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Let's solve the problem step by step:

### Part (a) - Write \( f(x) \) in the form \( (x + p)^2 + q \)

We are given the quadratic function:

\[
f(x) = x^2 + 4kx + (3 + 11k)
\]

We need to complete the square to express this function in the form \( (x + p)^2 + q \).

#### Step 1: Group the quadratic and linear terms
\[
f(x) = (x^2 + 4kx) + (3 + 11k)
\]

#### Step 2: Complete the square
To complete the square, we need to take half of the coefficient of \(x\) (which is \(4k\)), square it, and add/subtract it inside the function.

Half of \(4k\) is \(2k\), and squaring it gives \( (2k)^2 = 4k^2 \).

Thus:
\[
f(x) = (x^2 + 4kx + 4k^2) - 4k^2 + (3 + 11k)
\]

\[
f(x) = (x + 2k)^2 - 4k^2 + 3 + 11k
\]

#### Step 3: Simplify the constant term
Simplifying the constants:
\[
f(x) = (x + 2k)^2 + (-4k^2 + 11k + 3)
\]

Thus, the quadratic in completed square form is:
\[
f(x) = (x + 2k)^2 + (11k - 4k^2 + 3)
\]

Here, \( p = 2k \) and \( q = 11k - 4k^2 + 3 \).

### Part (b) - Sketch the graph for \( k = 1 \)

Now, we substitute \( k = 1 \) into the function and sketch it.

#### Step 1: Substitute \( k = 1 \) into the completed square form
\[
f(x) = (x + 2(1))^2 + (11(1) - 4(1)^2 + 3)
\]
\[
f(x) = (x + 2)^2 + (11 - 4 + 3)
\]
\[
f(x) = (x + 2)^2 + 10
\]

So the function is \( f(x) = (x + 2)^2 + 10 \).

#### Step 2: Analyze the graph

- **Vertex (minimum point):** The vertex occurs at \( x = -2 \) (from \( x + 2 = 0 \)) and \( y = 10 \). Therefore, the vertex is \( (-2, 10) \).
- **Y-intercept:** To find the y-intercept, set \( x = 0 \):
  \[
  f(0) = (0 + 2)^2 + 10 = 4 + 10 = 14
  \]
  So, the y-intercept is \( (0, 14) \).
- **X-intercepts:** Since the minimum value of \( f(x) = (x + 2)^2 + 10 \) is 10 (which is above the x-axis), there are no x-intercepts.

The graph is a parabola opening upwards with vertex \( (-2, 10) \) and y-intercept at \( (0, 14) \). There are no x-intercepts.

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### Here are five follow-up questions related to this problem:
1. How can we identify the vertex of a quadratic function from its completed square form?
2. What is the geometric significance of completing the square?
3. How do the values of \(k\) affect the shape and position of the parabola?
4. How can we derive the x-intercepts of a quadratic function?
5. Why does the parabola not have x-intercepts when \(k = 1\)?

### Tip:
When completing the square, always remember to carefully handle the constant terms to avoid errors in simplifying the equation.

f(x) = x^2 + 4kx + (3 + 11k). a) Write f(x) in the form (x + p)^2 + q, where p and q are expressions in terms of k. b) Given that k = 1, sketch the graph of y = f(x) showing clearly any intersections with the coordinate axes and labelling the minimum point.

Learn how to write the quadratic function f(x) = x^2 + 4kx + (3 + 11k) in completed square form and graph it when k = 1. This problem explores the method of completing the square and analyzing the parabola's vertex and intercepts.

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