Please enable JavaScript.
Coggle requires JavaScript to display documents.
Functions and Relations, CONNECTION, CONNECTION, CONNECTION, CONNECTON,…
Functions and Relations
1.1
Slope of a line: The slope of a line can be determined you can use the rise and the run of two points. The rise is the vertical change between two points, whereas the run is the horizontal change. The slope is calculated by dividing the climb by the run.
Formula: y2-y1/x2-x1.
Perpendicular and Paralell lines: Lines in a plane that are always the same distance apart are called parallel lines. Parallel lines never cross each other. Perpendicular lines are those that cross at a straight angle (90 degrees). Parallel lines have the same slope, (m1=m2). And perpendicular lines have Inverse Reciprocal slopes (m1)(m2)=-1. If the slopes are not the same or multiplied they equal -1, the line is neither parallel or perpendicular.
-
Applications
General form equation: Every line has a general form equation in this form. 0 = AX + BY + C.
A B and C represent real numbers and X an Y reprsents any points on the line
Point slope equation: Point-slope emphasizes the slope of the line and a point on the line to find the equation of a function.
Formula: y-y1=m(x-x1)
Point intercept equation: When the Y intercept (b value) is given this formula is used.
Formula: y = mx+b
Slope Intercept Form: The slope intercept form is the most common approach to express a line's equation. All you need to know to use slope intercept form is how to 1) get the slope of a line and 2) find the y-intercept of a line. Formula: y=mx+b
Increments: If a particle moves from point (X1,y1) to the point (x2,y2). The symbol most commonly used to represent increments is the letter delta (Δ) which means change in something.
Formula: Δx=x2-x1
1.4
Relations: A set of (x,y) values. If different x and y values are corresponding with a t value which is the parameter we can graph the function.
-
Elipses: Similar to a circle but it is stretched upwards which gives an oval shape. At the point (0,0) the standard form of an elipse is x^2/a^2 + y^2/b^2 = 1.
Cartesian Equation: This equation relates to y and x values only. Parametric equations for a curve have both an x and a y value as a function of a t value. Finding the single equation of a curve in standard form, with only x's and y's as variables, is a cartesian equation. To obtain this equation, you must solve all of the parametric equations at the same time.
Parametric equations: A parametric equation is one in which the dependent variables are defined as continuous functions of the independent variable (commonly represented by t), and the dependent variables are not dependent on any other variables. Parametric equations allow us to graph curves that are not functions.
Parametric Curve: A curve where x and y values are defined with another variable. Parametric equations can be used to describe any curve that can be represented on a plane, although they're most commonly used when curves on a Cartesian plane can't be defined by functions
1.2
Functions F(x): Functions are a way to show a connection between two values. Multiple sets of values together can be seen as a function. Functions are used everywhere in life. Examples are linear functions, quadratic functions and polynomial functions
Absolute Value functions: An absolute value function is a function that contains an algebraic expression within absolute value symbols. In these functions there cannot be any negative x or y values. Any negative value is changed to a positive value.
Composite Functions: Two functions that can be composed when a portion of the range of the first lies in the domain of the second. Composition of a function is done by substituting one function into another function.
-
-
-
Even and Odd Functions: A function y=f(x) is an even function if f(-x)=f(x). A function y=f(x) is an odd function if f(-x)=-f(x).To tell if the function is odd, even or neither, you take the function and replace x with –x, then simplify it. The function is even if you end up with the exact same function that you started with. The function is odd if you wind up with the exact opposite of what you started with.
Piecewise Functions: A piecewise function is a function built from pieces of different functions over different intervals. Each part of the function is stated by applying a different formulae for each piece of its domain.
-
Interval Notation:
The term "interval notation" refers to a method of describing continuous sets of real numbers using the numbers that connect them. When written down, intervals resemble ordered pairs. They are not, however, intended to designate a precise spot. Rather, they are intended to serve as a shortcut for expressing an inequality or a series of inequalities.
-
Interval notation is a faster way to represent the domain/range or solutions that are written as inequalities or numbers line solutions.
Viewing and Interpreting Graphs: To establish whether a graph reflects a function, use the vertical line test.
-
Domain and Range: The domain of a function f(x) is the set of all values for which it is defined, while the range is the set of all values that f(y) accepts
1.3
Exponential Growth: Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function, Without limit based on an exponential function. If a>1 then you would know that there is an exponential growth.
Exponential Decay: This is the opposite of exponential growth. The rate of change declines until it hits a point where it can't decrease further which means it has hit an asymptote of the function. If a is between 0 and 1 then you know an exponential decay is occuring
-
Number "e": The number e, often known as the natural number or Euler's number, is a significant mathematical constant with a value of about 2.71828. “e” is an irrational number meaning it cannot be written as a simple fraction.
Applications: Real life applications of exponential functions are bacteria growth, compound interest, radioactive decay and continuous compound interest. For continuous growth we use the formula A=Ao(e)^it as change is always constant For compound interest the formula used is A=Ao(1+i/n)^nt and for bacterial growth the formula used is A=Ao(B)^t/p.
1.5
One to one functions: It is a function which return a unique range for each element in their domain meaning that their answers will never be the same.
line test: The horizontal can be used on one to one functions to test if it is one. It tests that for every y value there is only one x value. If there is more than one then it isn't one to one. The vertical line test is a visual approach to determine if a curve is a graph of a function or not in mathematics. For each unique input, x, a function can only have one output, y. When all vertical lines intersect a curve just once, the curve will be a function.
Inverses: Inverse basically means the reciprocal of a function. To graph an inverse just graph the normal function and then switch all of the x and y the values and graph the inverse.
Logarithmic functions: To solve exponential equations and investigate the features of exponential functions, we can use logarithms. y = logax can be shown as the exponential function of y = a^x. With the equation y=logb(x+h)+k, the logarithmic function y=logbx can be shifted k units vertically and h units horizontally.
-
Properties of Logs:
-
-
-
-
-
Log rules: 
Applications: Sound (decibel measurements), earthquakes (Richter scale), star brightness, and chemistry are all examples of this (pH balance, a measure of acidity and alkalinity). Logarithms are useful for these things because they can solve exponentials.
1.6
-
Trigonometric Graphs: The basic sine and cosine functions of the form y = sin x and y = cos x can be graphed in two ways: by evaluating the function and using the unit circle, and by determining the matching y value.
-
-
Even and Odd Triganometric Ratios: Even functions are functions that when on a graph are symmetrical about the y-axis. Odd functions are not symmetrical on the y-axis. Cos x and sec x are even functions and the other 4 are all odd functions. A function is even if f(-x) = f(x) and it would be odd if f(-x)= -f(x)
Transformations of Trigonometric graphs: Transformations are changes to the basic sine and cosine graphs' amplitude, period, and midline. Altering the midline vertically shifts the graph; changing the amplitude vertically stretches or compresses the graph; and changing the period horizontally stretches or compresses the graph.
Inverse of Trigonometric functions: he inverse sine function, for example, can be expressed as sin1x or arcsinx. When two side lengths are known, the inverse trigonometric functions sin1(x), cos1(x), and tan1(x) are used to obtain the unknown measure of an angle in a right triangle.
6 Trig ratios: There are six functions for angles commonly used in trigonometry. Named sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).
Sinusoidal: Where A is the Amplitude, B is the period, C is the horizontal shift(phase shift), and D is the vertical shift (Mildline)
-
-
-
-
-
-