The problem asks us to evaluate certain expressions for the function $f(x) = 7 - 2x + 13x^2$. Let's solve each part step-by-step:

1. Calculate $f(a)$

Substitute $x = a$ into the function:

$f(a) = 7 - 2a + 13a^2$

2. Calculate $f(a + h)$

Substitute $x = a + h$ into the function:

$f(a + h) = 7 - 2(a + h) + 13(a + h)^2$

Simplify the expression:

$f(a + h) = 7 - 2a - 2h + 13(a^2 + 2ah + h^2)$

$= 7 - 2a - 2h + 13a^2 + 26ah + 13h^2$

3. Calculate $\frac{f(a + h) - f(a)}{h}$ for $h \neq 0$

Substitute the values of $f(a + h)$ and $f(a)$:

$\frac{f(a + h) - f(a)}{h} = \frac{(7 - 2a - 2h + 13a^2 + 26ah + 13h^2) - (7 - 2a + 13a^2)}{h}$

Simplify the numerator:

$= \frac{-2h + 26ah + 13h^2}{h}$

Factor out $h$:

$= \frac{h(-2 + 26a + 13h)}{h}$

Since $h \neq 0$, cancel $h$:

$= -2 + 26a + 13h$

Final Answers:

1. $f(a) = 7 - 2a + 13a^2$
2. $f(a + h) = 7 - 2a - 2h + 13a^2 + 26ah + 13h^2$
3. $\frac{f(a + h) - f(a)}{h} = -2 + 26a + 13h$

Would you like any further details, or do you have any questions? Here are some related questions:

1. What is the derivative of $f(x)$?
2. How does the result of $\frac{f(a + h) - f(a)}{h}$ relate to the derivative of $f(x)$?
3. How do you find the instantaneous rate of change of $f(x)$ at $x = a$?
4. How would you evaluate these expressions if the function were different?
5. How does this process help in finding limits?

Tip: The expression $\frac{f(a + h) - f(a)}{h}$ is crucial in calculus as it represents the difference quotient, which is used to find the derivative of a function.