#Functions #Year11 #Advanced >[!info]- [Working with functions | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n11/fa3d488ed2) >- MAV-11-01 applies algebraic techniques and the laws of indices and surds to manipulate expressions and solve problems >- MAV-11-02 uses functions and relations to model, analyse and solve problems ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ----------------------------------- | ---------------------------------------------------------------------------------- | ------------------------------------------------------------------------- | | [[Computation with Integers]] | - representation of integers on a number line | - absolute value | | [[Indices C]] | - index laws and surd laws | - *repeated content* | | [[Algebraic Techniques C]] | - expand, factorise, and simplify algebraic fractions | - *repeated content* | | [[Equations C]] | - solve quadratic equations | - *repeated content* | | [[Functions and other Graphs]] | - relations, functions, and notation.<br>- domain and range | - *repeated content*<br>- composite functions<br>- even and odd functions | | [[Linear Relationships C]] | - equation of a line<br>- coordinate geometry formulas<br>- simultaneous equations | - *repeated content*<br>- break-even analysis | | [[Non-Linear Relationships C]] | - quadratic, cubic, reciprocal functions, circles | - *repeated content* | | [[Variation and Rates of Change A]] | - direct and inverse variation | - *repeated content* | ## Algebraic techniques - Use index laws to simplify expressions and solve problems involving positive, negative, zero or fractional indices - Expand, factorise and simplify algebraic expressions - Simplify expressions involving algebraic fractions - Expand and simplify expressions involving surds - Identify the conjugate of $\sqrt a\pm\sqrt b$, and rationalise the denominators of expressions of the form $\frac{a}{\sqrt b\pm\sqrt c}$ and $(\sqrt a\pm\sqrt d)/(\sqrt b\pm\sqrt c)$, where $a, b, c$ and $d$ are positive rational numbers - Solve quadratic equations $ax^2+bx+c=0$ by factorisation, completing the square and using the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a, b$ and $c$ are real numbers and $a\neq0$ - Define the discriminant as $\Delta=b^2-4ac$ and use it to solve problems involving the number of real roots of a quadratic equation and determine the conditions for roots to be equal, distinct, real or rational ## Introduction to functions and relations - Describe a relation between two sets as an association between the elements of one set and the elements of the other set - Define a real function $f$ of a real variable $x$ as a relation where each element $x$ of a given set is associated with exactly one element $y$ of the second set - Recognise that a relation is a rule which can be represented by an algebraic formula, a table of values, a set of ordered pairs $(x,y)$ or a graph - Use the notation$f(x)$ to identify the unique value of $y$ associated with $x$ when working with functions, and refer to it as the value of $f$ at $x$ - Refer to $x$ in a function $y=f(x)$ as the independent variable of the function, and refer to $y$ as the dependent variable - Substitute numeric and algebraic expressions into the formulas of functions - Apply the vertical line test on the graph of a relation to determine whether it represents a function - Define the domain of a function $f$ as the set of real numbers on which $f$ is defined - Define the range of a function $f$ as the set of values of $f(x)$ obtained as $x$ varies over the domain of $f$ - Refer to a value in the domain of a function at which the function is 0 as a zero of the function - Recognise that the $x$-intercepts of the graph of $y=f(x)$ are its zeroes, the solutions of $f\left(x\right)=0$, and that the $y$-intercept is $f(0)$ ## Linear functions - Determine the equations of straight lines in gradient–intercept form $y=mx+c$ with gradient $m$ and $y$-intercept $c$ - Determine the equations of straight lines in general form $ax+by+c=0$, where $a$, $b$ and $c$ are constants - Determine the equation of a straight line passing through a point $(x,y)$ with gradient $m$ using the point–gradient formula $y-y_1=m\left(x-x_1\right)$ - Determine the equation of a straight line passing through two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ by calculating its gradient m using the formula $m=\frac{y_{2\ }-\ y_1}{x_{2\ }-\ x_1}$ - Determine the $x$-intercept and $y$-intercept of a straight line given its equation - Choose and apply appropriate techniques to graph a straight line given its equation - Find the equation of a line that is parallel or perpendicular to a given line - Solve linear inequalities and graph the solution on a number line ## Quadratic and cubic functions - Identify the $x$-intercepts of a parabola whose quadratic function is expressed in factored form - Use the discriminant to determine the number of $x$-intercepts on a parabola and justify its position in relation to the $x$-axis - Show by completing the square on the general quadratic $y=ax^2+bx+c$ that the axis of symmetry is $x=-\frac{b}{2a}$ where $a, b$ and $c$ are constants - Identify the axis of symmetry and vertex of a parabola by completing the square on its quadratic function - Choose and apply appropriate techniques to graph a parabola of the form $y={ax}^2+bx+c$ by identifying its $x$-intercepts if they exist, its $y$-intercept, its axis of symmetry using $x=-\frac{b}{2a}$ and its vertex - Find the equation of a parabola given sufficient graphical features - Use the fact that two quadratic functions are equal for all values of x if and only if the corresponding coefficients are equal to solve related problems - Recognise that solving $f\left(x\right)=k$ for some constant $k$ corresponds to finding the $x$-coordinate(s) of the intersection of the graphs $y=f\left(x\right)$ and $y=k$ - Solve problems by finding the solution to simultaneous equations involving a linear and a quadratic function, or two quadratic functions, both algebraically and graphically - Solve quadratic inequalities - Recognise and graph cubic functions of the form $f(x)=kx^3$ and $f\left(x\right)=k\left(x-a\right)\left(x-b\right)\left(x-c\right)$, where $a, b, c$ and $k$ are constants and $k\neq0$ ## Reciprocal functions - Graph functions of the form $f(x)=\frac{k}{x}$, where $k$ is a constant and $k\neq0$, and identify their hyperbolic shape and their asymptotes - Describe the behaviour of $f\left(x\right)=\frac{k}{x}$ as $x\rightarrow\infty$ and $x\rightarrow-\infty$ ## Constructing and using functions - Construct and use linear functions to model and solve problems in real-world situations, identifying the independent and dependent variables and any restrictions on these variables, and justify conclusions in the context of the problem - Use linear inequalities to model and solve problems in real-world situations, and justify conclusions in the context of the problem - Solve practical problems involving a pair of simultaneous linear equations both algebraically and graphically, with and without graphing applications, and justify conclusions in the context of the problem - Construct and use simultaneous equations to model and solve a problem where cost and revenue are represented by linear equations, identify and analyse the break-even point, and justify conclusions in the context of the problem - Model and solve practical problems involving quadratic functions and justify conclusions in the context of the problem ## Direct and inverse variation - Develop models of the form $y=kx^n$, where k is a non-zero constant, from descriptions of situations in which one quantity varies directly with another - Develop the model $y=\frac{k}{x^n}$, where k is a non-zero constant, from descriptions of situations in which one quantity varies inversely with another - Evaluate $k$ in the equations $y=kx^n$ and $y=\frac{k}{x^n}$, given one pair of values for the variables, and use the resulting formula to find other values of the variables - Analyse and solve problems involving direct and inverse variation ## Circles and semicircles - Derive the equation of a circle of radius $r$ with centre at the origin by considering Pythagoras’ theorem - Graph circles of the form $x^2+y^2=r^2$ from their equations - Determine the equation of a circle of the form $x^2+y^2=r^2$ given its graph - Identify and graph the semicircles $y=\sqrt{r^2-x^2}$, $y=-\sqrt{r^2-x^2}$, $x=\sqrt{r^2-y^2}$ and $x=-\sqrt{r^2-y^2}$ ## Properties of functions, relations and graphs - Extend the definitions of domain and range to relations - Recognise domains and ranges of functions and relations given in interval notation, as inequalities and as worded descriptions - Determine and describe the domain and range of functions and relations, using interval notation, inequalities or worded descriptions - Define a function to be even if its graph is unchanged under reflection in the y-axis, and odd if its graph is unchanged under rotation of 180° about the origin - Develop and use the tests that a function $f(x)$ is odd if $f\left(-x\right)=-f(x)$ and a function $f(x)$ is even if $f\left(-x\right)=f(x)$ - Solve problems involving even and odd functions - Use the composite function $f\left(g\left(x\right)\right)$, where the output of $g\left(x\right)$ becomes the input of $f\left(x\right)$ - Determine the equations of composite functions ## Piecewise-defined functions - Interpret piecewise-defined functions, where the function is defined differently in different parts of the domain - Graph piecewise-defined functions involving functions covered in the scope of the Mathematics Advanced course, test if they are even or odd, and determine the domain and range - Define informally that a function is continuous at a point if the curve can be drawn through the point without lifting the pen off the paper - Identify points where piecewise-defined functions and other functions are not continuous - Define a discontinuity of a function informally as a point where the function is not continuous ## Absolute value functions - Define the absolute value $\left|x\right|$ of a number $x$, also known as the magnitude of $x$, to be the distance from the origin to x on the number line - Establish and use the piecewise definition $|x|=\begin{cases}x, & \mathrm{for}\ x\geq0,\\-x,& \mathrm{for}\ x<0\\ \end{cases}$ - Show using numerical substitutions that $\sqrt{x^2}=\left|x\right|$ and use the result - Graph the function $y=\left|x\right|$, describe its symmetry, and identify its domain and range - Graph $y=\left|ax+b\right|$ with and without graphing applications, and identify its symmetry, domain and range - Solve absolute value equations of the form $\left|ax+b\right|=k$ algebraically and graphically