Functions and Graphs

Lines and linear relationships

A linear relationship looks like this.

Midpoint of two points: $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Direct proportion: $x$ and $y$ are in direct proportion if $y=kx$ for some value $k$.

Forms of a line

The slope-intercept form of a line is the most common. It looks as follows:

$$y=mx+c$$

where $m$ is the gradient of the line and $c$ is the y-intercept of the line. The x-intercept can be calculated by solving for the case that y = 0.

One other form that may appear is standard form:

$$ax+by=c$$

and this can easily be rearranged to be converted to slope-intercept form.

Parallel and perpendicular lines

Two lines are parallel i.e. never meet if their gradients are equal. In other words, the values of $m$ in their slope-intercept equations must be equivalent.

Two lines are perpendicular i.e. at right angles to each other if their gradients are negative reciprocals of each other. In other words, if $m_1$ and $m_2$ are the gradients of the lines:

$$m_1{m}_2=-1$$

Quadratic relationships

A quadratic relationship looks like this.

Forms of a quadratic

These include $y=a{x}^2+bx+c$, $y=a(x-b{)}^2+c$ and $y=a(x-b)(x-c)$.

Factorisation

One method of factorisation is a guess and check: one must find two numbers whose sum is equal to $b$ and whose product is equal to $c$ in the original equation.

For example, to determine roots of $y=x^2+3x-4$, one must find two numbers that sum to $3$ and multiply to $-4$. Two such numbers are $4,-1$ so the factorisation of the original quadratic is $(x+4)(x-1)$.

One other method of factorisation is completing the square. This relies on the facts that:

$$(a+b)(a+b)\equiv (a^2+2ab+b^2)(a+b)(a-b)\equiv (a^2-b^2)$$

To being factorisation by completing the square, one must find a number such that two times that number is equal to $b$. Using the example above of $y=x^2+3x-4$, one would perform calculations as follows as follows:

$$\begin{array}{cc}y& =x^2+3x-4\\ & =(x+1.5{)}^2-6.25\\ & =(x+1.5{)}^2-{\sqrt{6.25}}^2\\ & =(x+1.5+\sqrt{6.25})(x+1.5-\sqrt{6.25})\\ & =(x+4)(x-1)\end{array}$$

The discriminant

The discriminant of a quadratic is represented by the symbol Delta, $\Delta$.

The discriminant is equal to $b^2-4ac$.

Determining number of roots

The number of roots of a quadratic is determined by the discriminant.

If the discriminant is positive, the quadratic has two roots.

If the discriminant is zero, the quadratic has one root.

If the discriminant is negative, the quadratic has no real roots.

Line of symmetry and turning point

The line of symmetry of a quadratic is

$$\frac{-b}{2a}$$

The turning point can be calculated, then, by evaluating the quadratic at x-position $\frac{-b}{2a}$. Therefore the turning point is:

$$(\frac{-b}{2a},\frac{-b^2}{4a}+c)$$

(The y-coordinate can be calculated quite simply by merely evaluating the expression; this does not need to be committed to memory.) Note that the y-coordinate is equal to negative half of the discriminant.

Determining roots

A root is an x-intercept.

If the quadratic is in the form $y=a{x}^2+bx+c$, the quadratic formula may be used:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

If the quadratic is in the form $y=a(x-b)(x-c)$, the roots are $x=b$ and $x=c$, since $y$ can only be $0$ if one of the factors is $0$.

Inverse proportion

Here's a graph of inverse proportion:

Two variables are said to be in inverse proportion if one of them is in direct proportion to the reciprocal of the other, i.e. $x$ and $y$ are in inverse proportion if:

$$y=\frac{k}{x}$$

where $k$ is some number.

Asymptotes

An asymptote of an inverse graph (i.e. $y=\frac{a}{x-b}+c$) is a line that the graph 'tends to'. Since dividing by zero is undefined, the horizontal asymptote of the graph is $y=c$ and the vertical asymptote is $x=b$.

Powers and polynomials

A polynomial consists of the sum of a number of terms containing some number of powers of $x$.

An example of a polynomial:

$$P\left(x\right)=x^5+3x^4-2x^3+11x^2-5x-17$$

Degree

The degree of a polynomial is defined by the highest power of $x$ in the polynomial. For example in the polynomial above the degree is 5.

Coefficient

The coefficient of a term in a polynomial is equal to the 'multiplier' on the power of $x$. For example, in the polynomial above, the coefficient of $x^4$ $3$.

Constant term

A constant term has no power of $x$. For example in the polynomial above the constant term is $-17$.

Simplifying a polynomial

Bezout's Remainder Theorem may be used to factorise a polynomial.

$$(x-a)\text{is a factor of}P\left(x\right)\iff P\left(a\right)=0$$

In other words, a polynomial may be factorised by finding roots.

Roots of a polynomial

The Zero Factor Law can be used to find the roots of a polynomial.

If a polynomial is expressed in the form

$$P\left(x\right)=k(x-a)(x-b)(x-c)\dots$$

then you can easily find the roots of the polynomial: they are simply $a$, $b$, $c$, and so on.

Relations

A relation is represented as a collection of pairs, essentially points in the $xy$ plane. Every function is a relation.

Circle

A circle is defined by the formula:

$$(x-a{)}^2+(y-b{)}^2=r^2$$

where $(a,b)$ is the circle center and $r$ is the radius.

Rotated parabola

Here's an example:

The 'rotated parabola' is essentially a parabola on its side, of the form $y^2=x$. The turning point, line of symmetry and roots can be found quite easily.

1. Pretend that the $x$ and $y$ are swapped.
2. Solve as a normal parabola.
3. Swap $x$ and $y$ in the final solution.

Functions

A function is a relation where every variable maps to at most one other variable, i.e. every x-coordinate on a graph has either zero or one y-coordinates corresponding to it.

For example, the examples of relations given above are not functions, because they have two y-values for some x-value.

Vertical Line Test

The Vertical Line Test can be used to check if a graph is a function. Essentially if a vertical line cuts the graph more than once at any point, the graph is not a function.

Transformations

Assume $y=f\left(x\right)$ is the base graph.

$y=f\left(x\right)+a$ corresponds to "translate up $a$ units"

$y=f(x-a)$ corresponds to "translate right $b$ units"

$y=a\cdot f\left(x\right)$ corresponds to "dilate along the y-axis by a scale factor of $a$"

$y=f(a\cdot x)$ corresponds to "dilate along the x-axis by a scale factor of $\frac{1}{a}$"

Domain and Range

A function's domain is represented as follows:

$x\in R:x\ne 3$

where what comes after the colon is any expression representing the possible values of $x$.

Similarly range is represented as follows:

$y\in R:y\ne 0$

Natural Domain

The natural domain of a function is all of the values that the function is defined for.

For example, the function $y=\frac{1}{x-3}$ is undefined only for $x=3$, so its natural domain is:

$x\in R:x\ne 3$