Functions and Their Graphs
Core concepts
A function maps each input to exactly one output; know domain, range, and how to find them from a rule and from a graph. Be able to sketch and identify the key features of linear, quadratic, cubic, hyperbolic ($y=a/(x-h)+k$), square-root and circle relations, including asymptotes, intercepts, turning points and endpoints. Master transformations: $y=af(b(x-h))+k$ produces dilation by factor $a$ from the $x$-axis, dilation by factor $1/b$ from the $y$-axis, translation $h$ right and $k$ up. Understand inverse functions: they exist only when the original is one-to-one, the graph reflects in $y=x$, and domain and range swap. Know hybrid (piecewise) functions, modulus functions and how to solve equations and inequalities graphically.
Traps
- Applying transformations in the wrong order; horizontal changes act on $x$ before dilation, and inside-bracket operations behave 'backwards'.
- Forgetting that a dilation factor of $1/b$ means a horizontal stretch by $1/b$, not $b$.
- Stating the range of a restricted quadratic without checking the turning point.
- Assuming every function has an inverse; restrict the domain first.
- Omitting open circles or endpoints when sketching hybrid functions.
Worked example
For $y=-2\sqrt{x-3}+1$, the base curve $y=\sqrt{x}$ is translated 3 right and 1 up, reflected in the $x$-axis and dilated by factor 2 from the $x$-axis. Domain: $x\ge 3$. Range: since $\sqrt{x-3}\ge 0$, $-2\sqrt{x-3}\le 0$, so $y\le 1$.
Rule of thumb
Sketch the parent function first, mark its key points, then track each transformation on those points.