Linear Graphs and Equations
For any straight line, the gradient (M) is: dy/dx or difference in y/difference in x which is (y2-y1)/(x2-x1)
Equation of a line: y=mx+c which is used when the gradient and intercept is known or y-y1=m(x-x1) when the gradient and the co-ordinates (x1,y1) of a single point that the line passes through is known.
The mid-point of two graphs is found by (x1+x2)/2 , (y1+y2)/2 in the form (x,y)
Lines with the same gradient are parallel, while lines with gradients that are negative reciprocals of each other is perpendicular to it. (at a right angle to)
The point of intersection of lines that are not parallel can be found by simultaneous equations by:
- equating coefficients of the two lines
- substituing one equationinto the other
- equating both equations for y.
surd form is exact. they involve irrational roots, which are roots that cannot be expressed as fractions as they are irrational for example: √5
- √a x √b = √a x b = √ab
- √a / √b = √a/b
- √a+b ≠ √a+√b
It is often more useful when denominator of a fraction is rationalised. This is done by multiplying the top and bottom by the conjugate, as the product of two conjugates is always rationalised because (a+b)(a-b)=(a^2)-(b^2) and a surd^2 is always rational.
Quadratic Graphs and Further Equations
The graph of a quadratic is a parabola, they can be factorised into two linear factors.
A quadratic can be solved by:
- factorising (finding y=(x+r1)(x+r2) where r=root)
- completing the square (in the form y=k(x-p)^2 +q
- using the quadatic equation. -b+_(rb^2+4ac)/2a
the graph y=ax^2 +bx+c lets you find the y-intercept, c. Quadratics in completed square form have a translation of [p over q] with the vertex at (p,q) symmetry at x=p
The discriminant b^2-4ac can be used to find the number of roots:
- positive answer = 2 real roots
- 0 = one root / repeated roots.
- negative answer = no 'real' roots.
If a pair of simultaneous equations lead to a quadratic, then the discriminant shows the relationship between the graphs.
Multiplying/dividing by negative numbers means that the inequality symbol is reversed.
The discriminant can also be turned into an inequality. > 0, = 0, < 0, ≥0
The solution can be represented on a number line, graphically or using sign diagrams showing critical values for both linear and quadratic inequalities.
An index/indices is an expression like x^5
Cubic expression = 3 linear factors
y=x^3 translated by [3 over 1], replace x by (x-3) ; y by (y-1) so it should be y-1=(x-3)^3 = y=(x-3)^3 +1
Polynomial expressions = a+bx+cx^2 +dx^3 +ex^4.......etc
Factor Theorem: if the polynomial p(x) is divided by (x-a) with no remainders, p(a) is a factor of p(x)
Remainder Theorem: when p(x) is divided by (x-a) the remainder is p(a)
Equation of a Circle
Formula for a circle: (x-a)^2 +(y-b)^2 =r^2 with the centre at (a,b)
The expanded form x^2 +cx+y^2 +dy+e=0 can be sorted by completing the square for the first part x^2+cx then the y^2+dy+e
A tangent to the circle is a straight line that touches it once.
The normal is a line at right angles to the tangent, and as it will go through the centre of the circle the diameter would be a part of it.
The gradient of the normal is thenegative recipricol of the tangent.
The perpendicular bisector goes through the centre of the circle.
The discriminant can be used to show if the line:
- intersects the circle (chord) >0 (crosses twice)
- is at a tangant to the circle =0 (touches once)
- doesnt meet the circle <0
The gradient of a curve is constantly changing so the tangent is taken at a point, the gradient of that tangent is the gradient at that particular point.
The gradient function dy/dx is called the derivative also written as f'(x)
To diffrentiate a polynomial, look at it term by term (as the sum of functions = sum of seperate derivatives) and apply the rule for y=x^n , dy/dx= nx^n-1
The normal to the tangent on the curve is the negative reciprocal of M.
When the gradient of the tangent to the curve/derivative of a constant/y=k/gradient is 0, it is at a stationary/turning point.
The stationary point is either a maximum or minimum point. A maximum is the highest part of the curve while the minimum the lowest.
Although the gradient of all the stationary points are 0, you can tell the maximum and minimum apart by looking either side.
A maximum has positive gradients on the left, and a negative gradient on the right, as the curve now goes down after meaching the maximum.
A minimum has negative gradients on the left and a positive on the right.
To work out if the f(x) is increasing or decreasing, work out the derivative,f'(x), and see if its +ve or -ve. (+ve=increasing, -ve=decreasing)
If f(x) = +ve then minimum, if f(x) = -ve then its a maximum
Integration is the reverse of diffrentiation.
When integrating there is always an indefinate integral which includes a constant c, as 'c is indefinate. c is called the constant of integration.
The notation for indefinate integral is: ∫(......)dx
so if dy/dx=x^n (finding y in terms of x:) y is the indefinate integral; y=∫(x^n) = x^n +1/n+1 +c
You can find the constant if you substitue the (x,y) values with a co-ordinate on the graph.