Non-linear graphs

What you will learn

sketch parabolas in the form $y = ax^2 + bx + c$ and vertex form $y = a(x - h)^2 + k$ ,
identify the centre and radius of a circle from $(x - h)^2 + (y - k)^2 = r^2$ ,
recognise and sketch exponential curves $y = a^x$ (growth vs decay),
recognise the hyperbola $y = \tfrac{k}{x}$ and its key features.

Worked example 0 Real-world example: height of a kicked football

A football is kicked and its height in metres is modelled by $h = -5t^2 + 15t$ , where $t$ is time in seconds. Find the maximum height.

This is a parabola opening downward ( $a = -5 < 0$ ).
Vertex $t$ -coordinate: $t = -\dfrac{b}{2a} = -\dfrac{15}{2(-5)} = 1.5$ seconds.
Maximum height: $h = -5(1.5)^2 + 15(1.5) = -11.25 + 22.5 = 11.25$ metres.

Key idea: the vertex of a downward parabola gives the maximum value.

1. Parabolas

The graph of any quadratic function is a parabola.

Formula reference

General form: $y = ax^2 + bx + c$ . Vertex at $x = -\dfrac{b}{2a}$ .

Vertex (turning-point) form: $y = a(x - h)^2 + k$ . Vertex at $(h, k)$ .

If $a > 0$ the parabola opens upward (minimum at vertex).
If $a < 0$ the parabola opens downward (maximum at vertex).

Worked example 1 Sketching from vertex form

Sketch $y = 2(x - 3)^2 - 8$ .

Vertex: $(3, -8)$ .
Opens upward ( $a = 2 > 0$ ).
$y$ -intercept: set $x = 0$ : $y = 2(9) - 8 = 10$ , point $(0, 10)$ .
$x$ -intercepts: $0 = 2(x-3)^2 - 8$ , $(x-3)^2 = 4$ , $x - 3 = \pm 2$ , so $x = 1$ or $x = 5$ .
Plot vertex, intercepts, and draw a smooth U-shape.

Worked example 2 Converting general to vertex form

Write $y = x^2 - 6x + 5$ in vertex form.

Complete the square: $y = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4$ .
Vertex: $(3, -4)$ . The parabola opens upward.

2. Circles

Formula reference

(x - h)^2 + (y - k)^2 = r^2

Centre $(h, k)$ , radius $r$ .

Worked example 3 Identifying centre and radius

Find the centre and radius of $(x + 2)^2 + (y - 5)^2 = 49$ .

Rewrite: $(x - (-2))^2 + (y - 5)^2 = 7^2$ .
Centre $(-2, 5)$ , radius $7$ .

Worked example 4 Writing the equation of a circle

Write the equation of a circle with centre $(3, -1)$ and radius $4$ .

$(x - 3)^2 + (y + 1)^2 = 16$ .

3. Exponential functions

Formula reference

y = a^x

If $a > 1$ : exponential growth (curve rises steeply to the right).
If $0 < a < 1$ : exponential decay (curve falls toward zero).
The $y$ -intercept is always $(0, 1)$ (since $a^0 = 1$ ).
The $x$ -axis ( $y = 0$ ) is a horizontal asymptote.

Worked example 5 Sketching an exponential growth curve

Sketch $y = 2^x$ .

Key points: $(-2, \tfrac{1}{4})$ , $(-1, \tfrac{1}{2})$ , $(0, 1)$ , $(1, 2)$ , $(2, 4)$ , $(3, 8)$ .
Curve passes through $(0, 1)$ , rises steeply to the right.
As $x \to -\infty$ , $y \to 0$ (approaches the $x$ -axis but never touches it).

4. Hyperbolas (reciprocal functions)

Formula reference

y = \frac{k}{x}

Two branches: one in the first and third quadrants if $k > 0$ ; second and fourth quadrants if $k < 0$ .
Asymptotes: the $x$ -axis ( $y = 0$ ) and the $y$ -axis ( $x = 0$ ).
The graph never crosses either axis.

Worked example 6 Sketching a hyperbola

Sketch $y = \dfrac{6}{x}$ .

Key points: $(1, 6)$ , $(2, 3)$ , $(3, 2)$ , $(6, 1)$ , $(-1, -6)$ , $(-2, -3)$ .
$k = 6 > 0$ : branches in quadrants I and III.
As $x \to \infty$ , $y \to 0$ . As $x \to 0^+$ , $y \to \infty$ .

Three non-linear graphs on separate mini-axes: a parabola y = x squared minus 4, a circle centred at (0, 0) with radius 3, and an exponential y = 2 to the x.

Practice

Fluency

Tier 1: basic skills

State the vertex and direction (up/down) of $y = (x + 1)^2 - 9$ .
Find the $x$ -intercepts of $y = x^2 - 4x - 5$ by factorising.
State the centre and radius of $(x - 3)^2 + (y + 2)^2 = 25$ .
Write the equation of a circle with centre $(0, 4)$ and radius $6$ .
For $y = 3^x$ , find $y$ when $x = 0$ , $x = 2$ , and $x = -1$ .
For $y = \dfrac{4}{x}$ , find $y$ when $x = 1$ , $x = 4$ , and $x = -2$ .
Identify whether each function is a parabola, circle, exponential, or hyperbola: (a) $y = 5^x$ , (b) $x^2 + y^2 = 16$ , (c) $y = \tfrac{3}{x}$ , (d) $y = -x^2 + 2$ .
State the asymptote(s) of $y = \dfrac{-2}{x}$ .

Reasoning

Tier 2: mixed practice

Write $y = x^2 + 8x + 12$ in vertex form by completing the square, then state the vertex.
A ball is thrown upward with height $h = -4.9t^2 + 19.6t + 1.5$ . Find the maximum height and the time it is reached.
Determine whether the point $(3, 4)$ lies inside, on, or outside the circle $(x - 1)^2 + (y - 2)^2 = 9$ .
Sketch $y = 2^x$ and $y = \left(\tfrac{1}{2}\right)^x$ on the same axes. Describe the relationship between the two curves.
A hyperbola passes through $(2, 5)$ and has the form $y = \tfrac{k}{x}$ . Find $k$ and state the equations of the asymptotes.
The parabola $y = a(x - 1)^2 + 3$ passes through $(3, 11)$ . Find $a$ .

Reasoning

Tier 3: explain and apply

A tunnel has a parabolic cross-section. At ground level it is $8$ m wide and the maximum height is $5$ m. Taking the origin at the centre of the base, find the equation of the parabola and determine whether a truck $3$ m wide and $4$ m tall can pass through.
Show algebraically that the circle $x^2 + y^2 = 25$ and the line $y = x + 1$ intersect at two points. Find the coordinates.
Compare the graphs of $y = 2^x$ and $y = 3 \times 2^x$ . Explain how the multiplier affects the curve.
Explain why $y = \dfrac{k}{x}$ can never equal zero, no matter how large $x$ becomes.

Challenge

Reasoning

Harder reasoning

A quadratic $y = ax^2 + bx + c$ has vertex $(2, -3)$ and passes through $(0, 5)$ . Find $a$ , $b$ , and $c$ .
Find the two points where the parabola $y = x^2$ and the circle $x^2 + y^2 = 2$ intersect (for $y \geq 0$ ).
The curve $y = 2^x$ is reflected in the $y$ -axis and then shifted up by $3$ units. Write the equation of the resulting curve and state its asymptote.