A	- Solve 3-step linear equations → Solve simultaneous equations algebraically
Non-linear Relationships A	- Identify parabolas and exponential curves → Graph parabolas and identify their features - Identify parabolas and exponential curves → Graph exponentials and identify their features
Non-linear Relationships B	- Graph parabolas and identify their features (revisited content) - Graph exponentials and identify their features (revisited content)
Variation and Rates of Change A	- Describe and represent direct variation → Describe and represent inverse variation - Describe and represent inverse variation (revisited content) - Test for variation from a table of values (revisited content) - Recognise inverse variation from graphs (revisited content) - Solve variation problems using an equation (revisited content)
Equations C	- Solve simultaneous equations graphically (revisited content) - Solve simultaneous equations algebraically (revisited content) - Model word problems with simultaneous equations (revisited content)

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 $y=k{a}^x$ and $y=k{a}^{-x}$, 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 $y=a{x}^2+bx+c$ 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