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Algebraic Relationships · L4 of 6 ~50 min MST-12-S2-01 ⚡ +90 XP available

Quadratic Relationships — Parabolas

The parabola is the signature shape of $y = ax^2 + bx + c$. Understand the roles of $a$, $b$ and $c$, locate the vertex, axis of symmetry, and intercepts from a graph or equation, and use graphing technology to analyse quadratic relationships efficiently.

Today's hook — A ball is kicked into the air. Its height follows a parabola. The ball lands where the parabola crosses the $x$-axis. Can you sketch what that looks like before we start?

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Recall — your gut answer first

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A kicked ball traces a curved path. Its height $h$ (metres) at time $t$ (seconds) follows the rule $h = -5t^2 + 20t$. Without graphing, can you work out: (1) when is the ball at height zero (at ground level)? (2) when does it reach its peak height?

Write your thinking — there's no wrong answer here.

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The key features you need to own

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The quadratic $y = ax^2 + bx + c$ gives a parabola. Every parabola has three key features: the vertex (the turning point), the axis of symmetry (the vertical line through the vertex), and the intercepts ($x$-intercepts where the parabola crosses the $x$-axis, and $y$-intercept where it crosses the $y$-axis).

If $a > 0$: parabola opens upward (minimum vertex — a valley). If $a < 0$: opens downward (maximum vertex — a hill). The $y$-intercept is simply $c$ (substitute $x = 0$). The axis of symmetry is the vertical line midway between the $x$-intercepts.

$y$-intercept $= c$ | axis of symmetry: $x = \dfrac{x_1 + x_2}{2}$ (midpoint of $x$-intercepts)

Sign of $a$ determines direction

$a > 0$: opens up (U-shape, minimum). $a < 0$: opens down (∩-shape, maximum). This is the first thing to check when given a quadratic.

Vertex is the most important point

The vertex gives the maximum or minimum value of $y$. It lies on the axis of symmetry. Once you find the axis of symmetry ($x$-value), substitute back to find the $y$-value of the vertex.

Use graphing technology

NESA requires you to use graphing applications to graph quadratics. In Desmos or GeoGebra you can read off vertex, intercepts and axis of symmetry directly from the graph.

What you'll master

Know

Key facts

A quadratic relationship has the form $y = ax^2 + bx + c$
Its graph is a parabola: $a > 0$ opens up; $a < 0$ opens down
$y$-intercept $= c$; vertex is the minimum ($a>0$) or maximum ($a<0$)
Axis of symmetry: vertical line $x = \frac{x_1+x_2}{2}$ (midpoint of $x$-intercepts)

Understand

Concepts

How to identify a quadratic from any of its four representations
Why the parabola is symmetric about the axis of symmetry
How to determine the vertex from a graph without calculus
The meaning of $x$-intercepts in the context of a quadratic model

Can do

Skills

Identify quadratic relationships from equations, tables, ordered pairs or graphs
Graph $y = ax^2 + bx + c$ using a table or graphing technology
Read the $x$-intercepts, $y$-intercept, axis of symmetry and vertex from a graph
Find the axis of symmetry and vertex using the midpoint of $x$-intercepts

Key terms

ParabolaThe U-shaped (or ∩-shaped) curve produced by a quadratic relationship $y = ax^2 + bx + c$.

VertexThe turning point of a parabola — the minimum if $a > 0$, maximum if $a < 0$. Written as an ordered pair $(x_v, y_v)$.

Axis of symmetryThe vertical line passing through the vertex, dividing the parabola into two mirror-image halves. Its equation is $x = x_v$.

$x$-interceptsWhere the parabola crosses the $x$-axis ($y = 0$). Also called roots or zeros. A parabola may have 0, 1 or 2 $x$-intercepts.

$y$-interceptWhere the parabola crosses the $y$-axis ($x = 0$). For $y = ax^2 + bx + c$, the $y$-intercept is always $(0, c)$.

Concave up / downConcave up ($a > 0$): opens upward like a bowl. Concave down ($a < 0$): opens downward like a dome.

Recognising and Representing Quadratic Relationships

core concept

A quadratic relationship is recognisable in each of its four forms. The key difference from linear (straight line) and exponential (constant ratio) relationships is that in a quadratic table, the second differences between $y$-values are constant.

Identifying a quadratic from a table:

Consider $x = 0, 1, 2, 3, 4$ and $y = 1, 4, 9, 16, 25$:

First differences: $4-1=3$, $9-4=5$, $16-9=7$, $25-16=9$ — not constant (not linear)
Second differences: $5-3=2$, $7-5=2$, $9-7=2$ — constant! This confirms quadratic.

The four representations:

Equation: $y = x^2$ — the simplest quadratic
Table of values: $(0,0),\ (1,1),\ (2,4),\ (3,9),\ (-1,1),\ (-2,4)$
Ordered pairs: same as table, written in set notation $\{(0,0),(1,1),(2,4),\ldots\}$
Graph: a symmetric U-shaped curve (parabola) opening upward with vertex at the origin

Constant second differences = quadratic: This rule is the reliable test. If first differences increase (or decrease) by the same amount each time, the relationship is quadratic. The value of the second difference equals $2a$ (twice the leading coefficient).

What to write in your book

Identify quadratic from a table: constant second differences (first differences of the first differences).
Second difference $= 2a$ where $a$ is the leading coefficient.
Linear → constant first differences. Exponential → constant ratios. Quadratic → constant second differences.

Quick check: A table has $y$-values 3, 7, 13, 21, 31 for consecutive integer $x$-values. First differences are 4, 6, 8, 10. What type of relationship is this?

Graphing Parabolas — Key Features from Graphs

core concept

A parabola is completely described by its vertex, axis of symmetry, direction of opening, and intercepts. NESA requires you to graph using technology and read all four features directly from the graph.

Reading features from a graph:

$x$-intercepts: where the curve crosses the $x$-axis. Read the $x$-coordinate(s). A parabola can have zero (no crossing), one (touches the axis — double root), or two (crosses twice) $x$-intercepts.
$y$-intercept: where the curve crosses the $y$-axis. Read the $y$-coordinate. For $y = ax^2 + bx + c$, this is always $(0, c)$.
Vertex: the turning point — the lowest point if the parabola opens up, highest if it opens down.
Axis of symmetry: the vertical line $x = $ (vertex's $x$-coordinate), which divides the parabola into two identical halves.

Finding the axis of symmetry from two $x$-intercepts: If the $x$-intercepts are $(x_1, 0)$ and $(x_2, 0)$, the axis of symmetry is the midpoint:

$$x = \frac{x_1 + x_2}{2}$$

Substitute this $x$-value back into the equation to find the $y$-coordinate of the vertex.

Example: A parabola has $x$-intercepts at $x = 1$ and $x = 5$.
Axis of symmetry: $x = \frac{1 + 5}{2} = 3$.
If the equation is $y = -(x-1)(x-5)$, vertex at $x=3$: $y = -(3-1)(3-5) = -(2)(-2) = 4$.
Vertex: $(3, 4)$.

Technology instruction: In Desmos, enter the equation (e.g., y = -x^2 + 6x - 5). Click on the $x$-intercepts, vertex and $y$-intercept to read their coordinates. Use this to check answers found algebraically.

What to write in your book

$y$-intercept: substitute $x = 0$ — always equal to $c$.
Axis of symmetry from $x$-intercepts: $x = \frac{x_1 + x_2}{2}$ (midpoint).
Vertex: $x = $ axis of symmetry; $y = $ substitute back into equation.

True or false: A parabola with $x$-intercepts at $x = -3$ and $x = 7$ has its axis of symmetry at $x = 2$.

Reading Quadratic Graphs in Context

core concept

In context, every feature of a quadratic graph has a meaning. The $x$-intercepts, vertex and $y$-intercept all answer specific real-world questions. NESA will ask you to read and interpret these features from a given graph.

A ball is kicked and its height $h$ (m) at time $t$ (s) follows $h = -5t^2 + 20t$. Its graph is a downward parabola ($a = -5 < 0$, opens down). From the graph:

$y$-intercept (at $t=0$): $h = 0$ — the ball starts at ground level. This is the starting height.
$x$-intercepts: $t = 0$ and $t = 4$ — the ball is at ground level at $t = 0$ (kicked) and $t = 4$ seconds (lands). The ball travels for 4 seconds.
Vertex: Axis of symmetry: $t = (0 + 4)/2 = 2$ s. Height at $t=2$: $h = -5(4) + 20(2) = -20 + 40 = 20$ m. The vertex is $(2, 20)$ — maximum height of 20 m at 2 seconds.

Downward parabolas in context: When $a < 0$ (opens down), the vertex is the maximum — this represents peak height, maximum profit, greatest area, etc. The $x$-intercepts represent when the quantity returns to zero.

What to include in your answer: When asked to "identify the vertex" from a graph, always state: (1) the coordinates of the vertex, and (2) its meaning in context (e.g. "the maximum height of 20 m is reached at 2 seconds after the ball is kicked").

What to write in your book

$x$-intercepts = when quantity is zero (e.g. ball at ground, profit = 0, starting/ending values).
Vertex = maximum or minimum value — state the coordinates and the meaning in context.
$y$-intercept = value when $x = 0$ (initial value/starting condition).

Fill the blanks: For the parabola $y = -2x^2 + 8x$, the $y$-intercept is . The $x$-intercepts are at $x = 0$ and $x = $ . The axis of symmetry is $x = $ .

Worked examples · 3 problems

PROBLEM 1 · GRAPHING AND READING KEY FEATURES

For $y = x^2 - 4x - 5$: (a) Find the $y$-intercept. (b) Using technology, identify the two $x$-intercepts. (c) Find the axis of symmetry. (d) Find the coordinates of the vertex.

Part (a) — $y$-intercept
Substitute $x = 0$: $y = 0 - 0 - 5 = -5$
$y$-intercept: $(0, -5)$

The $y$-intercept is always the constant term $c = -5$ in $y = ax^2 + bx + c$.

PROBLEM 2 · CONTEXTUAL PARABOLA (PROJECTILE)

A ball is thrown and its height follows $h = -4t^2 + 16t$ where $h$ is in metres and $t$ in seconds. From the graph: (a) When does the ball land? (b) What is the maximum height and when does it occur? (c) What does the $y$-intercept represent?

Part (a) — when ball lands ($h = 0$)
$x$-intercepts from graph: $t = 0$ and $t = 4$
The ball lands at $t = 4$ seconds.

$h = 0$ means the ball is at ground level. From the graph, it starts at $t=0$ and lands at $t=4$.

PROBLEM 3 · TABLE OF VALUES AND FEATURES

A table gives: $x$ = 0, 1, 2, 3, 4 and $y$ = 8, 5, 4, 5, 8. (a) Show this is quadratic. (b) Identify the vertex and axis of symmetry.

Part (a) — first and second differences
First differences: $-3, -1, +1, +3$
Second differences: $2, 2, 2$ — constant!
Therefore quadratic.

Constant second differences confirm a quadratic relationship. The second difference is 2, so $2a = 2$, meaning $a = 1$.

Match each feature to its description:

Top 3 list: Name THREE things you should check or state when asked to "fully describe" a parabola from its graph.

Revisit your thinking

Ball: $h = -5t^2 + 20t$. $x$-intercepts at $t = 0$ and $t = 4$ s. Axis of symmetry: $t = 2$ s. Vertex: $h = -5(4) + 20(2) = 20$ m — the ball reaches 20 m at 2 seconds, then lands at 4 seconds. If you guessed the peak was at $t = 2$ (halfway), you had the right instinct — the symmetry of the parabola tells you exactly that.

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Short answer — exam-style questions

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ApplyBand 34 marks

SA 1. For the quadratic $y = x^2 - 6x + 8$: (a) State the $y$-intercept. (b) Using technology or otherwise, find the $x$-intercepts. (c) Hence find the axis of symmetry and the coordinates of the vertex. (4 marks)

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ApplyBand 44 marks

SA 2. The table below shows values of a quadratic relationship:

$x$	−2	−1	0	1	2	3
$y$	15	6	−1	−6	−9	−10

(a) Show using differences that this is a quadratic. (b) State the $y$-intercept. (c) The vertex is at $x = 2.5$. Find the $y$-coordinate of the vertex. (4 marks)

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AnalyseBand 43 marks

SA 3. A garden bed is designed so its area $A$ (m²) depends on its width $w$ (m) according to $A = -w^2 + 10w$. From a graph of this relationship: (a) Find the width that gives the maximum area, and the maximum area itself. (b) State the practical domain of the model (the valid values of $w$). (3 marks)

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📖 Comprehensive answers (click to reveal)

SA 1 (4 marks): (a) $c=8$, $y$-intercept $(0,8)$ [1]. (b) $x^2-6x+8=(x-2)(x-4)=0$, so $x$-intercepts at $(2,0)$ and $(4,0)$ [1]. (c) Axis: $x=(2+4)/2=3$ [1]. Vertex: $y=9-18+8=-1$, Vertex $(3,-1)$ [1].

SA 2 (4 marks): (a) First diffs: $-9,-7,-5,-3,-1$; Second diffs: $2,2,2,2$ — constant [1], confirms quadratic [1]. (b) $y$-intercept $= -1$ at $x=0$ [1]. (c) The equation is $y=x^2-4x-1$ (from $2a=2$, $a=1$; check values). At $x=2.5$: $y=(2.5)^2-4(2.5)-1=6.25-10-1=-4.75$ [1]. (Note: vertex could also be found as midpoint of $x$-intercepts if students find them first.)

SA 3 (3 marks): (a) $A=-w^2+10w$; opens down ($a=-1<0$), so max at vertex. $x$-intercepts: $-w^2+10w=0\Rightarrow w(10-w)=0\Rightarrow w=0$ or $w=10$. Axis: $w=(0+10)/2=5$ m [1]. Max area: $A=-25+50=25$ m² [1]. (b) Practical domain: $0 < w < 10$ (width must be positive and less than 10 for the garden to have positive area) [1].

Boss battle · The Parabola Inspector

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Algebraic Relationships · Maths Standard