If
f(x) = 2x2 − x + 3,
find the following f(a + h)

To find \( f(a + h) \) for the function \( f(x) = 2x^2 - x + 3 \), you substitute \( a + h \) for \( x \) in the function:

\[
f(a + h) = 2(a + h)^2 - (a + h) + 3
\]

Now, let's expand and simplify it:

1. Expand \( (a + h)^2 \):
   \[
   (a + h)^2 = a^2 + 2ah + h^2
   \]

2. Substitute this into the function:
   \[
   f(a + h) = 2(a^2 + 2ah + h^2) - (a + h) + 3
   \]

3. Distribute the 2:
   \[
   f(a + h) = 2a^2 + 4ah + 2h^2 - a - h + 3
   \]

So the final expression for \( f(a + h) \) is:

\[
f(a + h) = 2a^2 + 4ah + 2h^2 - a - h + 3
\]

Would you like to go over any particular details or have any further questions?

Here are 5 related questions for further exploration:

1. What is the value of \( f(2) \) if \( f(x) = 2x^2 - x + 3 \)?
2. How would you find \( f(a) + f(h) \) for the same function?
3. What is the derivative of \( f(x) = 2x^2 - x + 3 \)?
4. How does the value of \( h \) affect the expression \( f(a + h) \)?
5. What is the second derivative of the function \( f(x) = 2x^2 - x + 3 \)?

**Tip:** When substituting expressions into functions, always expand and simplify to ensure clarity in your results.

Learn how to compute f(a + h) for the quadratic function f(x) = 2x^2 - x + 3. Step-by-step solution involving expansion and simplification is provided. Suitable for students in Grades 10-12.

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