Functions and graphs: QCE Math Methods Unit 1 Year 11

What this dot point is asking

QCAA wants Year 11 students to recognise the major function families, identify their key features (intercepts, turning points, asymptotes and end behaviour), and apply the four standard transformations to sketch and describe graphs. This is the foundation for the Unit 3 and 4 calculus work, where the same families and transformations reappear with derivatives and integrals.

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

When analysing or sketching any function, work through a fixed checklist so nothing is missed:

• Domain and range (the set of allowed inputs and resulting outputs).
• $x$- and $y$-intercepts (set $y = 0$ and $x = 0$ respectively).
• Turning points or stationary points (maxima, minima, inflections).
• Asymptotes, both vertical and horizontal, where the graph approaches but never reaches a line.
• End behaviour, the direction the graph heads as $x \to \pm\infty$.

A function is a rule assigning exactly one output to each input, which is why graphs of functions pass the vertical line test: no vertical line meets the graph more than once.

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: $a < 0$ reflects in the $x$-axis, and $b < 0$ reflects in the $y$-axis.

Order of transformations

When several transformations combine, apply the operations on $x$ (inside the function) before the operations on $y$ (outside). For $y = af(b(x - h)) + k$, work from the inside out: first the horizontal dilation by $\tfrac{1}{b}$, then the horizontal translation by $h$, then the vertical dilation by $a$, then the vertical translation by $k$. Getting this order wrong is the most common source of misplaced graphs, because a horizontal shift applied before a horizontal dilation lands in a different place.

Reading key features off each family

Each family advertises its features in its equation. A quadratic in vertex form shows its turning point directly; an exponential $y = a^x + k$ has horizontal asymptote $y = k$; a logarithm $y = \log_a(x - h)$ has vertical asymptote $x = h$. Identifying the family first, then reading the relevant features, is faster and more reliable than plotting points blindly.