Chapter 3: Equations

1. PlySmlt 2 Root Finder

2. Exact answers
Factorisation
OR
Manually (formula)

3. D = b^2 - 4ac
D>0, 2 distinct roots
D=0, equal or repeated roots
D<0, no real root

4. ax^2 + bx + c always +ve
when a>0, D<0, curve above x axis

ax^2 + bx + c always -ve
when a<0, D>0, curve below x axis

OR

always +ve or -ve, complete square

a^2 can be smaller or equal to 0

1. GC
y intercept, press 2nd CALC 1, x=0
x intercept, press 2nd CALC 2, left bound and right bound
min pt, press 2nd CALC 3, left bound and right bound
max pt, press 2nd CALC 4, left bound and right bound
tangent, press 2nd DRAW 5, x value
intersection, press 2nd CALC 5, enter, enter
asymptote find manually
absolute sign, press MATHS > 1

Chapter 1: Graphing Techniques 1

1. Function:
a rule or relationship where for every input there is only one output

2. Complete Square
x^2 + bx + (b/2)^2 - (b/2)^2 + c
OR
y = a[x+(b/2a)]^2 + [c-(b^2/4a)]

3. Stationary point
dy/dx OR [(-b/2a),[c-(b^2/4a)]

4. Formula
x = [-b+(b^2-4ac)^1/2] / 2a
OR
x = [-b-(b^2-4ac)^1/2] / 2a

5. Cubic Function
positive x^3 --- graph starts from right, with "U" shape
negative x^3 --- graph starts from right, with inverted "U" shape

6. y = x^n , x is an element of real numbers (refer to notes for graphs)
Even n --- after 1, x^4 > x^2
below 1, x^4 < x^2

Odd n --- after 1, x^5 > x^3
below 1, X^5 < x^3

7. y = x^(1/n)
Even n --- after 1, x^(1/2) > x^(1/4)
below 1, x^(1/2) < x^(1/4)

Odd n --- after 1, x^(1/3) > x^(1/5)
below 1, x^(1/3) < x^(1/5)

8. y = 1/x , x is an element of real numbers and x is not 0
graph is hyperbola shape

9. y = a^x , x is an element of real number and a>0
asymptote at x axis
a>1 --- graph is exponential from left
0 < a < 1 --- graph is exponential from right
both graphs at upper axes

10. y = LOGaX , X>0 , a>0
asymptote at y axis
a>1 --- graph from negative y region curves to positive y region
0 < a < 1 --- graph from positive y region curves to negative y region
both graphs at right axes

11. y = e^x , y = lnx (reflection)
graphs of y = e^x and y = lnx are reflection of each other

12. Asymptotes
y = 1 - 1/x
horizontal asymptote (y asymp) --- 1
vertical asymptote (x asymp) --- equate x fn to 0 and solve for x

13. y = (x-a)(...)(...), where (x-a) is power 1
curve starts from right and cut through the x axis

y = (x-a)^2(...)(...), where (x-a) is power 2
curve starts from right and touches x axis first before cutting through the axis
maximum or minimum point

y = (x-a)^3(...)(...)
curve starts from right and it is a pt of inflexion first before cutting through
the axis
tve or -ve pt of inflexion

14. Ellipses
[(x-h)^2 / (a^2)] + [(y-k)^2 / (b^2)] = 1

15. Circles
(x-h)^2 + (y-k)^2 = r^2

16. Hyperbola
y = c/x
c>0 --- graph at 1st and 3rd quadrant
c<0 --- graph at 2nd and 4th quadrant

y = (ax+b)/(cx+d)
do long division
1. long division
2. asymptotes
3. intercepts

We love Mrs Khet because she is good at Maths and we suck at Maths so with she being good at Maths teaching us who are being bad at Maths we become good at Maths and we learn to love Maths.