The line y = 4x - 7 is tangent to a parabola that has a y-intercept of -3 and the line x = 1/2 as it’s axis of symmetry. Find the equation of the parabola.

Question

Working out please!

Step by step..

Thank you!

Answer 1

The parabola is Ax² + Bx + C

Since the y-intercept is -3, C=-3

-B/(2A) = 1/2 => B = -A

At the point of tangency the parabola and the tangent must have the same ordinate, i.e.

Ax² - Ax - 3 = 4x - 7 => Ax² -(A+4)x + 4 =0

and the slope of the parabola at the point of tangency must be 4, i.e.2Ax - A = 4

therefore

A = 4/(2x-1)

Plug this into the quadratic to get

[4/(2x-1)}x² - {[4/(2x-1) + 4}x +4 = 0

Divide through by 4 and write 1 as (2x-1)/(2x-1), then multiply by 2x -1 to get x² = 1 so that x= 1,

Then A = 4

I have intentionally left out some algebra for you to do!

Finally the parabola is 4x² - 4x -3