FORMULABANK

A function is called a quadratic function because the variable x is raised to the power of 2 (and there are no higher powers of x).

Formulas:

Standard Function

$\mathrm{f(x)}=a{x}^2+bx+c$

Discriminant

$D=b^2-4ac$

Vertex 

$x=\frac{-b}{2a}$ and $y=\frac{-D}{4a}$

Zeros 

$x=\frac{-b+\sqrt{D}}{2\cdot a}$

$x=\frac{-b+\sqrt{D}}{2\cdot a}$ and $x=\frac{-b-\sqrt{D}}{2\cdot a}$

• If $D<0$: No solutions, meaning the graph does not intersect the x-axis.
• If $D=0$: One solution, meaning the vertex lies on the x-axis:
• If $D>0$: Two solutions, meaning the graph intersects the x-axis at two points:

Properties of the Quadratic Function Based on Coefficients

• If a is greater than 0, the branches of the parabola open upwards 😊
• If a is less than 0, the branches of the parabola open downwards 😞
• The smaller the value of a, the wider the graph becomes.
• The larger the value of a, the narrower the graph becomes.

The Effect of the b Value

• If the b value is 0 or absent in the equation, the vertex lies on the y-axis.
• If the a and b values have the same sign (+a and +b, or -a and -b), the vertex is to the left of the y-axis.
• If the a and b values have opposite signs (+a and -b, or -a and +b), the vertex is to the right of the y-axis.

The Effect of the c Value

• The c value tells you where the graph intersects the y-axis.