Relations and Functions
Unit 1.1: Increments, Slope of Line, Parallel and Perpendicular Lines, Equations of Lines, and Application.
Parallel and perpendicular lines
Linear Equations
Parallel lines are when there are two line functions that are always the same distance apart. Algebraically if you have two functions both the m values would equal to each other. For example: y= 4x + 5 and y= 4x + 8. Both of the m values of this function are equal to each other meaning both of the lines are parallel.
Perpendicular lines are when there are two line functions and they intersect each other at a 90 degree angle. Algebraically if you have two functions and you multiply both the m values you would get -1. For example: y= -3x + 4 and y=1/3x + 3, -3*⅓ equals to -1, this means that both of the lines are perpendicular.
Slope Int form:
The slope int form is y=mx+b, this equation is used for a straight line. Inorder to use this equation you need to be able to find the slope of the line and the y intercept.
Slope Point form:
The slope point form is another way of writing a linear equation. You usually would use this form to solve for the m value when you are given two points. The point slope form is written like (y-y1)= m(x-x1).
Increments: are when points move from x1 to x2, and y1 to y2. The following is represnted by the following symbol ▵. The formula for the following is ▵x=x2-x1, and ▵y=y2-y1
1,3: Exponential Growth, Exponential Decay, Applications, and Number E
1.6: Radian Measure, Graphs of Trignometric Functions, Periodicity, Even and Odd Trig Functions, and Inverse Trig Functions
1.5: One to One Functions, Inverses, Finding Inverses, Logrithmic Functions, Properties of Logarithm
1,4: Relations, Circles, Ellipses, and Circles
Unit 1.2: Funtions, Domians, Viewing and Interpertating Graphs, Even and Odd functions, Absolute Value Funtions, and Composite Funtions
Functions can be represented by an equation and by a graph:
A graph is a function if it passes the vertical line test. You can test if an equation is a function by graphing it and seeing if it passes the vertical line test.
Another way to test if something is a function is when you have a set of the x and y values and the x values aren’t repeating.
Functions odd or even:
You can tell if a function is even if f(-x) gives you f(x), for example: y=x^2 + 3, if we put a negative in the x value we would get a positive answer.
You can tell if a function is odd if f(-x) gives you -f(x), for example y=x^3 + 5, if we put a negative in the for the x value we would get a negative answer.
Funtions Odd or Even:
You can tell if a function is even if f(-x) gives you f(x), for example: y=x^2 + 3, if we put a negative in the x value we would get a positive answer.
Y
ou can tell if a function is odd if f(-x) gives you -f(x), for example y=x^3 + 5, if we put a negative in the for the x value we would get a negative answer.
Interval notation: used for representing the domain and range, solutions that are written as inequalities, and number line solutions.
Close notation: Is used when there is an equal sign in the domain for example: a⪬x≥b, you would write the following as [a,b]
Open notation: is used when there is no equal sign in the domain for example: a﹤x﹥b, you would write the following as (a,b)
Piecewise function
Different functions put in one graph the domain and range is found by looking at all the functions
Each function have there own domain and then are plotted on one graph
Composition Functions
There are two functions and they can be composed where a portion of the range of the first function lies in the domain of the second function.
Exponential functions: y-b^2
Exponential growth: occurs when the b value is greater than one
Exponential decay: occurs when the b value is greater then 0 and less than one
Horizontal/Vertical translation:
A horizontal translation is when the graph shifts or left, for example, y= 2 ^(x-3) this equation has a horizontal translation of three units to the right.
A vertical translation is when the graph shifts upward or downward. For example 2^x +
4, this equation has a vertical translation 4 units up.
Horizontal/Vertical expansion/compression
A vertical expansion is when the graph is stretching for example y=3x^2 the graph is
going to have an expansion by three units.
A vertical compression is when the graph is shrinking vertically, for example, y=⅓ x^3, this equation has a vertical compression by ⅓ units.
A horizontal expansion is when the graph is expanding horizontally
A horizontal compression is when the graph is compressing horizontally
Formulas for exponential functions
Formula for compound continuously: A=Ao(e)^+it growth/A=A0(e)^-it
Formula for exponential growth and decay : A=Ao(B)^t/p
Formula for compound interest: A=Ao(1+i/n)
Inverse Funtion:
F(X)=3x+4, you would switch the x and y values and that would then be x=3y+4, x-4=3y, y=x/3-4/3, you would then write this equation as f-1(x)=x/3-4?3
Applications: We can use funtions to help determine temprature by comparing the two scales of ferhinite and celcius. We also use funtions for mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping airplanes in the air.
Applications: Slope is a measure of steepness so we can use it for building roads one for figuring out how steep the road will be. skiers/snowboarders need to consider the slopes of hills in order to judge the dangers, speeds, etc.
Applications: Not only can funtions help us determine the bacterical growth, compound interest, and bacterial decay. These functions are also used for loudness of sound, population increase, population decrease or radioactive decay
Relations:
Are a set of ordered pairs such as (x,y). This set of ordered pairs can be a function of a third variable t, known as a parameter.
Circle:
x^2+y^2=r^2 is the generic equation used for all circles. We can use a unit circle to represent the following equation
Ellipse: Are similar to a circle however they usually are stretched horizontally or vertically. The standard form of an ellipse would be centered at (0,0). The standard equation for an ellipse is:
Parametric Equations:
Parametric equations are used to obtain graphs of relations and functions. Parametric equations is a type of equation that can graph curves that are not functions. Each value of t represents a point (x,y)=(f(t),g(t)), that we can plot onto the graph. Parametric curves are the collection of points that we get by t being all possible values. They can be used to solve for curves on planes, however, they are mostly used when curves on a cartesian plane cannot be described by functions.
Cartesian equations:
is a type of equation only in terms of x and y. Whereas in a parametric curve both the x and y are functions of a third variable t. To solve for this equation y=5t, you would divide both sides with 5 and you would get y/5=t.
One to one function:
If there are two ordered pairs and the first coordinate is different then the second coordinate is the same. The function then satisfies as a one-to-one function. For example f(x1)=f(x2). To check if an equation is a to one function graphically you can do a horizontal line test and if the graph touches the line more than once then it is not a one-to-one function. If it only touches once then it satisfies as a one-to-one function.
Inverse:
An inverse function is when there is one function and the x a and y values are switched to an inverse function. For example y=3x+4, that would then be x=3y+4 you then have to manipulate the y. You would then get x/3-4=y. You would then write this as the inverse of x, f-1(x)= x/3-4.
Logarithmic function:
Functions that are inverse of an exponential function. For example, if you have y=a^x the inverse of this function is y =logbaseA(x). The logarithmic function can help us solve for the value of x or any other unknown variable. We can use logarithmic functions to solve for sound(decibel measures), earthquakes(righter scale), the brightness of stars, etc.
Properties of log:
Log base 10 is known as the common log.
Log base e is known as the natural log
Application: Log is used to help measure measure the acidic, basic or neutral of a substance that describes a chemical property in terms of pH value.
Radian measure:
is the ratio of the from the length of the central arc to the radius of the arc. We can convert radians to degrees and degrees to radians. For example degree pi/180, this is the conversion from degree to radian. Radian 180/, this is the conversion from radian to degree.
Trigonometric ratios:
3 Basic Ratios:
SinΘ=y/r=opp/hyp
CosΘ=x/r=adj/hyp
TanΘ=y/x=opp/adj
3 Reciprocal ratios:
CscxΘ= r/y=hyp/opp
SecΘ=r/x=hyp/adj
CotΘ=x/y=adj/opp
Periodic Function:
a function is considered a periodic function if there is a positive number p. For example f(x+p)= f(x) for every value of x.
Functions:
The following image is the generic way of writing a trig function. Where a equals the vertical stretch/shrink or reflection about the x axis. F representing sin, cos, or tan. B representing horizontal stretch/shrink or reflection about the y axis. C representing a horizontal shift. D represents a vertical shift.
Graphing: In Order to graph trigonometric ratios we look at the ABCD values.
The d value is the vertical displacement; this value represents the midline of the graph.
The b value represents the the new period We calculate the period by diving 2pi/b
The a value represents the amplitude, which gives you the max and min value of the graph
The c value represents the phase shift
The following is an visual example of how you would graph a trig function.
Inverse trig functions:
Inverse functions of trig ratios have arc in front of the trig ratio. Another way of knowing if a trig ratio is an inverse is when the function has been moved to the other side. For example sin x = ½, then you would do y= Sin-1 (½). The following are inverse graphs.