Topic 7: Non-linear Relationships, Functions and Their Graphs

Function

Relation where each input x has exactly one output y.

Function notation

f(x).

Vertical Line Test

A relation is a function if any vertical line cuts the graph once.

Domain

Set of permissible x values.

Range

Set of permissible y values.

Inverse Function

Swap x and y, rearrange; reflection about y=x.

Parabolas

Graphs of y = ax² + bx + c.

Key Features of Parabolas

Turning point (vertex), axis of symmetry, roots, y-intercept.

Sketching Parabolas

Using transformations, factorisation, completing the square, quadratic formula.

Graphs of Circles

(x - h)² + (y - k)² = r²; can include half circles.

Exponential Functions

y = a^x or y = a^(-x).

Features of Exponentials

Horizontal asymptote (usually x-axis), y-intercept (usually (0,1)).

Hyperbolic Functions

y = k/x.

Features of Hyperbolas

Asymptotes (x and y axes).

Cubic Functions

y = ax³ + d.

Transformations

Vertical/horizontal shifts, reflections, dilations.

Graphs to Describe Change

Distance-time and rate of change.

Non-linear relationship

A relationship that doesn't form a straight line when graphed

Quadratic equation

y = ax² + bx + c

Shape of quadratic graph

Parabola

Parabola opens upwards

a > 0

Parabola opens downwards

a < 0

Vertex of parabola

Turning point (maximum or minimum)

Axis of symmetry

x = −b / (2a)

y-intercept of parabola

c (from y = ax² + bx + c)

x-intercepts of parabola

Solve y = 0 using factorising, completing the square or quadratic formula

Quadratic formula

x = (−b ± √(b²−4ac)) / 2a

Discriminant (b²−4ac)

Tells number of x-intercepts

• Discriminant > 0

Two real solutions

• Discriminant = 0

One real solution (touches x-axis)

• Discriminant < 0

No real solution (no x-intercepts)

Sketching parabola steps

1. Shape (a), 2. y-intercept, 3. x-intercepts, 4. Vertex, 5. Axis of symmetry

Completing the square form

y = a(x − h)² + k (vertex is (h, k))

Sketching from vertex form

Identify vertex (h, k) and direction of opening

Quadratic (basic form)

y = ax² + bx + c

Quadratic (turning point form)

y = a(x – h)² + k (turning point: (h, k))

Quadratic (factored form)

y = a(x – r₁)(x – r₂)

Cubic (basic form)

y = ax³ + bx² + cx + d

Cubic (factored form)

y = a(x – r₁)(x – r₂)(x – r₃)

Exponential growth

y = a × b^x (a > 0, b > 1)

Exponential decay

y = a × b^x (a > 0, 0 < b < 1)

Hyperbola (rational)

y = a / (x – h) + k (asymptotes at x = h, y = k)

Circle (standard form)

(x – h)² + (y – k)² = r² (centre: (h, k), radius: r)

Semicircle (upper)

y = √(r² – (x – h)²) + k

Semicircle (lower)

y = –√(r² – (x – h)²) + k