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Unit 1 Mind Map - Coggle Diagram
Unit 1 Mind Map
Functions and Graphs
Function:
Definition: from a set domain(D) to a set range(R) is a rule that assigns a unique element in R to each element in D.
Symbol: f(x) --> this means y equals x and can symbolize specific values of a function
Domain:
Definition:set of values which cannot be surpassed on the x axis.
Symbol: D
How to find domain:
1) look at the f(x) equation and isolate the x value and the number without a variable.
2) isolate x so that the number with no variable is no the right side
3) replace x with the greater than or equal to sign
Range :
Definition: set of values which cannot be surpassed on the y axis
Symbol: R
How to find range:
1) look at f(x) equation and replace x with the 0
2) solve for y
3)replace equal sign with greater than or equal to sign
Interval Notation: it is the fastest way to represent the domain and range when they are written as inequalities or number line solution
How to write interval notation:
Closed intervals: [ ], closed dot, greater than equal to sign or less than equal to sign
Open intervals: ( ), open dot, greater or less than sign, negative or positive infinite
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Vertical Line test: if the line passes through more than one point, then the graph/line(s) are not a function
How to do the vertical line test:
1) draw a vertical line at any random spot through the graph
2) if the line passes through the same x value more than once then it is not a function
Piecewise function:
Definition: a function built from pieces of different functions over different intervals
How to solve a piecewise function:
1) identify each different equation and its domain
2) make a table of values for each equation
3) fill the table of values for each equation (ensure all y and x values are available on the chart)
4)using the values from the table of values, graph all of the equations
5) determine domain and range
Example: 
Absolute Functions:
Definition: an algebraic expression within absolute value symbols (I I). An absolute value of a number is the distance from 0 on the number line.
How to solve:
1) identify the expression within the symbol
2) convert negative into positive if the expression within the symbol is negative otherwise keep the expression the same.
Composite Functions: Definition: when two functions are written inside one anotherHow to solve:
1) identify f(x) and g(x) equation --> should look like f(g(x))
2)substitute the answer from g(x) into f(x) and solveExample:
Exponential Functions
Exponential Growth:
Definition: a process in which a quantity increases over time (ie grows faster and faster)
Base equation: y = B^x and B must always be greater than 1 for exponential growth
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Exponential Decay: Definition: This is the opposite of growth meaning a quantity is decreasing over time until it reaches a limit where it cannot further decrease, meaning it reached it asymptote.
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The number E:
Definition: it is a mathematical constant which is approximately valued at 2.71.
How to label/express E:
It can be expressed as the sum of n equals to 0 through infinity of 1/n. It could also be a base for the natural logarithm of ln.
Applications:
Definition: real life applications of exponential functions
How can they be applied to real life:
They could be used to find the compound interest continuously or once, used to calculate growth of bacteria, etc.
Formulas:
1) A=Ao)1+i/n)^nt --> compound interest
2) A=Ao(B)^(t/p) --> exponential growth/decay
Functions and Logarithms
One to One functions: Definition: A one to one function occurs when a function passes the vertical line test.
similarly to the vertical line test, the vertical line test proves the graph is a one to one function if it does not go through more than one y value.
Inverse:
Definition: An inverse of an equations essentially means switching the x with the y and vice versa. Meaning, all of the original x values become the new y values whereas the original y values become the new x values.
Base equation: f(x)--> f-1(x)
How to find the inverse:
1) make a table of values and label it with the original equation
2) make another table of values and this time flip all of the x values to y and vice vera and label this chart with the inverse equation of the original.
How to graph inverse:
1) sketch the original equation
2) using the table of values of the inverse equation, sketch the inverse graph
Applications:
Definition: Real world applications/scenarios
Some of the real world applications of Functions and Logarithms could be the pH scale which is used in chemistry. Another could be to measure earthquakes and sound.
Properties of Logarithms: The main properties of logs are loga(x) which is used for log functions. This determines the amount of times the base is multiplied to itself until it reaches another number. Other properties of logarithms include log(xy)=loga(x)+logb(y), log(x/y)=log(x)*log(y), logb(x)=loga(x)/loga(b), and logb(xk)=klongb(x).
Parametric Equations
Relations: Definition: A set of paired values (x and y) that correspond with a T value which is called a parameter.
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What is the Parametric Equation: x=f(t), y=g(t) these equations can be graphed by using a graphing calculator
Circles: The equation which is applicable to all circles and could be used to graph circles is x^2 + y^2 = r^2. The R = radius of the circle. An example of this is the unit circle.
Ellipses: An ellipse is similar to a circle however it is more oval shaped meaning it is more stretched either vertically or horizontally. To find an ellipse, the equation used is x^2/a^2 + y^2/b^2 = 1.
Parametric Curve: They are not actual functions instead it is a curve where the x and y variables are defined the T variable. Similar to the Cartesian equation, we only use two variables.
Trigonometric Functions
Radian Measure: is the measure of the angle ACB at the centre of the unit circle which equals the length of the arc that ACB cuts from the unit circle. Equation: a=thetaR (theta must always be in radians)
Periodicity: Due to the even function graph being symmetrical to the y axis ,as stated in the Functions and Graphs lesson, cos(X) and sec(x) would be even graphs. Due to odd graphs being symmetrical at the original, the last four would be odd functions (sec(x),tan(x),sin(x), cot(x))
Inverse Trig Functions: The arc functions are the inverse trig functions. This can be found as the value of sin(theta) is given and you are usually asked to find theta.
Graphs of Trig Functions: Period is the length of a wave of a function. This can be calculated by using the formula 2pi/b. Formula: sine function: y=Asin(B(x-C))+D cosine function: y=Acos(B(x-C))+D (these can be graphed)
Transformations of Trig Graphs: If the A,B,C,D values change, there will be a shift vertically and horizontally, reflection n x/y axis, and horizontal or vertical compression or expansion.
Lines
Increments:
DEFINITION: if a particle moves from point (x1,y1) to the point (x2,y2).
How to find:
change in x(delta x)=x2-x1 and change in y(delta y)=y2-y1
Slope of a line:
DEFINITION:
Let P1(x1,y1) and P2(x2,y2) be points on a nonvertical line, L. The slope of L is m=rise/run.
HOW TO FIND SLOPE:
A slope of a line can be calculated from the increments in coordinates. The slope can be found by doing rise over run. The rise is found by calculating change in y=y2-y1. The run can be found by calculating the change in x=x2-x1.
Equation of a line:
Definition: You can write an equation of any non vertical line L if the slope m and coordinates of one point are available.
HOW TO FIND EQUATION OF A LINE:
1) y-y1/x-x1=m --> let m be the slope
2) y-y1=m(x-x1) --> point slope form (equation of a line that passes through point (x1,y1) with slope m.
Slope intercept form: is the equation of a line with slope m and y intercept b
Equation: y=mx+b
How to find the equation:
1)find slope m by following steps for the point slope equation
2)
General Form: is a general linear equation in x and y.
Equation: Ax+By=C (A and B cannot both be 0)
How to find equation:
1)find the slope intercept form or point slope form equation
2)rearrange equation so that it looks like the general form equation
Perpendicular lines: Definition: lines that intersect at a right angle.
Equation: (m1)(m2) = -1
How to find perpendicular lines:
1) identify the m value for both the equations
2) replace the values in the equation
3) if both the values do not equal -1, they are not perpendicular lines
Parallel lines:
Definition: two lines that never intersect each other no matter how far they are extended on either side
Equation: m1=m2
How to find parallel lines:
1) identify m value for both equations
2) replace values in the equation
3) if the both m values are not equal, then they are not parallel lines