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Formulas to Memorize for the ACT

 

 

length of a line  

 

midpoint of a line

  

 

 slope of line  

 

equation of a line  y – y1 = m(x – x1)

 

slope of a perpendicular line   -1/m 
(invert slope of original line, and then change sign)

 

equation of a circle 
centre is at point (h, k), r is radius

 

equation of a line tangent to a circle

 (1) find slope of the circle’s radius, (2) use the equation y – y1 = m(x – x1) with m now the inverted-and-sign- changed slope of the radius, and (x1,y1) the point of contact of the line with the circle

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roots of a quadratic equation

 

For x2 -2x +6, a=1, b=-2, c=6

a 'root' is the number you substitute for x which makes an equation correct. For x2 -2x +6, the roots are 2 and 3.

Zero Rule= If two things multiplied = zero, then each one = zero.

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Exponents
Rule Example
x0 = 1 60 = 1
xrxs = xr + s 2324 = 23 + 4 = 27
xr
xs		  =  xr - s,  x    0
64
63		 = 64 - 3 = 61 = 6
(xr)s = xrs (23)4 = 23·4 = 212
(xy)r  =  xryr (2·6)2 = 2262 = 4·36 = 144
( x
y		 ) r  =  xr
yr		 ,  y    0
( 6
2		 ) 2  =  62
22		  =  36
 4		  =  9
x-r  =   1 
xr		 ,  x    0
2-3  =   1 
23		  =  1
8
( x
y		 ) -r  =  ( y
x		 ) r ,  x,y    0
( 2
6		 ) -2  =  ( 6
2		 ) 2  =  9

x1/2 =   ( the square root of x)            161/2 = 4

          x -1/2 =                                                                  16-1/2 = 1/4
 
          
          x2/3 = the cube root of x2                              22/3 = cube root of  22 = cube root of 4

 

13 = 1;  23 = 8;  33 = 27;  43 = 64;  53 = 125

 

Scientific Notation
Scientific Notation is used to express decimal numbers in a form such that there is a number with one non-zero digit to the left of the decimal point multiplied by an appropriate power of 10.
In Scientific Notation Not in Scientific Notation
3.426 × 106 3426000.0
3.426 × 10-6 0.000003426

 

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complex numbers

i = 

i² = -1

Argand Diagram: normal graph, but x axis = real (a),
y-axis = imaginary (bi). Plotting points is the same as on any graph.

absolute value (length of line in Argand diagram) =  

complex conjugate of (a + bi) = (a – bi) used for division

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For right triangles 

Sin = opp/hyp

Cos = adj/hyp

Tan = opp/adj

 

Sine Rule

area of triangle = (½)(a)(b)(sin C)

Cosine Rule  a² = b² + c² - (2)(b)(c)(cos A)

 

Note: When using your calculator for trigonometry questions, remember that the calculator works with decimals (numbers divided by 100), while degrees are divided by 60 (minutes). Thus if your calculator says 27.6º, that equals 27º 36' (.6 x 60 = 36). Most schools are now allowing decimals instead of minutes.

 

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Differentiation All of the following are the same question: "find the derivative", "differentiate", "find dy/dx"; "solve f ’(x)"; "find g' (x)"

To differentiate axn
multiply a (the coefficient) by n (the exponent), and then subtract 1 from the exponent.

For questions involving speed--such as with a firework question, find the derivative of the time (t) formula given to you in the question.

Chain Rule
used for expressions such as y = (2x² + 3)4

Differentiate as usual, but then multipy your answer by the differential of what's inside the brackets

dy/dx = 4(2x²+3)³(4x)