**Equations Formula:**

What is linear equation ?

when equation between two variables that gives a straight line when plotted on a graph its called linear equation.

ax + b = 0, a ≠ 0 ⇒ x = – b / a

**Two Linear Equations :**

#### a_{1} x + b_{1} y = c_{1} a_{1} / a_{2} ≠ b_{1} / b_{2}

a_{2} x + b_{2} y = c_{2}

**Exponential Equation :**

- a
^{x}= c, a > 0 - If c > 0, a ≠ 1 : So one root x = log
_{a}c

If c = 1, a = 1, So many roots,

If c ≠ 1, a = 1, So no solution,

If c ≤ = 0, no So no solution,

**Logarithmic Equations :**

loga_{a} x = c (a > 0, a ≠ 1)

x = a ^{c}

**Function Transformation Rule:**

**Horizontal Shifting :**

**g(x) = f(x – c)**

**Vertical Shifting :**

## g(x) = f(x) + c

## Stretching :

## g(x) = c f(x), 0 < c < 1

## h(x) = D f(x), D > 1

**Reflection :**

## g(x) = f(-x)

## h(x) = -f(x)

**Exponential Inequation :**

- α
^{A}(x) > a^{B}(x) - α > 0, A(x) > B(x)
- 0< α < 1, A(x) < B(x)

**Logarithmic Inequation :**

loga α(x) = loga β(x)

- a > 1:
- β(x) > 0 , α(x) > β(x)

- 0<a<1:
- α(x) > 0, α(x) < β(x)

**Other Functions Graph:**

## Square Root :

**Parabolic:**

**Reciprocal :**

**Cubic :**

**Sec :**

**Cosec :**

**Absolute :**

**Constant :**

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