\(D=b^2-4ac\quad(a\ne0)\)
Conditions for all real values of x
| Statement | \(y=ax^2+bx+c\) | \(y=a(x-h)^2+k\) | Sketch |
|---|---|---|---|
| Always positive: \(y>0\) for all real \(x\) | |||
| Always negative: \(y<0\) for all real \(x\) | |||
| Never negative: \(y\ge 0\) for all real \(x\) | |||
| Never positive: \(y\le 0\) for all real \(x\) |
Example
| Curve | Shape test |
|---|---|
| \(y=2x^2-3x+4\) |
Check both the leading coefficient and the discriminant.
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