g(x)=x2+4x+1 find following 1:g(2) 2:g(T) 3:(x+2)

Given the function \( g(x) = x^2 + 4x + 1 \), let's find the following values:

### 1. \( g(2) \)
Substitute \( x = 2 \) into the function:

\[
g(2) = (2)^2 + 4(2) + 1 = 4 + 8 + 1 = 13
\]

So, \( g(2) = 13 \).

### 2. \( g(T) \)
Substitute \( x = T \) into the function:

\[
g(T) = T^2 + 4T + 1
\]

So, \( g(T) = T^2 + 4T + 1 \).

### 3. \( g(x + 2) \)
Substitute \( x = x + 2 \) into the function:

\[
g(x+2) = (x+2)^2 + 4(x+2) + 1
\]

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

\[
(x+2)^2 = x^2 + 4x + 4
\]
\[
4(x+2) = 4x + 8
\]

Now, combine all terms:

\[
g(x+2) = x^2 + 4x + 4 + 4x + 8 + 1 = x^2 + 8x + 13
\]

So, \( g(x+2) = x^2 + 8x + 13 \).

### Summary:
1. \( g(2) = 13 \)
2. \( g(T) = T^2 + 4T + 1 \)
3. \( g(x+2) = x^2 + 8x + 13 \)

Would you like further details or explanations on any of these steps?

Here are five related questions to expand your understanding:

1. What is \( g(0) \) for the given function?
2. How would you find the roots of \( g(x) = 0 \)?
3. What is the vertex of the parabola represented by \( g(x) \)?
4. How do you determine whether the parabola opens upwards or downwards?
5. What is the axis of symmetry for the function \( g(x) \)?

**Tip:** When substituting expressions into functions, ensure you fully expand and simplify the expression to avoid errors.

Learn how to evaluate and transform the function g(x) = x^2 + 4x + 1. This problem covers function evaluation at specific points and transformation by substitution. Suitable for students in Grades 9-12.

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