Functions and Graphs

Functions

A function assigns exactly one output to each input. We write $y = f(x)$.

• Domain: the set of allowed inputs $x$.
• Range: the set of outputs $y$.

To find $f(-2)$ when $f(x) = x^2 - 3x$, replace every $x$ with $(-2)$, brackets included: $f(-2) = (-2)^2 - 3(-2) = 10$.

Graph of a quadratic function

For $y = ax^2 + bx + c$:

• $a > 0$: opens upwards, has a minimum point.
• $a < 0$: opens downwards, has a maximum point.
• $y$-intercept is $c$; $x$-intercepts are the real roots of $ax^2 + bx + c = 0$.

Completing the square

Rewrite $y = ax^2 + bx + c$ as $y = a(x - h)^2 + k$. Then:

• the vertex is $(h, k)$,
• the axis of symmetry is $x = h$,
• the maximum or minimum value is $k$.

Example: $y = x^2 - 4x + 7 = (x^2 - 4x + 4) + 3 = (x - 2)^2 + 3$. The vertex is $(2, 3)$ and the minimum value is $3$.

Common mistake: for $y = (x + 3)^2 + 2$ the vertex is $(-3, 2)$, not $(3, 2)$. The sign flips inside the bracket.

Transformations of graphs

New function	Effect on $y = f(x)$
$y = f(x) + k$	translate up $k$ units
$y = f(x) - k$	translate down $k$ units
$y = f(x + k)$	translate left $k$ units
$y = f(x - k)$	translate right $k$ units
$y = -f(x)$	reflection with respect to the $x$-axis
$y = f(-x)$	reflection with respect to the $y$-axis
$y = kf(x)$	stretch vertically by factor $k$
$y = f(kx)$	shrink horizontally by factor $k$

Exam tip: changes inside the bracket act on $x$ and go the "opposite" way: $f(x + 3)$ moves the graph 3 units to the left.