Algebra II: Linear and Quadratic Functions

Algebra II: Linear Functions

*Note: This section covers key concepts related to linear functions.*

Forms of Linear Equations

  • Slope-Intercept Form: y = mx + b, where ‘m’ represents the slope (steepness of the line), ‘b’ is the y-intercept (where the line crosses the y-axis), ‘x’ is the independent variable, and ‘y’ is the dependent variable.
  • Point-Slope Form: y – y₁ = m(x – x₁), where ‘m’ is the slope, and (x₁, y₁) is a given point on the line. ‘x’ and ‘y’ remain as variables.
  • Standard Form: Ax
Read More

Understanding Statistical Significance and Portfolio Diversification in Finance

Understanding Statistical Significance in Regression Analysis

Recall that the probability of rejecting a correct null hypothesis is equal to the size of the test, denoted α. The possibility of rejecting a correct null hypothesis arises from the fact that test statistics are assumed to follow a random distribution, and hence they will take on extreme values that fall in the rejection region some of the time by chance alone. A consequence of this is that it will almost always be possible to find significant

Read More

Proof of Triangle Congruence: SAS Postulate

Theorem 8: Triangle Congruence via SAS Postulate

Statements

  1. In \( \triangle ABC \leftrightarrow \triangle GET \)
  • i) \( BC = WE \)
  • ii) \( \angle B = \angle GET \)
  • iii) \( BA = GE \)
\( \therefore \triangle ABC \cong \triangle GEF \) \( \therefore AC = GF \) & \( \angle A = \angle GET \) But \( DF = A \) \( \therefore GF = DF \) \( \therefore \) In \( \triangle DEG \), \( m \angle 1 = m \angle \) Similarly, in \( \triangle GFD \), \( m \angle 2 = m \angle 4 \) \( \therefore m \angle 1 + m \angle 2 Read More

Understanding Lines and Angles in Analytic Geometry

Length of a Segment from a Point to a Line

The shortest distance from a point P (x1, y1) to a line Ax + By + C = 0 is given by the following expression:

e4-1.gif

In this expression, (x,y) represent the point from which we want to measure the distance, and A, B, and C are the coefficients of the line.

The conditions to consider are the following:

  • If the value of C is not equal to 0, the sign of the radical is the opposite of C.
  • If C=0 and B is not equal to 0, the radical will have the same sign as B.
  • If C=B=0,
Read More

Understanding Surveying Levels and Measurement Errors

Components of a Surveying Level

The image below illustrates the components of a typical surveying level:

  1. Ocular
  2. Reticulum Focus
  3. Image Focus
  4. ASA (presumably a brand or model designation) dismantled, with screws
  5. RS232 serial interface
  6. Leveling mechanism
  7. Screw
  8. Lens with electronic distance meter (EDM) integrated
  9. Exit measurement beam
  10. Display
  11. Spherical level
  12. Keyboard
  13. Trigger button
  14. Switch
  15. Fine measurement, horizontal

Types of Surveying Levels

Levels can be classified as follows:

  • Tilt Levels: These levels are adjusted
Read More

Principal Component Analysis and Cluster Analysis

Principal Component Analysis (PCA)

PCA is a mathematical method that uses an orthogonal transformation to convert a set of correlated variables into uncorrelated variables called the principal components. The first component has the highest variance (it captures the most variation in the data), followed by the second, third, and so on. The components must be uncorrelated (remember orthogonal direction). Normalizing data becomes extremely important when the predictors are measured in different units.

Read More