Mathematical Functions and Sequences Notes

Linear Equations

• Linear Slope Formula
  $m = \frac{y<em>1 - y</em>2}{x<em>1 - x</em>2}$

  • This formula calculates the slope (m) of a line by taking two points on the line, (x1, y1) and (x2, y2).

• Slope-Intercept Form
  $y = mx + b$

  • In this form, m represents the slope and b represents the y-intercept of the line.

• Point-Slope Form
  $y - y<em>1 = m(x - x</em>1)$

  • This is useful when you know one point on the line and the slope.

• Linear Standard Form
  $Ax + By = C$

  • A, B, and C are constants, and this form can represent lines where A and B are not both zero.

Quadratic Functions

• Quadratic Standard Form
  $y = ax^2 + bx + c$

  • This form is used to express a quadratic function where a is not zero.

• Vertex Form
  $y = a(x - h)^2 + k$

  • In this form, (h, k) is the vertex of the parabola represented by the quadratic equation.

• Quadratic in Intercept Form
  $y = (x - p)(x - q)$

  • This form shows the roots of the quadratic function at points p and q, indicating where the graph intersects the x-axis.

• Axis of Symmetry
  $x = -\frac{b}{2a}$

  • This equation provides the x-coordinate of the vertex of the parabola.

• Quadratic Formula
  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

  • This formula is used to find the roots of a quadratic equation.

Exponential Functions

• Exponential Form
  $y = a(b)^x$

  • In this expression, a is the initial amount, b is the base, and x is the exponent, typically describing exponential growth or decay.

• Exponential Growth
  $y = a(1 + r)^x$

  • Where r is the rate of growth, applied when the quantity increases over time.

• Exponential Decay
  $y = a(1 - r)^x$

  • Where r is the rate of decay, applied when the quantity decreases over time.

Sequences

• Arithmetic Sequence (Explicit)
  $a<em>n = a</em>1 + d(n - 1)$

  • This formula provides the nth term of an arithmetic sequence, where a1 is the first term, d is the common difference, and n is the term number.

• Geometric Sequence (Explicit)
  $a<em>n = a</em>1 imes r^{n-1}$

  • This formula provides the nth term of a geometric sequence, where a1 is the first term and r is the common ratio.

• Arithmetic Sequence (Recursive)
  a1 = # \quad an = a_{n-1} + d

  • The first term is provided, and subsequent terms are calculated based on the previous term plus the common difference.

• Geometric Sequence (Recursive)
  a1 = # \quad an = a_{n-1} \times r

  • The first term is provided, and subsequent terms are calculated based on the previous term multiplied by the common ratio.