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Functions 11

Chris Kirkpatrick, Barbara Alldred, Andrew Dmytriw, Shawn Godin, Angelo Lillo, David Pilmer, Susanne Trew, Noel Walker · Grade 11 · Study material generated by AI from this edition. Not affiliated with the publisher.

Cheat sheet

Key terms

Term	Meaning
Function	Relation where each input has exactly one output
Domain / range	Set of all inputs / set of all outputs
Vertical-line test	A graph is a function if no vertical line hits it twice
Inverse $f^{-1}$	Undoes $f$ : point $(a,b)$ becomes $(b,a)$ ; graph reflected in $y=x$
Equivalent expressions	Simplify to the same expression; same graph
Restriction	Value that makes a denominator 0; excluded from domain
Vertex / axis of symmetry	Turning point $(h,k)$ / vertical line $x=h$ through it
Exponential function	$f(x)=ab^x$ , $b>0$ , $b\neq1$ ; variable in the exponent
Horizontal asymptote	Line a graph approaches but never reaches ( $y=0$ for $b^x$ )
Principal angle	Counterclockwise angle from the initial arm, $0^\circ$ to $360^\circ$
Related acute angle	Acute angle between the terminal arm and the x-axis
Period	Change in $x$ for one full cycle (peak to peak)
Amplitude	Distance from axis to max or min; never negative
Arithmetic / geometric sequence	Constant difference $d$ / constant ratio $r$
Series	Sum of the terms of a sequence
Simple / compound interest	Interest on principal only / on principal plus earlier interest
Present value	Amount to invest now to reach a future amount
Annuity	Equal payments at regular intervals (ordinary: end of each period)

Formulas and rules

Quadratics

Form	Equation	Shows
Standard	$f(x)=ax^2+bx+c$	y-intercept $c$
Vertex	$f(x)=a(x-h)^2+k$	vertex $(h,k)$
Factored	$f(x)=a(x-r)(x-s)$	zeros $r,s$ ; axis $x=\frac{r+s}{2}$

$a>0$ : opens up, minimum; $a<0$ : opens down, maximum.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ ; vertex at $x=-\frac{b}{2a}$
Revenue $R(x)=p(x)\cdot x$ ; profit $P(x)=R(x)-C(x)$
$a^2-b^2=(a+b)(a-b)$ (a sum of squares does not factor)

Exponents ( $b\neq0$ ) $b^0=1,\quad b^{-n}=\frac{1}{b^n},\quad b^mb^n=b^{m+n},\quad \frac{b^m}{b^n}=b^{m-n},\quad (b^m)^n=b^{mn},\quad b^{m/n}=(\sqrt[n]{b})^m$ Example: $27^{2/3}=3^2=9$ .

Trigonometry ( $r=\sqrt{x^2+y^2}$ )

$\sin\theta=\frac yr,\ \cos\theta=\frac xr,\ \tan\theta=\frac yx$ ; $\csc\theta=\frac1{\sin\theta},\ \sec\theta=\frac1{\cos\theta},\ \cot\theta=\frac1{\tan\theta}$
Sine law: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ ; cosine law: $c^2=a^2+b^2-2ab\cos C$
Pythagoras: $a^2+b^2=c^2$

Angle	sin	cos	tan
$30^\circ$	$\frac12$	$\frac{\sqrt3}{2}$	$\frac{\sqrt3}{3}$
$45^\circ$	$\frac{\sqrt2}{2}$	$\frac{\sqrt2}{2}$	$1$
$60^\circ$	$\frac{\sqrt3}{2}$	$\frac12$	$\sqrt3$

Quadrant	Angle	Positive ratio
1	$\theta$	All
2	$180^\circ-\theta$	Sine
3	$180^\circ+\theta$	Tangent
4	$360^\circ-\theta$	Cosine

Sinusoidal: $f(x)=a\sin(k(x-d))+c$ . Axis $y=\frac{\max+\min}{2}$ ; amplitude $\frac{\max-\min}{2}$ . $\sin$ and $\cos$ have period $360^\circ$ , range $-1$ to $1$ . A point rotated from $(r,0)$ lands at $(r\cos\theta,\ r\sin\theta)$ .

Sequences and series

	Arithmetic	Geometric
Term	$t_n=a+(n-1)d$	$t_n=ar^{n-1}$
Sum	$S_n=\frac{n(t_1+t_n)}{2}=\frac{n[2a+(n-1)d]}{2}$	$S_n=\frac{a(r^n-1)}{r-1}$ , $r\neq1$

Fibonacci: $t_n=t_{n-1}+t_{n-2}$ (1, 1, 2, 3, 5, 8...). Row $n$ of Pascal's triangle gives the coefficients of $(a+b)^n$ .

Finance

Simple (linear)	Compound (exponential)
$I=Prt$ , $A=P(1+rt)$	$A=P(1+i)^n$ , $PV=\frac{A}{(1+i)^n}$

Processes

Inverse: write $y=f(x)$ , swap $x$ and $y$ , solve for $y$ (use $\pm\sqrt{\ }$ for quadratics). E.g. $y=3x-4\Rightarrow f^{-1}(x)=\frac{x+4}{3}$ .
Factor $ax^2+bx+c$ : take out the GCF; find two numbers with product $ac$ and sum $b$ ; split the middle term; group. $10x^2-x-3=(2x+1)(5x-3)$ .
Complete the square: factor $a$ from the $x$ -terms; add and subtract $(\frac{b}{2})^2$ inside; factor the square; move the extra term out (times $a$ ). $-5(t^2-8t+16-16)+100=-5(t-4)^2+180$ .
Solve for angles $0^\circ$ to $360^\circ$ : use the sign to pick quadrants (CAST); find the related acute angle; build each principal angle. $\sin\theta=-\frac{\sqrt3}{2}\Rightarrow240^\circ,300^\circ$ .
Classify from a table: constant first differences means linear; constant second differences means quadratic; a constant ratio means exponential.
Compound setup: $i=$ annual rate ÷ periods per year; $n=$ years × periods per year. 4.8% quarterly for 21 years: $i=0.012$ , $n=84$ .

Watch out

✗ $(3x+2)^2=9x^2+4$ . ✓ $9x^2+12x+4$ : square by multiplying, so cross terms appear.
✗ Subtracting changes only the first sign. ✓ Change every sign in the second bracket.
✗ Equal values at one input prove equivalence. ✓ One test can only disprove it.
✗ The inverse of a quadratic is always a function. ✓ It is only a function if the domain is restricted to one side of the vertex.
✗ $f^{-1}(x)=\frac{1}{f(x)}$ . ✓ The inverse reverses the operations; it is not a reciprocal.
✗ In $(x-3)^2$ the graph shifts left. ✓ $h=3$ , so it shifts right 3.
✗ $y=-2\sin x$ has amplitude $-2$ . ✓ Amplitude is 2; the negative sign reflects the graph.
✗ Use the annual rate with monthly periods. ✓ Convert to $i$ and $n$ first.

Mnemonics

"All Silly Turtles Crawl" — quadrants 1–4 positive ratios: All, Sine, Tangent, Cosine.
"Root rests on the bottom" — in $b^{m/n}$ the denominator is the root and the numerator is the power.
"h hides its sign" — $(x-h)$ shifts the opposite way to the sign you see.
"Simple adds, compound multiplies" — simple interest is arithmetic (a line); compound is geometric (a curve).
"1-2-ratio" — flat 1st differences: line; flat 2nd differences: parabola; flat ratio: exponential.

Functions 11

Cheat sheet

Functions 11: cheat sheet

Key terms

Formulas and rules

Processes

Watch out

Mnemonics

Cheat sheet

Functions 11: cheat sheet

Key terms

Formulas and rules

Processes

Watch out

Mnemonics