Comprehensive Guide to Transforming Functions: Quadratic, Rational, Cubic, and More
Transformations of Functions
Quadratic Function:
The graph of the basic quadratic function Y=x^2 can be transformed using the following parameters:
g(x) = af(b(x+c)) + d
- a:
- a > 1: Vertical stretch by a factor of ‘a’.
- 0 < a < 1: Vertical compression by a factor of ‘a’.
- a = -1: Vertical reflection across the x-axis.
- b:
- b > 1: Horizontal compression by a factor of 1/b.
- 0 < b < 1: Horizontal stretch by a factor of 1/b.
- b = -1: Horizontal reflection across the y-axis.
- c:
- c > 0: Horizontal shift
A Comprehensive Guide to Measurement, Finance, and Geometry
MEASUREMENT AND STATISTICS
Accuracy and Precision
Accuracy: How close a measured value is to the actual value.
Precision: The smallest measurement possible on a measuring tool.
Measures of Central Tendency
Mean: Average (sum of all values divided by the total number of values).
Median: Middle value when data is arranged from least to greatest.
Mode: Most frequent value.
Trimmed Mean: Mean calculated after discarding a certain percentage of the highest and lowest values.
Weighted Mean: Mean calculated by
Read MoreStatistical Hypothesis Testing and Confidence Intervals: A Comprehensive Analysis
1. Analyzing Beats Per Minute in Dance Songs
Hypotheses and Parameter
The parameter of interest is μ = mean beats per minute for all dance songs.
- H0: μ = 120.5 beats per minute
- Ha: μ > 120.5 beats per minute
T-Procedure Validity
Even with a slightly skewed distribution of beats per minute, the t-procedure remains valid due to its robustness to deviations from normality.
Test Statistic and P-value
Assuming necessary assumptions are met, the test statistic and P-value are calculated using a t-test (
Read MoreComprehensive Guide to Transforming Mathematical Functions
Transformations of Quadratic Functions
The general form of a quadratic function is:
g(x) = af(b(x + c)) + d
This formula describes how to transform the graph of the basic quadratic function y = x². Let’s break down each parameter:
Vertical Transformations
- a > 1: Vertical Stretch: The graph is stretched vertically by a factor of a. For example, if a = 2, the graph is twice as tall.
- 0 < a < 1: Vertical Compression: The graph is compressed vertically by a factor of a. For example, if a = 0.5, the
Statistics Cheat Sheet: Key Concepts and Formulas
Find the probability that exactly four of the sampled people own a tablet computer. We input 15 for n, .3 for p, and 4 for x = binompdf(15,.3,4) 15 is sample size .30=30% || To compute P(x), select poissonpdf and enter the values for λt and x separated by commas and press ENTER. || To compute P(Less than or equal to x), select poissoncdf and enter the values for λt and x separated by commas and press ENTER. || Let A and B be events with P(A) = 0.7, P(B) = 0.4, and P(B|A) = 0.2. Find P(A and B)
Read MoreUnderstanding Uncertainty and Reliability in Structural Design
Q1. Uncertainty and Safety Factors in Structural Design
(a) Types of Uncertainty
i. Aleatory Uncertainty
Aleatory (Random or Objective) uncertainty, also known as irreducible or inherent uncertainty, is the intrinsic randomness of a phenomenon. Think of rolling dice – the outcome is inherently unpredictable.
ii. Epistemic Uncertainty
Epistemic (Subjective) uncertainty is reducible uncertainty stemming from a lack of knowledge or data. As we gather more information, this type of uncertainty decreases.
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