The line is tangent to a parabola The line $y = 4x-7$ is tangent to a parabola that has a $y$-intercept of $-3$ and the line $x=\frac{1}{2}$ as its axis of symmetry. Find the equation of the parabola. I really need help solving this question. THx

Let the parabola be defined by the function $y = f(x)$ and let $g(x) = f(x) - 4x + 7.$
Then $y = g(x)$ is a parabola passing through $(0,4)$
and it has slope $-4$ at $x = \frac 12.$
In addition, the parabola described by $y = g(x)$ is tangent to the $x$-axis
so $g(x)$ can be written $g(x) = a(x - p)^2.$
We can also write the derivative of $g$ with respect to $x$ as
$g^\prime(x) = 2a(x - p).$
This gives us $g(0) = ap^2 = 4$ and $g^\prime\left(\frac 12\right) = a(1 - 2p) = -4.$
Then
$$ \frac{ap^2}{a(1 - 2p)} = \frac{4}{-4}, $$
and we can simplify both sides of this to conclude that
$$ \frac{p^2}{1 - 2p} = -1. $$
Rearranging terms,
$$ p^2 = -1(1 - 2p), $$
$$ p^2 - 2p + 1 = 0 $$
Solve for $p;$ that is the $x$-coordinate of the (unique) point where $g(x) = 0.$
It's also the $x$-coordinate of the unique point where $f(x) = 4x - 7.$
Given $p,$ use one of the previous equations (such as $ap^2 = 4$) to solve for $a.$
You can then expand $g(x)$ as a simple polynomial.
Use the fact that $f(x) = g(x) + 4x - 7$ to write $f(x).$
You can solve the problem without introducing the additional parabola $y = g(x),$
but I like the geometric intuition of this method. The key idea is that by a
change of coordinates ($y \rightarrow y - 4x + 7$), we transform the parabola
$y = f(x)$ into one with a (somewhat) easier geometry (or at least so I think).
It seems a mere coincidence that the point of tangency of $y = f(x)$ and $y = 4x - 7$
just happens to be a point that we could already know is on $y = f(x)$ by virtue of
the axis of symmetry and the one given point of that parabola.
That's not a necessary condition for this problem to be solvable.
It does suggest a clever insight that for this particular problem that
leads to a solution more quickly than the steps above, because you can find
the point of tangency almost right away.
