Mathematical Formulas for Linear and Quadratic Equations

Linear Equations

  • Linear Slope Formula:

    • Formula: m=y<em>1−y</em>2x<em>1−x</em>2m = \frac{y<em>1 - y</em>2}{x<em>1 - x</em>2}
    • Description: This formula calculates the slope (m) of a line given two points (x1, y1) and (x2, y2).

  • Slope-Intercept Form:

    • Formula: y=mx+by = mx + b
    • Description: This form represents the equation of a line where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

  • Point-Slope Form:

    • Formula: y−y<em>1=m(x−x</em>1)y - y<em>1 = m(x - x</em>1)
    • Description: This form is useful when you have a slope and a point on the line (x1, y1).

  • Linear Standard Form:

    • Formula: Ax+By=CAx + By = C
    • Description: In this form, A, B, and C are integers, where A should be positive.
Quadratic Equations

  • Quadratic Standard Form:

    • Formula: y=ax2+bx+cy = ax^2 + bx + c
    • Description: This represents the equation of a quadratic function where a, b, and c are coefficients.

  • Vertex Form:

    • Formula: y=a(x−h)2+ky = a(x - h)^2 + k
    • Description: This form is useful for finding the vertex of the parabola, which is at the point (h, k).

  • Quadratic in Intercept Form:

    • Formula: y=(x−p)(x−q)y = (x - p)(x - q)
    • Description: This form highlights the roots (p and q) of the quadratic equation.

  • Axis of Symmetry:

    • Formula: x=−b2ax = -\frac{b}{2a}
    • Description: This formula gives the x-coordinate of the vertex of the parabola.

  • Quadratic Formula:

    • Formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
    • Description: This formula is used to find the roots of the quadratic equation ax² + bx + c = 0.
Exponential Functions

  • Exponential Form:

    • Formula: y=a(b)xy = a(b)^x
    • Description: This represents the general form of an exponential function where a is a constant and b is the base.

  • Exponential Growth:

    • Formula: y=a(1+r)xy = a(1 + r)^x
    • Description: This describes situations where the quantity increases by a certain percentage rate (r).

  • Exponential Decay:

    • Formula: y=a(1−r)xy = a(1 - r)^x
    • Description: This describes situations where the quantity decreases by a certain percentage rate (r).
Sequences

  • Arithmetic Sequence - Explicit Formula:

    • Formula: a<em>n=a</em>1+d(n−1)a<em>n = a</em>1 + d(n - 1)
    • Description: This formula gives the nth term of an arithmetic sequence, where d is the common difference.

  • Geometric Sequence - Explicit Formula:

    • Formula: a<em>n=a</em>1⋅rn−1a<em>n = a</em>1 \cdot r^{n-1}
    • Description: This formula gives the nth term of a geometric sequence, where r is the common ratio.

  • Arithmetic Sequence - Recursive Formula:

    • Formula: a<em>1=first term,a</em>n=an−1+da<em>1 = \text{first term}, \quad a</em>n = a_{n-1} + d
    • Description: This defines the nth term in terms of the previous term plus the common difference.

  • Geometric Sequence - Recursive Formula:

    • Formula: a<em>1=first term,a</em>n=an−1⋅ra<em>1 = \text{first term}, \quad a</em>n = a_{n-1} \cdot r
    • Description: This defines the nth term in terms of the previous term multiplied by the common ratio.