Linear, quadratic, cubic and quartic polynomial functions, basic exponential functions $y = a^x$, logarithmic functions $y = \log_a(x)$, and the standard transformations (dilation, reflection, translation)

What this dot point is asking

VCAA wants you to recognise the standard function families introduced in Unit 1, identify their key graphical features (intercepts, turning points, asymptotes), and apply the four standard transformations (dilation, reflection, translation in $x$ and $y$).

Function families

Linear $y = mx + c$. Gradient $m$, $y$-intercept $c$. Sketched as a straight line.

Quadratic $y = a x^2 + bx + c$ or vertex form $y = a(x-h)^2 + k$. Parabola opening up if $a > 0$, down if $a < 0$. Turning point at $(h, k)$ in vertex form.

Cubic $y = a x^3 + b x^2 + c x + d$. Either monotonic or with two turning points. Inflection at the average of the turning points.

Quartic $y = a x^4 + \ldots$. Either two or four sign changes; up to three turning points.

Exponential $y = a^x$ for $a > 0, a \neq 1$. Always positive. Horizontal asymptote $y = 0$. Passes through $(0, 1)$. Increasing if $a > 1$, decreasing if $0 < a < 1$.

Logarithmic $y = \log_a(x)$ for $a > 0, a \neq 1$. Defined only for $x > 0$. Vertical asymptote $x = 0$. Passes through $(1, 0)$. Inverse of $y = a^x$.

Key graphical features

For each function:

• Domain. Set of allowed $x$ values.
• Range. Set of resulting $y$ values.
• Axis intercepts. Where the graph crosses the $x$- and $y$-axes.
• Turning points / stationary points. Local maxima and minima.
• Asymptotes. Lines the graph approaches but never meets.
• End behaviour. What happens as $x \to \pm \infty$.

Sketching requires all relevant features labelled.

The four transformations

Given $y = f(x)$, the transformations:

Translation in $y$. $y = f(x) + k$ shifts up by $k$ (down if $k < 0$).

Translation in $x$. $y = f(x - h)$ shifts right by $h$ (left if $h < 0$). Note the sign convention: $(x - h)$ means shift right.

Dilation in $y$. $y = a f(x)$ stretches vertically by factor $a$ (compresses if $0 < a < 1$, reflects if $a < 0$).

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

Combined transformations. $y = a f(b(x - h)) + k$ combines all four with vertex at $(h, k)$.

Reflections. Special case of dilations:

• IMATH_51 : reflection in $x$-axis.
• IMATH_53 : reflection in $y$-axis.

Worked examples

Example 1. Linear

$y = 2x - 3$. Gradient 2, $y$-intercept $-3$. $x$-intercept: $0 = 2x - 3$, $x = 1.5$. Sketch as straight line through $(0, -3)$ and $(1.5, 0)$.

Example 2. Quadratic transformation

Start with $y = x^2$ (parabola, vertex at origin). Apply $y = 2(x - 1)^2 - 5$. This is:

• Dilation by 2 in $y$ (steeper).
• Translation right by 1.
• Translation down by 5.

Vertex at $(1, -5)$. Opens up. Solve for intercepts as in the worked past question.

Example 3. Exponential transformation

$y = 2 \cdot 3^x + 1$. Start with $y = 3^x$. Apply dilation by 2 (stretches vertically), then translation up by 1. New horizontal asymptote: $y = 1$. $y$-intercept: $2 \cdot 1 + 1 = 3$.

Domain, range and inverse

For inverse functions (covered in Unit 4), the domain and range swap. For Unit 1, observe:

• IMATH_72 has domain $\mathbb{R}$, range $(0, \infty)$.
• IMATH_75 has domain $(0, \infty)$, range $\mathbb{R}$.

The domains and ranges are reflections in the line $y = x$.

Common errors

Translation sign error. $y = f(x - 3)$ shifts right by 3 (positive direction), not left.

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

Forgetting the asymptote on exponential / log graphs. Exponentials have horizontal asymptotes; logs have vertical asymptotes. Mark them.

Confusing domain with range. Domain is the set of valid inputs; range is the set of outputs. For logs, the domain is restricted to positive $x$.

In one sentence

Unit 1 introduces the major function families (linear, quadratic, polynomial up to quartic, exponential, logarithmic) and the four standard transformations (translation in $x$ and $y$, dilation in $x$ and $y$); sketching requires all key features (intercepts, turning points, asymptotes, end behaviour) labelled, and transformations applied in the correct order (inside-the-bracket first, outside second).