Find the function of a line passing through three points

D is the vertex of a parabola. The parabola cuts the x-axis at A and at C(4, 0). The parabola cuts the y-axis at B(0,?4). The area of ABC is 4.
Determine the area of DBC.

Next, we determine the equation of the parabola.
The parabola has x-intercepts 2 and 4, so has equation y = k(x ? 2)(x ? 4).
Since the parabola passes through (0,?4) as well, then ?4 = k(?2)(?4) so k = ?1/2 .
Therefore, the parabola has equation y = ?1/2 (x ? 2)(x ? 4).

I do not understand the things in bold. Can you please explain? Thank you in advance!

The parabola has equation \(\displaystyle \frac{-1}{2}(x-2)(x-4)\)

Completing the square gives us \(\displaystyle y=\frac{-1}{2}(x-3)^{2}+\frac{1}{2}\)

This is in the form \(\displaystyle y=a(x-h)^{2}+k\), where (h,k) are the coordinates of the vertex.

Thus, the vertex has coordinates \(\displaystyle (3,1/2)\)

Now, use the three points to find the area of DBC

\(\displaystyle D(3,1/2), \;\ B(0,-4), \;\ C(4,0)\)

Anna55 said:

D is the vertex of a parabola. The parabola cuts the x-axis at A and at C(4, 0). The parabola cuts the y-axis at B(0,?4). The area of ABC is 4.
Determine the area of DBC.

Next, we determine the equation of the parabola.
The parabola has x-intercepts 2 and 4, so has equation y = k(x ? 2)(x ? 4).
Since the parabola passes through (0,?4) as well, then ?4 = k(?2)(?4) so k = ?1/2 .
Therefore, the parabola has equation y = ?1/2 (x ? 2)(x ? 4).

I do not understand the things in bold. Can you please explain? Thank you in advance!

Click to expand...

In your studies, you will learn that there are several forms for the equation of a parabola.

ONE of those forms is usually called "intercept form".....if a parabola intersects the x-axis at points (p, 0) and (q, 0), then the equation of the parabola can be written in the form

y = k(x - p)(x - q)

Note that if x = p, or if x = q, then y = 0.

So, if the parabola in your problem has x-intercepts 2 and 4, we can substitute 2 and 4 for p and q in the above general equation to get

y = k(x - 2)(x - 4)

You should also be aware that the coordinates of any point on the graph of the parabola should satisfy the equation of the parabola. If the parabola passes through the point (0, -4), then the equation for the parabola should be true when x = 0 and y = -4.

We've got this equation so far:

y = k(x - 2)(x - 4)

We know the point (0, -4) is on the graph, so we can substitute 0 for x and -4 for y:

-4 = k(0 - 2)(0 - 4)
-4 = k(-2)(-4)
-4 = 8k
-4/8 = k
-1/2 = k

That's the missing piece of information needed to complete the equation of the parabola:

y = k(x - 2)(x - 4)
y = (-1/2)*(x - 2)*(x - 4)

Thank you very much! I understand everything now!