Any Position of the Lines before Curves
|Definition
The position of the line have three possible position showed picture below.
For instance the line g : y = mx + n and parabolas h : y = ax2 + bx + c.
When the equation of the line g is substituted over the parabola’s equation h, we get a new equation that is quadratic equation.
yh = yg
ax2 + bx + c = mx + n
ax2 + bx – mx+ c – n = 0
ax2 + (b – m)x + (c – n) = 0………….a new equation quadratic
The Discriminant (D) of the new equation is:
D = (b – m)2 – 4a(c – n)
By seeing the value of the discriminant’s equation quadratic, it show the position of the line g before parabolas h without to draw it graph first. This is main criteria :
- If D > 0, then the equation quadratic has two real values, so the line g across parabolas h in two any point.
- If D = 0, then the equation quadratic has two real same values, so the line g offend the parabolas h
- If D < 0, then the equation quadratic has no real values, so the line g has no intersect nor offend to the parabolic h.
Example 1
(Indonesian National Test)
The graph of quadratic function is defined offend the line
. The value b is fulfilled by…
Alternative answer
First step we have to make equal both of them. As we know that , thus we get equation
{quadratic equation}
The coefficient of quadratic equation is :
The condition is the equation quadratic has two real same values, so the line offend the parabolas
Discriminant = 0
So the values of b needed is b=3
Example 2
The line y = –2x + 3 offend a parabolas f(x) = x2 + (m – 1)x + 7, then the values m requirement is …
Alternatif answer
First step we have to make equals both of them. As we know that , thus we get equation
The coefficient of quadratic equation is :
The condition is the equation quadratic has two real same values, so the line offend the parabolas
Discriminant = 0
then we get a quadratic equation in variable m. By using factor’s solving we get
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