MATH 536 CHEAT SHEET INFO |
Composition of Functions:(click here)
To find f [ g(x) ] first replace g(x) with its expression and then apply f to it.
Example
f (x) = x^{ 2} + 3x – 2 | g (x) = 1 – x | f [ g(x) ] = f ( 1 – x ) = ( 1 – x )^{ 2 } + 3(1 – x) – 2 |
Absolute Value Function: (click here)
Basic Rule of Inequalities:
Absolute Value Inequalities:(click here)
Square Root Function:(click here)
vertex at (h, k) | a > 0 up | a < 0 down | |
b = +1 right | b = – 1 left |
ex: becomes
so vertex is at (– 3, 7) curve moves up and left from vertex.
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Square Root Inequalities:(click here)
Same basic rule of inequalities.
ex: u 9(x + 1) < (2x + 3)² u 9x + 9 < 4x² – 12x + 9.
NOTE: we're squaring SIDES of the equation if the side is a binomial we get a trinomial!
We must always check our solutions in square root equations and inequalities.
They don't all work! We have to respect the restrictions on the radical expression.
Hint: To find the rule for a square root function, if given the vertex and another point,
decide if the curve moves left or right so you know if b is +1 or – 1.
Then include it in the rule of correspondence.
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GREATEST INTEGER FUNCTION (STEP FUNCTION) :(click here)
f (x) = a [ b(x – h) ] + k
note : [ a ] means greatest integer of a.
if a > 0 & b > 0 steps up | if a < 0 & b > 0 steps down |
if a > 0 & b < 0 steps up | if a > 0 & b < 0 steps down |
b > 0 open on right | b < 0 open on left |
for steps up open on the left, make a < 0 & b < 0 | |
vertical separation of steps = | a | | horizontal length of steps = |
domain: R depends on reality (could be R > 0) | start at (h, k) |
range: an + k, n Z (or strictly positive or negative) |
note: left of 0, be careful. ex [ – 9.2] = – 10
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RATIONAL FUNCTIONS (FRACTIONS) :(click here for equations) (click here for function)
general form: | standard form: |
Intercepts decide increasing or decreasing: x-intercept: , y-intercept: , |
if a < 0 the curve is always increasing if a > 0 the curve is always decreasing |
vertical asymptote (x that makes denominator = 0) general form: , standard form: x = h |
horizontal asymptote (ratio of the coefficients of x) general form: standard form: y = k, |
note: asymptotes are LINES so we write an equation not a constant!!!
With the general form of the rule, we find the intercepts (set x = 0 and y = 0) so we can tell if the curve increases or decreases over its domain.
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Example: we'll sketch the graph of
Vertical Asymp : x = – 1 | Horizontal Asymp: y = 4 | |
y-int = – 8 (set x = 0) | zero = 2 (set y the numerator = 0) | |
Dom: R; x ! – 1 | Rng: R; y ! 4 | always increasing |
f (x) is negative ] – 1, 2 ] | f (x) is positive ] – º, – 1 [ 4 [ 2, º [ |
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Rational Inequalities:(click here)
Since an even # of negatives multiplied or divided make a positive,
and an odd # of negatives multiplied or divided make a negative,
once we establish the zeros of the fraction's factors,
we test each interval created for the result we need -- positive or negative.
Example:
solve
the zeros of the factors are x = – 2 and x = 5 we set up a number line like this
______– 2__________ 5 ____
which creates 3 intervals -- left of – 2, between – 2 and 5, and right of 5.
set x = – 3 the fraction is positive ( > 0)
set x = 0 , the fraction is negative (< 0)
set x = 6 the fraction is positive ( > 0)
The solution is x – 2 and x > 5
If equality is included, the solution only includes the zero of the numerator since the denominator can't be equal to 0. Division by zero is still undefined.
EXPONENTIAL FUNCTIONS :(click here)
f(x) = a(c) ^{b ( }^{x – h}^{ )} + k raise C ^{b} for the new base to eliminate the b.
Example: f (x) = – 5 (3)^{ 2 ( x – 5 )} + 7 becomes f (x) = – 5( 9)^{ ( x – 5 )} + 7 use 3^{ 2} or 9 as c.
now use your bookmark or your booklet for the rest.
Example: list the properties of f (x) = – 5(½)^{ – 2 ( x + 3 )} – 1; [(½)^{ – 2} = 4]
the function becomes f (x) = – 5(4)^{ – 2 ( x + 3 )} – 1
analysis: base > 1, a < 0 so curve is always decreasing
HA: y = – 1, y-int = – 321
Dom: R | Rng: ] , – 1 [ | no zero | always decreasing. |
SOLVING EXPONENTIAL EQUATIONS : click here.
Log Functions: click here.
Expanding, Contracting & Evaluating Log Expressions:(click here)
Example: find an equivalent expression for
log_{ a} 1 + log_{ }_{a} a^{ 2} + log_{ a} (a/2)
0 + 2 + log_{ a} a – log_{ a} 2 = 3 – log_{ a} 2.
Log Equations:(click here)
Example: solve log^{ }_{5} (x - 1) + log^{ }_{5} (x + 3) – 1 = 0 becomes log^{ }_{5} (x^{ }2 + 2x – 3) = 1
now go to exponential form, solve the quadratic x^{ 2} + 2x – 3 = 5
Example: find the numerical value of 2 log_{ b }1 + log_{ b }b^{ 2} + log_{ b} (1/b)
2(0) + 2 + log_{ b} 1 – log_{ b }b = 2(0) + 2 + 0 – 1 = 1
Optimization: (click here for graphing inequalities) (click here for optimization)
Note: watch for situations where only integer or positive coordinates suit the reality of the situation. We can't rent ½ a bus or hire ½ a person. We can't have negative time values.
Finding Inverse Functions:(click here)
switch the x's and y's, then solve for y.
When we find an inverse function, we switch the domain and range, so restrictions on the domain and range of f (x), determine restrictions on f^{ – 1}(x). The domain and range restrictions are reversed.
Example: find f^{ – 1}(x) if f (x) = , x – 5 and y ¼
= f^{ – 1}(x) so y – 5 and x ¼.
Example: find f^{ – 1}(x) if f (x) = 2, we get x = 2
so y = and since x 1 in f (x), y 1 in f^{ – 1}(x).
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Euclidean Geometry of the Circle and Right Triangle:
these lessons are in the Geometry MathRoom logID: plgm4, plgm5 and plgm6
see pg. 322 in Book 1 for the list of theorems.
Statistics:
mean deviation: where x_{i} are the data values, is the mean, n = # of data items
standard deviation: for a sample; for a population.
Z-Score or Standard score: measures the # of standard deviations between
Bivariate Stats:(2 variables)
Scatter Plots, Correlation, and Line of Regression:
To get the "a" and "b" for the line of regression y = ax + b , enter the 2 lists of data values, then use 2-var stats and linreg from the Stats menu on your calculator. If asked to find the value for a different data value, plug this x-value into the equation for the line.
To find the correlation coefficient r, use either the formula for the rectangle or use your calculator. Strong correlation comes from values of r close to .
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Trigonometry:(click here)
Know your identities (on your bookmark), the unit circle, the two triangles, the restraints on the sine, cosine and tangent functions, and the definitions of the six trig functions.
Circles: (click here)
standard forms for the equation of a circle | |
with center (0, 0), radius = r | with center (h, k), radius = r |
equation is x^{ 2} + y^{ 2} = r^{ 2} . | equation is (x – h)^{ 2} + (y – k)^{ 2} = r^{ 2} . |
To get to standard form from general form x^{ 2} + y^{ 2} - 2hx - 2ky + c = 0 ,
take – ½ of the coefficients of x and y to find (h, k), then set c = h^{ 2} + k^{ 2} – r^{ 2} to solve for r.
Equation of a Tangent :(click here)
since the tangent is perpendicular to the radius at the point of contact, find the slope of the radius, take the negative reciprocal to get the slope of the tangent.
Write the equation of the tangent line using the slope and the point of contact.
For a full lesson on conics, ellipse, hyperbola, parabola, click here
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Ellipses: The sum of the distances from a point on the ellipse to the foci = 2a.
standard forms for the equation of an ellipse | |
center (0, 0) | center (h, k) |
a is the semi-MAJOR axis | b is the semi-minor axis |
c = focal distance (between center and a focus), c^{ 2} = a^{ 2} – b^{ 2} . |
With the ellipse, the larger of a^{ 2} and b^{ 2} always goes under the variable of the major axis.
So, if the ellipse is vertical, the larger value goes under the y term.
latus rectum: is the chord of the ellipse, perpendicular to the major axis through the focus.
To find its length, find c and set x (or y) = c for the corresponding values of the other variable. Then find the length.
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Hyperbolae: The difference between the distances from a point on the hyperbola to the two foci is constant and is equal to 2a.
Hyperbola Equations:
center (0, 0): | center (h, k): |
where a is always horizontal and b is always vertical.
If the hyperbola is vertical, use – 1.
The axis through the vertices is the transverse axis.
The axis perpendicular to it is the conjugate axis.
The foci are always on the transverse axis,
the focal distance is the distance between the center and the focus.
asymptotes: The equations of the asymptotes to the hyperbola are:
Parabolas: every point on the path is equidistant from the focus and the directrix.
opening up or down | opening left or right |
(x – h)^{ 2} = ! 4c (y – k) | (y – k)^{ 2} = ! 4c (x – h) |
TEXT BOOK STUDY GUIDE: (Carousel/Mathematical Reflections Series)
Book 1:
Book 2: