Functions and graphs introduced in Year 11, including linear, quadratic, cubic, polynomial, exponential and logarithmic functions; their key features, intercepts and transformations

What this dot point is asking

QCAA wants Year 11 students to recognise the major function families, identify their key features, and apply the four standard transformations. Foundation for Unit 3 / 4 calculus work.

Function families

Linear $y = mx + c$. Straight line, gradient $m$, $y$-intercept $c$.

Quadratic $y = ax^2 + bx + c$ or vertex form $y = a(x - h)^2 + k$. Parabola, turning point $(h, k)$.

Cubic $y = ax^3 + bx^2 + cx + d$. Up to two turning points; can be monotonic.

Polynomial. Degree $n$ polynomial has up to $n - 1$ turning points.

Exponential $y = a^x$ for $a > 0, a \neq 1$. Always positive, horizontal asymptote $y = 0$.

Logarithmic $y = \log_a(x)$. Defined for $x > 0$, vertical asymptote $x = 0$, inverse of exponential.

Key features to identify

• Domain and range.
• IMATH_16 - and $y$-intercepts.
• Turning points / stationary points.
• Asymptotes (vertical, horizontal).
• End behaviour.

The four standard transformations

Given $y = f(x)$:

Translation in $y$. $y = f(x) + k$ shifts up by $k$.

Translation in $x$. $y = f(x - h)$ shifts right by $h$.

Dilation in $y$. $y = af(x)$ stretches vertically by factor $a$.

Dilation in $x$. $y = f(bx)$ compresses horizontally by factor $1/b$.

Reflections are special cases ($a$ or $b$ negative).

Worked example

$y = 3(x - 2)^3 - 5$. Start with $y = x^3$. Apply:

• Translation right by 2.
• Vertical stretch by 3.
• Translation down by 5.

Inflection point at $(2, -5)$.

Common errors

Translation sign error. $f(x - h)$ shifts right by $h$, not left.

Wrong transformation order. Apply inside-the-bracket first (operations on $x$), then outside (operations on $y$).

Forgetting asymptotes. Exponentials have horizontal asymptotes; logs have vertical asymptotes. Mark them.

In one sentence

Unit 1 introduces the major function families (linear, quadratic, polynomial, exponential, logarithmic) and the four standard transformations (translation and dilation in both $x$ and $y$); sketching requires all key features (intercepts, turning points, asymptotes, end behaviour) labelled.