Math 436 Cheat-Sheet Info List |
binomial squared: (a + b)^{ 2} = a^{ 2} + 2ab + b^{ 2} (get a TRINOMIAL!)
root squared: ; remember that anything squared is POSITIVE!
roots expressed as powers:
, so
rationalizing radicals :
1) monomial denominator : (don't forget to reduce!)
2) binomial denominator :
based on difference of squares : (a + b)(a – b) = a^{ 2} – b^{ 2}
negative exponents: turn things into fractions.
, and
factoring: 1) common factor: ab + ac = a (b + c)
fractions: FACTOR IMMEDIATELY
quadratic equations: relate to zero, then either:
** watch for these easy ones from the standard form: – (x + 7)^{ 2} – 19 = 0
make it (x + 7)^{ 2} = 19 take plus or minus sq. rt & transpose the 7.
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*** Sometimes points are given in function form: f(– 2) = 5 is the same as (– 2, 5) ***
mid point: average the coordinates
distance between points: (it's Pythagoras!)
distance from pt to line: P(x_{1} , y_{1}) line: Ax + By + C = 0
distance between parallel lines: find a point (intercept) on 1 line and use formula above.
point of division: make similar triangles of part to whole. Watch for fractions instead of ratios.
Equation of a line:
find slope then write slope.
for general form: cross multiply, put everything on one side = 0
for standard form: multiply by x – x_{ 1} and transpose y_{ 1} .
symmetric form: , a = x-intercept and b = y-intercept.
Line // x-axis has equation y = b
Line // y-axis has equation x = a.
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Analytic Geometry Mathroom Index
data collection:
data collection methods:
Data can be collected by phone interviews, direct observation, printed questionnaires, internet surveys, mechanical devices (counters in turnstiles), or from archived data -- such as weather data, medical records about birth and death, etc.
representative sampling: the data must be gathered from samples that represent all parts of the target population. In questions where we have to find how many of a specific group to include in the sample, we use proportion.
sources of bias: inappropriate sampling methods such as poor wording of the question, lack of relevance, the range of the sample (too local), the sample size.
Measurement Errors during data analysis and lack of data because of a low response rate cause bias as well.
Measuring and Displaying Data:
measures of position:
Quartiles: section the distribution into 4 approximately equal quarters (25%)
Quintiles: section the distribution into 5 approximately equal fifths (20%)
Percentiles: section the distribution into 100 approximately equal hudredths (1%)
definition: A function is a relation between an independent and a dependent variable such that each value of the independent variable produces one and only one value for the dependent variable. (ie: for each x there is only one y)
Line test for a function: if any vertical line crosses the curve more than once, the curve doesn't represent a function.
domain: all the possible values for x that will yield a real value for y. Start at the left end and go right. Watch for fractions and even roots.
For fractions, rule out values of x that make denominator = 0
For even roots, make sure the expression under the root is m 0 .
range: all the possible values for y that result from plugging in x-values. Start at the bottom and go up to the top.
For increasing, decreasing, positive and negative use x-values.
For range and extreme value, use y-values.
composition of functions: A function is a rule telling you what to do to the variable caught in the brackets. To find f [g(x)] first replace g(x) with its expression, then apply f.
ex:
f(x) = 3x – 1 | g(x) = – 2x^{ 2} + 3x – 2 |
then f [g(x)] = f(– 2x^{ 2} + 3x – 2) = | 3(– 2x^{ 2} + 3x – 2) – 1 |
The Quadratic Function (Parabolas)
There are 3 forms for the rule of correspondence (equation) of a parabolic function:
1) General Form: f(x) = ax^{2} + bx + c | 2) Standard Form: f(x) = a (x – h)^{2} + k |
3) Zeros Form: f(x) = a (x – x_{1}) (x – x_{2}) NB: the "a" is the same in the 3 forms! |
Changing from General to Standard Form:
k = f(h) | so f(x) = ax^{2} + bx + c becomes |
Example: Find the vertex of the parabola f(x) = – 2x^{2} + 8x – 7
rewrite the equation in standard form.
h = – 8 / – 4 = 2 | f(2) = 1 | so the vertex is at (2, 1) |
The standard form of this equation is f(x) = – 2(x – 2)^{2} + 1
Since a < 0, the curve opens downward so y has a maximum value of 1.
The zeros form:
The zeros form is useful in solving optimization problems since it gives us the zeros and we know that h is the mid-value of the zeros. Since the maximum or minimum happens at x = h, we can find the max or min easily.
example: A farmer has 1000 metres of fencing to enclose a rectangular field for his sheep. He wants to enclose a maximum area. What dimensions should he make the field and what is the maximum area?
In most questions, we will have to find the rule of correspondence and then find either an
x-value or a y-value that corresponds to the info given.
** for finding zeros, solve the quadratic equation by factoring or quadratic formula.
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similarity theorems:
1) AA -- 2 angles of 1 triangle = 2 angles of another triangle
2) SAS -- sides proportional about contained equal angles
3) SSS -- 3 sides of 1 triangle proportional to 3 sides of another.
NB: if the sides are proportional or all angles are equal, any two shapes (trapezoids, octagons) are similar.
Be careful to compare corresponding sides. In a case where the lines are not //, mark the equal angles as shown below.(I used n, & o for the = angles, c for the common angle)
Now it is easy to see which sides correspond by noting the angles which "contain" the side. So, to solve for x we use ADE ~ ACB: we compare small to big to find x.
the ratio of sides is 4 : 12 = 1 : 3
note: x and 12 contained by angles n & o; 3.5 and (4.8 + 5.7) = 10.5 contained by angles c & o. Now compare big to small to find y.
so y = 10.9
(3.5 + y) and 4.8 both contained by angles c & n so the ratio is 3 : 1.
equivalent: for 2 dimensional shapes means equal areas.
for 3 dimensional solids means equal volumes.
similar solids: if two solids are similar, and :
if
if you know level 3 and need level 2 -- take cube root, then square. Watch for statements like the area of the larger is 4 times the area of the smaller. It tells you the level 2 ratio of 4 : 1.
( Plane Geometry MathRoom Index )
For right triangles use definitions (Soh Cah Toa) to solve for sides.
A = 90° – 47° = 43°
b = 12 sin 47° = 8.78
a = 12 cos 47° = 8.18
when we're given only sides, to find angles, we use arcsin, arccos or arctan depending on which ratio we have.
, so angle B = 60.24°
angle A = 90° – 60.24° = 29.76°
sin 60.24° = cos 29.76° , so b = 7.17
Oblique Triangles (no right angle !!)
Law of Sines: use if given either 2 angles & 1 side; or 2 sides & 1 angle opposite a given side.
solving for sides: use the appropriate pair.
solving for an angle: then take arcsin.
*** if asked for an obtuse angle -- find the acute angle then subtract from 180°. ***
Law of Cosines:
use if given 2 sides & contained angle, or given all 3 sides, no angles.
1) solving for sides:
a^{ 2} = b^{ 2} + c^{ 2} – 2bc cos A | b^{ 2} = a^{ 2} + c^{ 2} – 2ac cos B | c^{ 2} = a^{ 2} + b^{ 2} – 2ab cos C |
** don't forget to take square root.**
2) solving for an angle:
then use arccos -- Find the biggest angle first -- then use law of sines to find the next angle.
Use subtraction to find the 3rd angle.
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Vertically opposite angles theorem | Isosceles triangle theorem | Parallelogram theorem |
30°, 60°, 90° triangle theorem | Parallel line theorem |
Congruence thms.: SSS, SAS, ASA,
RIGHT TRIANGLE CONG. THM. hypotenuse & 1 side = hypotenuse & 1 side of right triangle
Area and Volume Formulas will be given to you.
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(all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).