Advanced Math

Quadratic Functions : General Form

y = ax^2 + bx + c

Quadratic Functions: Sum of roots

-\tfrac{b}{a}

Quadratic Functions: Product of Roots

\tfrac{c}{a}

Quadratic Functions : Vertex X

-\tfrac{b}{2a}

Quadratic Functions: y

Plug the X into desmos

Quadratic Functions: vertex form

a(x-h)²+k

Quadratic Functions: Completing the square step 3

y=a(x+\frac{b}{2a})2+(c-\frac{b^2}{4a})

Quadratic Functions: Completing the square step 2

\;\; = a\Bigl[\bigl(x + \tfrac{b}{2a}\bigr)^2 - \bigl(\tfrac{b}{2a}\bigr)^2\Bigr] + c

Quadratic Functions: Completing the square step 1

y = a\bigl(x^2 + \tfrac{b}{a}x\bigr) + c

Quadratic Functions: Completing the square step 4

identify the vertex Vertex (h,k) where h = -\tfrac{b}{2a} and k = c - \tfrac{b^2}{4a}.

Quadratic Functions: a>0

opens up

Quadratic Functions: a<0

opens down

Quadratic Functions: ac method

two numbers multiply to ac and adding up to b

Quadratic Functions: Quadratic formulas

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Quadratic Functions: Discriminant

D = b^2 - 4ac

Quadratic Functions": D>0

two distinct real solutions.

Quadratic Functions: D=0

one solution

Quadratic Functions: D<0

no real solutions

Exponential Functions: Form

y= a * b^x

Exponential Functions: Growth

b>1

Exponential Functions: Decay

0<b<1

Nonlinear Equation Techniques: Quadratics

solve by factoring, completing the square, or quadratic formula.

Nonlinear Equation Techniques: nonlinear equation

substitute one equation into the other or graph both curves to find intersection points.

Equivalent Expressions

Simplify by combining like terms, factoring, expanding.

Rational Expressions

factor numerator & denominator, then cancel common factors.