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Transcript of Advanced Functions
Narita and Tharanya
Unit 1: Characteristics and Properties of Functions
1.1 How do you know if it's a function?
each number on domain corresponds to only one number in range
1.2 Absolute values
f(x) = |x| describes the distance, f(x), of any number x from the origin on a number line
-one zero located at origin
-compromised of 2 linear functions and is defined as follows:
f(x) = x, if x>0
-x, if x<0
-symmetric about the y-axis
-absolute means positive
1.3 Properties of Graphs of Functions
f(x) = 1/x
f(x) = root x
Functions can be categorized based on characteristics:
-intervals of increase/decrease
1.4 Sketching graphs of functions
use f(x) = af(k(x-d))+c to perform transformations
use mapping to find corresponding points (x/k + d, ay + c)
- a determines verticle stretch, reflection in x-axis, and amplitude for sinx
-k determines horizontal compression, reflection in y-axis, and period for sinx
-d determines horizontal shift, or y-asymptote
-c determines verticle shift or equation of axis
1.5 Inverse Relations
Not all inverse are functions. To find inverse, switch position of x and y value, then solve for y
Graph reflected in y = x
and inverse is represented with f of prime 1
1.6 piecewise functions
to graph, graph each piece of the function over the given interval. the function can be either continuous or not. Continuous if endpoint of each piece are touching.
1.7 exploring operations with functions
if 2 functions have domains that overlap, they can be added, subtracted or multiplied to create a new shared function.
Unit 2: Understanding Rates of Change
2.1 Determining avg ROC
2.2 Estimating Instantaneous ROC from tables of values and equations
ways to calculate ROCinst:
Preceding interval, following interval, centered interval, and difference quotient.
best solution is to use difference quotient where h is a small number (ie. 0.001).
2.3 Exploring ROC inst. using graphs
Slope of secant line equivalent to average ROC over intercal defined by x-coordinated of 2 points that are used to define secant line. Slope of tangent at a point is ROC inst. at that point
2.4 Using ROC to create graphical model
in problem that involves movement, graphs can show displacement (distance, height, or depth) versus time.
Increasing line means displacement increasing as time increases, and vice versa. Horizontal means there is no movement as time increases.
At a max or min point, the slope is 0, therefore the inst ROC is 0.
If value is positive before the point, and negative after the point, a max occurs, and vice versa
Unit 3: polynomial functions
3.1 Exploring poly functions
A function in the form
where "a" are real numbers, and "n" is a whole number
A degree is the highest exponent in the expression, and the number of curves on the graph is one less the degree.
(ie, function with degree of 4 will have 3 turning points)
3.2 Characteristics of Poly functions.
For odd degree, if leading coefficient is (-), functions extends from second quad to fourth. if positive, it extends from 3rd to 1st quad.
For even degree, if leading coefficient (-), it extends from 3rd to 4th quad. if positive, it extends from 2nd quad to 1st quad.
Odd degree must have at least 1 zero. Even can have none.
Odd functions are symmetrical about the origin f(-x) = -f(x)
Even functions are symmetrical about the axis f(-x) = f(x)
If no symmetrical properties, the function is neither off or even.
3.3 Characteristics of Poly Functions in Factored Form
zeroes of a poly functions y = f(x) are the same as roots of the related poly equation f(x) = 0
To make an equation from graph, take zeroes and put them in factored form, then solve for A by using an additional coordinate.
You can find turning points by determining the degree
3.4 transformations of cubic and quartic funcs.
Parent function is y = x^n
Same as any transformation of other functions
Unit 4: Poly Equations and Inequalities
4.1 Solving Poly Equations
4.2 Solving linear inequalities
4.3 solving poly inequalities
4.4 ROC in poly functions
3.5 dividing polys
Can be divided using long division like normal numbers, or, a faster was is synthetic version. Synthetic can only be used when divisor is linear.
degrees should be arranged in descending order before dividing. Leave a 0 if any coefficient for a power is not present.
if the remainder is 0, the divisor is a factor of the polynomial.
3.6 factoring polys
To factor a poly of degree 3 or greater:
-use factor theorem to determine factor of f(x)
-divide f(x) by x - a
-factor quotient if possible
May have to use factor theorem more than once, but not all poly functions are factorable.
factor theorem: x - a is a factor of f(x) if and only if f(a) = 0
remainder theorem: when a poly, f(x), is divided by x - a, the remainder is equal to f(a)
3.7 factoring a sum or difference of cubes
An expression containing 2 perfect cubes that are added together is sum of cubes. if one is subtracted from the other, it is difference of cubes.
can be solved using factoring, Table of values, transformations, or graphing calculator.
Solutions to a polynomial equation f(x) = 0 are zeroes of the corresponding poly function, y = f(x)
if dividing of multiplying by (-) number, the sign must be flipped
most linear equations have 1 solution where linear inequalities have many solutions
Number lines are used to show the inequality. closed circle means include, open circle means exclude.
First determine roots of corresponding poly equation, then consider the sign of the poly in each interval created by roots. Solution is what satisfies the inequality.
To determine positive or negative intervals, you can use a factor table, a number line, or a graph.
ROC methods mentioned previously can apply here
Average ROC and inst ROC are main methods used here
Unit 5: Rational Functions, equations, and inequalities
5.1 graphs of reciprocal functions
5.2 Exploring quotients of poly funtions
5.3 graphs of reciprocal functions
5.4 Solving rational equations
5.5 solving rational inequalities
5.6 ROC in rational functions
-y coordinates of reciprocal functions are reciprocals of y coordinates of original function
-graph of reciprocal has v-asymptote at each zero of original function
-if original is linear or quadratic, horizontal asymptote for reciprocal will always be y = 0
-reciprocal has same + and - intervals as origin
-if original has points on 1 or -1, that is where the reciprocal will intersect
-Function has a hole when p(x) and q(x) contain a common factor (x - a)
-has vertical asymptote at "a" on denominator
-has horizontal asymptote if degree of p(x) is less than or equal to the degree of q(x)
-is oblique when degree of p(x) is greater than q(x) by exactly 1
Choose a value on the left and right of y-asymptote. if the value is (-), the line is decreasing as it approaches the asymptote. if the value is (+), the line is increasing as it approaches the asymptote
zeroes of rational function are zeroes of function in numerator. reciprocal functions don't have zeroes.
all functions in the for f(x) = 1/ g(x) has the x-axis as the horizontal asymptote.
when adding inequality, multiply numerator by opposing denominator to make it easier to solve
Find all possible values of variable to satisfy inequality. Move all variables to one side to solve.
Remember to reverse sign if dividing or multiplying by negative number
Cannot find ROC inst. at a point where the graph is discontinuous or has a hole/v-asymptote
Unit 6: Trig functions
6.1 Radian Measure
6.2 Radian measure and angles on cartesian plane
6.3 Exploring graphs of primary trig functions
6.4 Transformation of trig functions
6.5 Exploring graphs of reciprocal trig functions
6.6 Modelling with trig functions
6.7 ROC in trig functions
Radian is alternative way to represent size of an angle. arc length, a, of circle is proportional to its radius, r, and the central angle subtends, theta, by the formula:
To convert degrees to radians, multiply by pi, and divide by 180 degrees.
To convert radians to degrees, multiply by 180 degrees and divide by pi.
Angles in special triangles can be expressed in radians. Radians is used to determine exact values of trig ratios
Trig ratios for any principle angle, theta, in standard position can be determined by finding the related acute angle, beta, using coordinates of any point that lies on the terminal arm of the angle
f(x) = asin(k(x-d))+c are for the parent function y = sin (x)
and f(x) = acos(k(x-d))+c are for y = cos (x)
-|a| gives the amplitude
gives the period
-y = c gives Equation of axis
Table of values, graphs, and equations of sinusoidal functions can be used as mathematical models when solving problems. They're the most efficient strategy because calculations can be made using the equation.
Apply what you know here, and you will succeed! It's the same!