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Functions, Statistical Analysis, Calculus, Calculus II - Coggle Diagram
Functions
Absolute Value Equations and Inequations
Application of Functions
Sketching Graphs
Roots indicate shape
Equation
Do two equations
Make it negative value
Same equation
Check by seeing if substituted numbers produced same numbers LHS and RHS
Inequality
Using two results put numbers on timeline
Check by substituting x in original equation
less than
greater than
or between
Absolute Value
Number will be positive
Using negative inequalities switch the sign
Absolute Value Graphs
Making Graph
Create table of axis
Find central x-value
Find what number makes equation = 0
Input numbers accordingly
Finding Intersection
Make two equations equal each other
Graphical Solutions Involving Absolute Values
Finding no solutions
When second equation meets og equation and only intercepts 1 point anything further away will produce no solutions
Intersecting Linear Inequalities
Finding Satisfied Regions
Sub in Origin
If inequality satisfied = true
Regions of the Number Plane
Finding Region Satisfying both Conditions
Sub in Origin
If true shade region that satisfy condition
If false shade region above/opposite
Graphs of Radical and Rational Functions
Radical Functions
Domain Restrictions
Can't take square root of a negative number
Denominator can never be zero
negative square root = bottom half of parab
Transformation of Functions
Sketching Parabolas
Identify
x -intercept
y = 0
y-intercept
x = 0
Vertex
middle of x-intercepts
Parabola
Negative x
Concave down
Positive x
Concave up
Transforming the Parabola
y = ax^2 + c
Graph y axis
Variable C
Moves up or down c values on the y axis
Graphing Exponentials
Graph y axis
Exponential
Positive exponential
Left to right
Negative exponential
Right to Left
Graphs of the Form y = kx^n
Types of Exponential Graphs
Linear
Directly Proportional
Parabola
Vertex at the Origin
Cubic Curve
Hyperbola
Inversely Proportional
Circular
Radical
Graph Transformations
Vertical Translations
y = f(x) + a
Shifts graph vertically by
a
units
Positive a
up
Negative a
down
Horizontal Translations
y = f(x+a)
Shifts graph horizontally by a units
Positive a
left
Negative a
right
Reflections
y = -f(x)
Reflects graph across x-axis
y = f(-x)
Reflects over y-axis
Dilations
a
f(x)
Vertical Dilation
Narrower across the y-axis
f(
b
x)
Horizontal Dilation
Narrower across the x-axis
Statistical Analysis
The Normal Distribution
Definition
Bell shaped curve which is symmetrical about the mean
Mean, median and mode
Majority of values are around the centre of the curve
Normal Distribution Percentage
Standard Deviation
Definition
Measure of how widespread a set of scores is from the mean
Finding Standard Deviation
Calculate Variance
Square root variance = SD
Calculate variance by
Terms
S.D. (population)
S.D. (sample)
Percentages
68% lie within 1 S.D.
95% lie within 2 S.D.
99.7% lie within 3 S.D.
Equation
We say:
Max at:
Graph
Draw standard bell-shaped curve
Mean is in the center
Place numbers across x - axis using SD
Z-scores
Definition
Measure of how far in SD a score is above or below the mean
Z-score
Calculating and Scaling
Probability Density Functions
PDF
Conditions
otherwise
Mean and Mode for Continuous Random Variables
for continuous random variable X w a PDF
Continuous Random Variables
Calculus
Differentiation of Trigonometric, Exponential and Logarithmic Functions
Small Angle Results
Limits
x on top needs to be same as bottom
Trigonometric Differentiation
Examples
Rules
Differentiation of Logs and Exponentials
Examples
First and Second Derivatives
Increasing and Decreasing Rates of Change
Rate of Change
Rate of Increase
Steep = Increasing at an increasing/faster rate
Less steep = Decreasing at a decreasing rate
Gradient
Negative = Decrease
Positive = Increase
Increasing and Decreasing Functions - Stationary Points
describes the curve's gradient (slope)
Gradients
Positive
Increasing Curve
Negative
Curve is Decreasing
Zero
Stationary point
Types
Maximum pt
1 more item...
Minimum Turning Point
1 more item...
Horizontal Inflexion Point
1 more item...
To find the turning point sub the zeroes into the equation and see if the satisfy number line
Inflexion Points
describes the concavity of the curve
Examples
Concave up
Concave down
Inflexion point
This only occurs if there's a change of concavity
check if above and below numbers result in negative and positive
Using x value in 2nd deriv sub into the intiial equation
Rules of Differentiation
Differentiation Laws
Differentiation Methods
Rules
Function of a Function Rule
Product Rule
Inside + Outside
Quotient Rule
Calculus II
Anti-Derivative
Integration
Examples
Integration Involving Trig Functions
Standard Integrals
Same set out for all other equations too
Integration of Logs and Exponentials
Examples
derivative of ax
Graphs of Original Functions
Graph
x-axis which is 0 = max/min
max/min may be point of inflexion
above x-axis gradient is positive
below x-axis gradient is negative
Equation of a Curve given it's Derivative
Steps
Integrate gradient function
Substitute point (y, x)
y = ax^n + C
If satisfies equation make C result
Motion in a Straight Line
Distance, Velocity and Acceleration
Terms
Position
make t = 0 to find initial position
Velocity
to find when stationary make v = 0
Acceleration
Key Phrases
Initially at rest,
Stationary
t = 0
v = 0
Rates of Change
Definition
A rate is often a derivative with respect to time
dt at the bottom
Areas and Definite Integrals
Riemann Sums and Definite Integrals
Steps
Find width
Use Run / Rise
Find Y for each X
Place numbers across x axis between given numbers using width
Find Estimated Area
Width x Sum of all Heights
Trapezoidal Rule
Area
Width of Each Strip
First y- value
Middle y-values
might be more than 1 of those middle values
last y-value
Steps
Number of strips
= Function values - 1
Sub-intervals
Make table finding y-values
