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 completly stumped on this one... Any help would be greatly appreciated!
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.
Start from the 2 point formula for a line. [tex] \frac {y - y_1} {x - x_1} = \frac {y_2 - y_1} {x_2 - x_1} [/tex] The formula for your parabola is [tex] y = x^2 [/tex] So we can write [tex] y_1 = x_1^2 [/tex] and [tex] y_2 = x_2^2 [/tex] Use this information in the 2 point formula to get [tex] \frac {y - y_1} {x - x_1} = \frac {x_2^2 - x_1^2} {x_2 - x_1} [/tex] Note that the numerator on the Right Hand Side is the differenc of squares and can be factored to get [tex] \frac {y - y_1} {x - x_1} = \frac {(x_2 - x_1) (x_2 + x_1)} {x_2 - x_1} [/tex] Cancel like factors in the RHS [tex] \frac {y - y_1} {x - x_1} = (x_2 + x_1) [/tex] Now rearrange this to get [tex] y - y_1 = (x - x_1) (x_2 + x_1)[/tex] Simplify to get: [tex] y = x (x_2 + x_1) - x_1 x_2[/tex] 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.