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.

y_h = y_g

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 :

1. If D > 0, then the equation quadratic has two real values, so the line  g  across parabolas h in two any point.
2. If D = 0, then the equation quadratic has two real same values, so the line g offend the parabolas h
3. 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