| Shape |
\(a\neq 0\) for any quadratic
\(a\gt 0\) means parabola opens up and the vertex is a minimum
\(a\lt 0\) means parabola opens down and the vertex is a maximum |
\(x\)-intercepts \((x_0)\)
\(y=0\) |
Quadratic formula
\(x_0=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) |
Read directly
\(x_0=r\:;\) or \(x_0=s\) |
Solve for \(x\)
\(x_0=h\pm\sqrt{\dfrac{-k}{a}}\) |
\(y\)-intercept \((y_0)\)
\(x=0\) |
Read directly
\(y_0=c\) |
Substitute with \(x=0\); solve for \(y_0\)
\(y_0=ars\) |
Substitute with \(x=0\); solve for \(y_0\)
\(y_0=ah^2+k\) |
Vertex
\((x_v,y_v)\) |
\(x_v=\dfrac{-b}{2a}\)
Substitute to get \(y_v\) |
\(x_v=\dfrac{r+s}{2}\)
Substitute to get \(y_v\) |
Read directly
\((h,k)\) |
| Number of real roots |
Discriminant
\(\Delta=b^2-4ac\) |
|
|
None |
\(\Delta \lt 0\) |
Form does not exist |
\(\frac{k}{a}\gt 0\) |
|
One |
\(\Delta = 0\) |
\(r=s\)
form becomes \(a(x-r)^2\) |
\(k=0\)
form becomes \(a(x-r)^2\) |
|
Two |
\(\Delta \gt 0\) |
\(r\neq s\) |
\(\frac{k}{a}\lt 0\) |
| Interconversion |
|
|
To standard |
N/A |
Expand and simplify |
Expand and simplify |
|
To factored |
- Quadratic factoring
- Calculate roots and write directly
(Remember \(a\))
- May not exist (check \(\Delta\))
|
N/A |
- Through standard form
- Calculate roots and write directly
(Remember \(a\))
- May not exist (check \(\frac{k}{a}\))
|
|
To vertex |
- Completing the square
- Calculate the vertex and write directly
(Remember \(a\))
|
- Through standard form
- Calculate the vertex and write directly
(Remember \(a\))
|
N/A |