Algebraic Relationships

Algebraic relationships is the most heavily examined focus area in Standard 2. It was worth 15 marks in NESA's sample HSC paper for the 2024 syllabus, so it deserves the most revision time. Almost every question is set in a real context, such as a business, a population or a ball's flight, and asks you to read a graph, substitute into a formula, or explain what a number means.

• Simultaneous linear equations: solve a pair graphically (where the lines cross) and algebraically; write a pair of equations from a worded problem; find a break-even point for cost and revenue.
• Exponential models $y=k{a}^x$ and $y=k{a}^{-x}$: growth and decay, the meaning of the $y$-intercept, and the limits of the model.
• Quadratic models $y=a{x}^2+bx+c$: the parabola, its intercepts, axis of symmetry and vertex (maximum or minimum), and which values make sense.
• Reciprocal models $y=\frac{k}{x}$: inverse variation, the hyperbola shape, and finding $k$.
• Graphing with and without technology, and using a spreadsheet for break-even problems.

Two linear equations are simultaneous when you want the values of $x$ and $y$ that make both true at once. On a graph, that is the point of intersection of the two lines.

Solving graphically

Draw both lines on the same axes (a table of three values for each line is enough) and read off where they cross. Graphing apps give an exact point; a hand-drawn graph gives an estimate, so check it by substituting into both equations.

Solving algebraically

If both equations start "$y=\dots$", set the right-hand sides equal (this is substitution). Otherwise use elimination: make one variable's numbers match, then subtract.

Example. A florist sold 50 bunches of flowers. Small bunches cost $6 and large bunches $9, and she took $360. Let $x$ be the number of small and $y$ the number of large bunches.

Multiply the first equation by 6 to get $6x+6y=300$, then subtract it from the second equation:

She sold 30 small and 20 large bunches. Check: $6\times 30+9\times 20=360$

Break-even point

A business has costs $C$ (fixed costs plus a cost per item) and revenue $R$ (money from sales). The break-even point is where $C=R$: the business makes neither a profit nor a loss. To the right of it, revenue is above cost and the business makes a profit $=R-C$; to the left it makes a loss.

Worked example: a coffee cart

A coffee cart has fixed costs of $2400 a month, and each coffee costs $3.50 to make. Coffees sell for $9.50.

(a) Write equations for the cost and revenue for $x$ coffees.

(b) Find the break-even point. Set $R=C$:

At 400 coffees, $R=9.5\times 400=3800$ The break-even point is (400, 3800): 400 coffees a month, with cost and revenue both $3800.

(c) Find the profit from selling 550 coffees.

The profit is $900. On a graph, this is the vertical gap between the revenue line and the cost line at $x=550$. In a spreadsheet, put $x$ in column A and formulas for $C$ and $R$ in columns B and C; the break-even point is the row where the two columns are equal.

An exponential relationship changes by the same factor each step, not the same amount.

• $k$ is the initial value, the $y$-intercept (the value when $x=0$).
• $a$ is the growth or decay factor. A 6% increase each year gives $a=1.06$; a 20% decrease gives $a=0.8$.
• The graph of $y=a^x$ (with $a>1$) passes through $(0,1)$ and rises more and more steeply. The graph of $y=a^{-x}$ is its mirror image in the $y$-axis: it falls quickly, then flattens.
• Both graphs get closer and closer to the $x$-axis on one side but never touch it. The $x$-axis is an asymptote.

Worked example: a medicine leaving the body

The amount of a medicine in a patient's blood is modelled by $A=400(0.8{)}^t$, where $A$ is in milligrams and $t$ is the time in hours after the dose.

(a) What does 400 represent? It is the value of $A$ when $t=0$: the dose was 400 mg.

(b) What does 0.8 tell you? Each hour, 80% of the medicine remains, so the amount falls by 20% an hour.

(c) Find the amount after 3 hours. Use the power key on your calculator.

(d) After how many whole hours is the amount first below 100 mg? Use a table of values (trial and error):

The amount is first below 100 mg after 7 hours. Show the values either side of the target, as above; that is the working that earns the marks.

Limitations

Exponential growth can't continue forever: a population runs out of food or space, and a price can't double every year forever. Exponential decay never reaches exactly zero in the model, but in reality the medicine is eventually all gone. Questions often ask you to explain one such limitation in context.

A quadratic relationship $y=a{x}^2+bx+c$ has a parabola as its graph.

• If $a<0$ the parabola is "upside down" (concave down) and has a maximum. If $a>0$ it has a minimum.
• The $y$-intercept is $c$ (put $x=0$).
• The $x$-intercepts are where $y=0$; read them from the graph.
• The axis of symmetry is the vertical line halfway between the $x$-intercepts.
• The vertex (turning point) lies on the axis of symmetry. Substitute that $x$-value to find its height.

Worked example: a water jet

A garden sprinkler sends a jet of water along the path $h=-0.25x^2+3x$, where $h$ is the height and $x$ is the horizontal distance, both in metres. The graph crosses the $x$-axis at $x=0$ and $x=12$.

(a) Find the axis of symmetry and the maximum height.

The axis of symmetry is $x=6$ and the maximum height is 9 m, reached 6 m from the sprinkler.

(b) Interpret the intercepts. At $x=0$ the water leaves the sprinkler at ground level, and at $x=12$ it lands, 12 m away.

(c) For which values of $x$ is the model valid? Only $0\le x\le 12$. Outside this the formula gives negative heights, which make no sense for water above the ground.

On a calculator, type brackets around negatives: $-0.25\times 6^2$ must give $-9$. Typing $(-0.25\times 6{)}^2$ is a common error.

Two quantities vary inversely if one doubles when the other halves. Their product is constant:

$k$ is the constant of variation. The graph is a hyperbola: in practical problems you only see the part where $x>0$, a curve that falls steeply then flattens, approaching both axes without touching them.

Example. A group hiring a bus for $1200 shares the cost equally, so the cost per person is $C=\frac{1200}{n}$ for $n$ people. With 30 people, $C=\frac{1200}{30}=40$, so $40 each. For the cost to fall to $25 each, solve $25=\frac{1200}{n}$, giving $n=\frac{1200}{25}=48$ people.

If $k$ isn't given, find it first from one pair of values ($k=xy$), then use the formula. A limitation: $n$ must be a whole number and the bus has a maximum number of seats, so the model only makes sense for some values.

• Reading the break-even point as only the $x$-value. Give both coordinates and say what each means.
• Writing a 15% decrease as $a=0.15$ instead of $a=0.85$.
• Forgetting brackets when squaring a negative or entering a power: $0.8^3$ is not $0.8\times 3$.
• Reading the vertex height from a hand-drawn graph instead of substituting the axis of symmetry into the equation.
• Confusing direct variation ($y=kx$, a straight line through the origin) with inverse variation ($y=\frac{k}{x}$, a hyperbola).
• Giving a non-whole answer, like 6.2 hours or 47.3 people, when the question asks for whole units.

• Interpret or explain what … represents means put the number into words in context, with units: "400 mg is the initial dose", not "it's the $y$-intercept".
• Show at least two lines of algebra for simultaneous equations; a correct answer from a guess earns little.
• In graph-reading questions, rule lines from the axis to the curve and state the value with units.
• For a "first exceeds" or "first falls below" question, show the values for the whole numbers on both sides.
• Limitations questions want a real-world reason the model stops working, not "the graph goes on forever".

• Linear relationships (Year 11): gradient, intercept and direct variation are the starting point here.
• Investment and loans: compound interest and declining-balance depreciation are exponential models.
• Formulas and equations (Year 11): substitution and solving equations are used in every question in this topic.
• Bivariate data analysis: a line of best fit is a linear model built from data.