Questions: Use the definition (mtan =lim h rightarrow 0 fracf(a+h)-f(a)h) to find the slope of the line tangent to the graph of (f) at (P). Determine an equation of the tangent line at (P). (f(x)=7+4 x^2 ; P(0,7)) (mtan =) (square) (Type an integer or a fraction.)

Find the slope of the line tangent to the graph of $f$ at point $P$.

Calculate $f(a+h)$ and $f(a)$ for $a = 0$.

We have $f(a+h) = f(0+h) = 7 + 4(h)^2 = 4h^2 + 7$ and $f(a) = f(0) = 7$.

Use the limit definition to find the slope $m_{\tan}$.

Using the limit definition, we find:
$$m_{\tan} = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} = \lim_{h \rightarrow 0} \frac{(4h^2 + 7) - 7}{h} = \lim_{h \rightarrow 0} \frac{4h^2}{h} = \lim_{h \rightarrow 0} 4h = 0.$$

The slope of the tangent line is $\boxed{0}$.

Determine an equation of the tangent line at point $P$.

Use the point-slope form of the line equation.

The point-slope form is given by $y - y_1 = m(x - x_1)$. Here, $m = 0$ and $(x_1, y_1) = (0, 7)$. Thus, the equation becomes:
$$y - 7 = 0(x - 0) \implies y = 7.$$

The equation of the tangent line is $\boxed{y = 7}$.

The slope of the tangent line is $\boxed{0}$.
The equation of the tangent line is $\boxed{y = 7}$.