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:

• $y = -f(x)$: reflection in $x$-axis.
• $y = f(-x)$: reflection in $y$-axis.

Domain, range and inverse

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

• $f(x) = a^x$ has domain $\mathbb{R}$, range $(0, \infty)$.
• $f^{-1}(x) = \log_a(x)$ has domain $(0, \infty)$, range $\mathbb{R}$.

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

Examples in context

Example 1. Transforming a quadratic. Starting from $y = x^2$, the graph of $y = 2(x - 3)^2 - 5$ is dilated vertically by factor $2$, translated $3$ right and $5$ down. Its vertex is at $(3, -5)$, and the $y$-intercept is $y = 2(0 - 3)^2 - 5 = 18 - 5 = 13$. The parabola opens upward.

Example 2. Exponential vs logarithmic features. The graph of $y = 2^x$ has horizontal asymptote $y = 0$, $y$-intercept $1$, and range $y > 0$. Its inverse $y = \log_2 x$ has vertical asymptote $x = 0$, $x$-intercept $1$, and domain $x > 0$. They are reflections of each other in the line $y = x$.

Try this

Q1. Describe the transformations taking $y = x^2$ to $y = (x + 1)^2 + 4$, and state the vertex. [2 marks]

• Cue. Translate $1$ left and $4$ up; vertex $(-1, 4)$.

Q2. State the asymptote, $y$-intercept and range of $y = 3^x - 2$. [3 marks]

• Cue. Asymptote $y = -2$; $y$-intercept $3^0 - 2 = -1$; range $y > -2$.

Q3. The graph of $y = a(x - h)^2 + k$ has vertex $(2, -1)$ and passes through $(0, 3)$. Find $a$. [2 marks]

• Cue. $3 = a(0 - 2)^2 - 1 = 4a - 1$, so $a = 1$.