Find and simplify

*f*(*a* + *h*) − *f*(*a*)

*h*

,    (*h* ≠ 0)

for the following function.

*f*(*x*) = 2*x*2 − 2*x* + 7

We need to find and simplify the expression

\[
\frac{f(a + h) - f(a)}{h}
\]

for the function \( f(x) = 2x^2 - 2x + 7 \), where \( h \neq 0 \).

### Step 1: Compute \( f(a + h) \)
Substitute \( x = a + h \) into the function \( f(x) = 2x^2 - 2x + 7 \):
\[
f(a + h) = 2(a + h)^2 - 2(a + h) + 7
\]
Expand \( (a + h)^2 \):
\[
(a + h)^2 = a^2 + 2ah + h^2
\]
Now substitute back:
\[
f(a + h) = 2(a^2 + 2ah + h^2) - 2(a + h) + 7
\]
Distribute the terms:
\[
f(a + h) = 2a^2 + 4ah + 2h^2 - 2a - 2h + 7
\]

### Step 2: Compute \( f(a) \)
Substitute \( x = a \) into the function:
\[
f(a) = 2a^2 - 2a + 7
\]

### Step 3: Subtract \( f(a) \) from \( f(a + h) \)
Now subtract \( f(a) \) from \( f(a + h) \):
\[
f(a + h) - f(a) = (2a^2 + 4ah + 2h^2 - 2a - 2h + 7) - (2a^2 - 2a + 7)
\]
Simplify by canceling like terms:
\[
f(a + h) - f(a) = 4ah + 2h^2 - 2h
\]

### Step 4: Divide by \( h \)
Now divide by \( h \):
\[
\frac{f(a + h) - f(a)}{h} = \frac{4ah + 2h^2 - 2h}{h}
\]
Factor out \( h \) from the numerator:
\[
\frac{f(a + h) - f(a)}{h} = \frac{h(4a + 2h - 2)}{h}
\]
Since \( h \neq 0 \), we can cancel out \( h \):
\[
\frac{f(a + h) - f(a)}{h} = 4a + 2h - 2
\]

### Final Answer:
The simplified expression is:

\[
\frac{f(a + h) - f(a)}{h} = 4a + 2h - 2
\]

Would you like further details or have any questions about the process? 

Here are 5 questions related to the topic:
1. What is the significance of the expression \( \frac{f(a + h) - f(a)}{h} \) in calculus?
2. How would the answer change if \( f(x) \) had an additional linear term?
3. Can you explain the role of the limit as \( h \) approaches 0 in finding the derivative?
4. How do we expand binomials like \( (a + h)^2 \)?
5. What would the expression look like if \( f(x) \) was a cubic function?

**Tip:** In calculus, the expression \( \frac{f(a + h) - f(a)}{h} \) is key to understanding the concept of derivatives. It represents the average rate of change of the function over the interval from \( a \) to \( a + h \).

Learn how to simplify the expression (f(a + h) - f(a)) / h for the function f(x) = 2x^2 - 2x + 7. Step-by-step solution included, covering calculus concepts like derivatives and function evaluation. Suitable for Grades 11-12.

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