If f (x) = x2 − 4x + 6, find (a) f (0); (b) f (3); (c) f (−2). Show that f( )  = f( )  and f (2 − h) = f (2 + h).

Let's go step by step to solve each part of this problem.

### Function:
Given the function:  
\[
f(x) = x^2 - 4x + 6
\]

### Part (a) - Find \( f(0) \)
To find \( f(0) \), substitute \( x = 0 \) into the function:

\[
f(0) = (0)^2 - 4(0) + 6 = 0 - 0 + 6 = 6
\]

So, \( f(0) = 6 \).

### Part (b) - Find \( f(3) \)
To find \( f(3) \), substitute \( x = 3 \) into the function:

\[
f(3) = (3)^2 - 4(3) + 6 = 9 - 12 + 6 = 3
\]

So, \( f(3) = 3 \).

### Part (c) - Find \( f(-2) \)
To find \( f(-2) \), substitute \( x = -2 \) into the function:

\[
f(-2) = (-2)^2 - 4(-2) + 6 = 4 + 8 + 6 = 18
\]

So, \( f(-2) = 18 \).

### Part (d) - Show that \( f(x) = f(4 - x) \)
Let's substitute \( 4 - x \) into the function:

\[
f(4 - x) = (4 - x)^2 - 4(4 - x) + 6
\]

First, expand \( (4 - x)^2 \) and \( -4(4 - x) \):

\[
f(4 - x) = (16 - 8x + x^2) - (16 - 4x) + 6
\]
\[
f(4 - x) = 16 - 8x + x^2 - 16 + 4x + 6 = x^2 - 4x + 6
\]

Notice that this is the same as the original function \( f(x) = x^2 - 4x + 6 \). Therefore:

\[
f(x) = f(4 - x)
\]

### Part (e) - Show that \( f(2 - h) = f(2 + h) \)
We need to evaluate \( f(2 - h) \) and \( f(2 + h) \) and show that they are equal.

1. **Find \( f(2 - h) \):**

\[
f(2 - h) = (2 - h)^2 - 4(2 - h) + 6
\]
Expanding \( (2 - h)^2 \) and \( -4(2 - h) \):

\[
f(2 - h) = (4 - 4h + h^2) - (8 - 4h) + 6 = 4 - 4h + h^2 - 8 + 4h + 6
\]
Simplify:

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

2. **Find \( f(2 + h) \):**

\[
f(2 + h) = (2 + h)^2 - 4(2 + h) + 6
\]
Expanding \( (2 + h)^2 \) and \( -4(2 + h) \):

\[
f(2 + h) = (4 + 4h + h^2) - (8 + 4h) + 6 = 4 + 4h + h^2 - 8 - 4h + 6
\]
Simplify:

\[
f(2 + h) = h^2 + 2
\]

Since \( f(2 - h) = f(2 + h) \), the statement is proven.

### Conclusion:
- \( f(0) = 6 \)
- \( f(3) = 3 \)
- \( f(-2) = 18 \)
- \( f(x) = f(4 - x) \)
- \( f(2 - h) = f(2 + h) \)

Would you like a further breakdown of any part of the solution?

### 5 Related Questions:
1. How do you derive the vertex form of a quadratic function?
2. What is the symmetry property of quadratic functions in general?
3. How can you find the minimum or maximum of \( f(x) = x^2 - 4x + 6 \)?
4. What is the axis of symmetry for the function \( f(x) = x^2 - 4x + 6 \)?
5. How do transformations affect the graph of a quadratic function?

### Tip:
When proving that two expressions are equal, always expand and simplify both sides carefully to make sure they match.

If f(x) = x^2 − 4x + 6, find (a) f(0); (b) f(3); (c) f(−2). Show that f(x) = f(4 - x) and f(2 − h) = f(2 + h).

Discover how to solve quadratic function f(x) = x^2 − 4x + 6 through step-by-step calculations. This guide shows you how to find f(0), f(3), f(−2), and prove the symmetry properties f(2 − h) = f(2 + h) and f(x) = f(4 − x). Ideal for students in Grades 9-10.

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