Mathematics

Module 1 for Mathematics Advanced (HSC)

Annotated version of the standard math formula sheet.

Exponents and Logarithms for 2U and 3U Mathematics.

Roots of multiplicity r A root of a polynomial is a value of \(x\) for which \(P(x)=0\) For example, \(P(x)=x^2+6x+9\) can be expressed as \((x+3)^2\) In this case, \(-3\) is a root of multiplicity 2 of \(P(x)\) Roots of multiplicity 1 are also known as “single roots” Roots of multiplicity 2 are also known as “double roots” Roots of multiplicity 2 are also known as “triple roots” Curve Sketching Graphs with multiple roots have specific rules for sketching Rules for Leading Coefficient and Degrees The table below explains what happens as a graph approaches infinity and negative infinity, based on the leading coefficient and degree: Degree: OddDegree: EvenLeading Coefficient: PositiveUp Arrow: 1st Quadrant

Important Formulas (That You Probably Should Memorise) \(\frac{d}{dx}(a)=0\) \(\frac{d}{dx}(x)=1\) \(\frac{d}{dx}(ax)=a\) \(\frac{d}{dx}(x^{n})=nx^{n-1}\) \(\frac{d}{dx}(f(x)+f(g))=\frac{d}{dx}(f(x))+\frac{d}{dx}(g(x))\) \(f`(x)=\lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}\)

Definition A function is a relation between two sets of data where each input has 1 or less potential outputs Horizontal Lines, Parabolas, Linear Equations, Hyperbolas, Exponentials, Polynomials and Cubic Graphs are all examples of functions Circles and Vertical Lines are NOT functions In other words, functions can be one-to-one or many-to-one relationships, but not one-to-many relationships (In reference to input and output values) Notation There are 3 methods of expressing functions: \(y=123\) \(f(x)=123\) \(f:x→123\) All of the above methods say the same thing: When \(x\) is the input, \(123\) is the output For example: \(y=2x\) \(f(x)=2x\) \(f:x→2x\) All state that when \(x\) is the input, \(2x\) is the output Vertical Line Test The vertical line test is a quick way to test if a graph is a function If a vertical line can cut the function TWICE OR MORE, the graph is not a function In the graph below, the red graph is a function, but the blue line is not, because the green vertical line cuts the blue line at 2 points Set Notation In set notation, different types of brackets have different meanings: “(” and “)” are used to write a set where the boundaries are EXCLUDED “[” and “]” are used to write a set where the boundaries are INCLUDED \(\infty\) means Infinity while \(- \infty\) means Negative Infinity \(x\in[1,\infty)\) means that “\(x\) is in the set of all numbers between 1 and infinity” Domain And Range All functions have a Domain and Range The domain of a function is all the valid input values The range of a function is all the valid output values Some input values are INVALID and therefore not part of the Domain For Example: In \(g(x)=\sqrt{x}\), only positive values of \(x\) are possible (because negative numbers have no graphable roots) Therefore, \(x\) must be greater than or equal to zero (0) This can be expressed as \(x \geq 0\) OR \(x\in(0,\infty)\) Some output values are INVALID and therefore not part of the Range y-asymptotes are not part of the range All y values above/below the minimum/maximum y of a graph are not part of the range Transformations of a Function (from \(f(x)\)) Vertical Translation Up \(c\) units: \(f(x)+c\) Vertical Translation Down \(c\) units: \(f(x)-c\) Horizontal Translation Left \(c\) units: \(f(x+c)\) Horizontal Translation Right \(c\) units: \(f(x-c)\) Odd and Even functions Even Functions: Symmetrical about the y-axis Rules: \(f(-x)=f(x)\) If \((x,y)\) is a valid solution to \(f(x)\), \((x,-y)\) is in the same function Odd Functions: Symmetrical about the origin \((0,0)\) Rules: \(f(-x)=-f(x)\) If \((x,y)\) is a valid solution to \(f(x)\), then \((-x,-y)\) is also a valid solution Proving/Solving Odd and Even Functions: Find \(f(-x)\) Simplify \(f(-x)\) If \(f(-x) = -f(x)\), the function is ODD If \(f(-x) = f(x)\), the function is EVEN If \(f(-x) \neq f(x)\) AND \(f(-x) \neq -f(x)\), the function is NEITHER ODD NOR EVEN

Straight Line Graphs Standard Form: \(y = mx + b\) Features of a Straight Line Graph X-intercept: substitute \(y=0\) Y-intercept: value of \(b\) Transformations of a Straight Line Graph Vertical Translation Up: increase \(b\) Vertical Translation Down: decrease \(b\) Increase steepness: increase \(m\) Decrease steepness: decrease \(m\) Reflect in y-axis: \(m \times -1\) Reflect in x-axis: \(y \times -1\) AND \(m \times -1\) Reflect in Main Diagonal \((y=x)\): switch y and x Horizontal Translation Left: increase \(b\) Horizontal Translation Right: decrease \(b\) Lines Parallel to the Axis Standard Form (parallel to x-axis): \(y=b\) Standard Form (Parallel to y-axis): \(x=a\) Transformations of Lines Parallel to the Axis Vertical translation up: Increase \(b\) Vertical Translation Down: Decrease \(b\) Horizontal Translation Left: Decrease \(a\) Horizontal Translation Right: Increase \(a\) Parabolas General Form: \(y=ax^2 + bx + c\) Features of a Parabola X-Intercepts (not always present): intersects of parabola and \(y=0\) Y-Intercept: intersects of parabola \(x=0\).