How do I find a constant $C$, when $x = a$, through the tangent line?

Question

The full question is as follows:

For some constant $C$, the equation of the tangent line to the graph of

$y = f(x)$ = $4x^4$+C

at the point where $x=a$ is

$y = −78.608x − 106.1252$

Find $C$.

My teacher does not go by the textbook, and I can't find any information online, so I'm completely stuck on how to solve this

Answer 1

This is what I believe the answer to be, but confirmation that it is correct would be helpful.

1. The original equation is $f(x) = 4x^4 + C$
2. As $x=a$, this can be written as $f(a) = 4a^4 + C$
3. Take the derivative of $f(a)$, giving $f'(a) = 16a^3$
4. $f'(a)$ is simply the slope of the tangent line, which can be used to create the equation $-78.608 = 16a^3$, which, when solved, gives $a = -1.7$
5. Now that $a$ is known, plug it back into the original equation to give the equation $f(a)=4(-1.7)^4+C$
6. Further, because a graph and its tangent line at a specific point are equal at that point, the following equation can be made: $-78.608(-1.7)-106.1252 = 4(-1.7)^4 + C$
7. Solve the equation for C, and C = -5.9