IEB Grade 12 Functions and Graphs Cheat Sheet

The key features of every Paper 1 graph, inverses and how to read information from a sketch.

The graphs

• Parabola $y=a(x+p{)}^2+q$: turning point $(-p;q)$, axis $x=-p$.

• Hyperbola $y=\frac{a}{x+p}+q$: asymptotes $x=-p$, $y=q$; axes of symmetry $y=\pm (x+p)+q$.

• Exponential $y=a{b}^{x+p}+q$: asymptote $y=q$; $b>1$ grows, $0<b<1$ decays.

• Log $y={\log }_{b}x$: domain $x>0$, $x$-intercept $(1;0)$, asymptote $x=0$.

Transformations of $y=f(x)$

• $f(x-a)$: $a$ right. $f(x)+a$: $a$ up.

• $-f(x)$: reflect in the $x$-axis. $f(-x)$: reflect in the $y$-axis.

• $af(x)$: vertical stretch by $a$.

Inverses

• Swap $x$ and $y$, make $y$ the subject. The graph reflects in $y=x$.

• $y=b^x⟺x={\log }_{b}y$, so $f(x)=b^x$ has $f^{-1}(x)={\log }_{b}x$.

• $y=a{x}^2$: restrict the domain ($x\ge 0$ or $x\le 0$) so the inverse is a function; choose $+\sqrt{}$ or $-\sqrt{}$ to match.

• Domain of $f$ = range of $f^{-1}$, and the other way round.

• Horizontal line test: if a horizontal line cuts $f$ twice, $f^{-1}$ is not a function.

Reading graphs

• Intersections: set $f(x)=g(x)$.

• Vertical distance: top $-$ bottom; maximise with the turning point or calculus.

• $f(x)>g(x)$: $f$ above $g$. $f(x)\cdot g(x)<0$: opposite sides of the $x$-axis.

• $f(x)=k$ has as many roots as the times $y=k$ cuts the graph.

Know by heart

• Asymptotes and symmetry axes of the hyperbola

• Log definition $y={\log }_{b}x⟺x=b^y$

• Transformation rules

On the formula sheet

• No function formulae are given; everything here must be known

Common traps

• Shifting $f(x-3)$ left.

• Turning point of $a(x+p{)}^2+q$ written as $(p;q)$.

• Forgetting $x>0$ for log inequalities.

• Writing $f^{-1}(x)$ as $\frac{1}{f(x)}$.