Why does multiplying the positive x values give the y-intercept on a parabola?

[Menos]

Interesting feature of this graph. Consider 2 points on the parabola, I'll take (-2,4) and (4,16). By multipling the positive x values (2*4=8), you can get the y-intercept of the line from (-2,4) to (4,16). Proof: The line including (-2,4) and (4, 16) is written as y=2x+8. Thus, the y-intercept is 8. My question is why does this work? I've been trying to figure it out for a while, and I am completely stumped on this one... Any help would be greatly appreciated!

 

[0rthodontist]

First find the general equation for the slope b in terms of x1, y1, x2, and y2. Then use the fact that y1 = x1^2 and y2 = x2^2.

Actually it's not the product of the absolute value of the x values, it's the opposite of the product of the x values.

 

[Integral]

Start from the 2 point formula for a line.

$$\frac {y - y_1} {x - x_1} = \frac {y_2 - y_1} {x_2 - x_1}$$

The formula for your parabola is

$$y = x^2$$

So we can write

$$y_1 = x_1^2$$
and
$$y_2 = x_2^2$$

Use this information in the 2 point formula to get

$$\frac {y - y_1} {x - x_1} = \frac {x_2^2 - x_1^2} {x_2 - x_1}$$

Note that the numerator on the Right Hand Side is the differenc of squares and can be factored to get

$$\frac {y - y_1} {x - x_1} = \frac {(x_2 - x_1) (x_2 + x_1)} {x_2 - x_1}$$

Cancel like factors in the RHS
$$\frac {y - y_1} {x - x_1} = (x_2 + x_1)$$

Now rearrange this to get

$$y - y_1 = (x - x_1) (x_2 + x_1)$$
Simplify to get:
$$y = x (x_2 + x_1) - x_1 x_2$$

Clearly you are correct for the simple parabola, in addition it can be seen that the slope of the line is the sum of the x coordinates.