Revision sheet

Quadratic patterns

Constant second differences mean quadratic.

Watch out: growing differences are not linear.

Standard and factored form

Factored form gives the $x$-intercepts; standard form gives the $y$-intercept.

Watch out: the factor $x + 5$ gives the intercept $-5$.

Graphs of quadratics

Sign of $a$: up or down; $c$: $y$-intercept; vertex midway between the zeros.

Watch out: the $y$-intercept is $c$, not $a$.

Vertex form

$y = a(x - h)^2 + k$: vertex $(h, k)$.

Watch out: $(x + h)$ means the vertex is at $-h$.

Quadratic functions

1. Vertex at $x = -\dfrac{b}{2a}$.
2. Its height $f\left(-\dfrac{b}{2a}\right)$ is the minimum ($a > 0$) or maximum ($a < 0$).
3. Vertex form: $a(x - h)^2 + k$, vertex $(h, k)$.

Watch out: the constant term $c$ is $f(0)$, not the extreme value.

Piecewise functions

1. Find the piece whose condition the input satisfies.
2. Use that piece's formula.
3. $|x| = x$ for $x \ge 0$, and $-x$ for $x < 0$.

Watch out: at a boundary, read $<$ and $\ge$ carefully.

Shifts and reflections

• $f(x - h)$: $h$ units right. $f(x) + k$: $k$ units up.
• $-f(x)$: flip across the $x$-axis. $f(-x)$: flip across the $y$-axis.
• Inside $f(\dots)$ acts on $x$; outside acts on $y$.

Watch out: $f(x + 3)$ moves the graph left.

Inverse functions

1. Write $y = f(x)$.
2. Solve for $x$, undoing the last operation first.
3. Rename: $f^{-1}(x)$. Check that $f(f^{-1}(x)) = x$.

Watch out: $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$.